Kernel regression utilizing heterogeneous datasets

Abstract

Data analysis in modern scientific research and practice has shifted from analysing a single dataset to coupling several datasets. We propose and study a kernel regression method that can handle the challenge of heterogeneous populations. It greatly extends the constrained kernel regression [Dai, C.-S., & Shao, J. (2023). Kernel regression utilizing external information as constraints. Statistica Sinica, 33, in press] that requires a homogeneous population of different datasets. The asymptotic normality of proposed estimators is established under some conditions and simulation results are presented to confirm our theory and to quantify the improvements from datasets with heterogeneous populations.

Keywords:

• Conditional expectation
• constraints
• data coupling and integration
• external data
• heterogeneous populations
• kernel estimation

1. Introduction

With advanced technologies in data collection and storage, in modern statistical analyses we have not only a primary random sample from a population of interest, which results in a dataset referred to as the internal dataset, but also some independent external datasets from sources such as past investigations and publicly available datasets. In this paper, we consider nonparametric kernel regression (Bierens, 1987; Wand & Jones, 1994, December; Wasserman, 2006) between a univariate response Y and a covariate vector $\mathit{U}$ from a sampled subject, using the internal dataset with the help from independent external datasets. Specifically, we consider kernel estimation of the conditional expectation (regression function) of Y given $\mathit{U}=\mathit{u}$ under an internal data population, (1) ${\mu }_1\left(\mathit{u}\right)=E(Y\mid \mathit{U}=\mathit{u},D=1),$(1) where D = 1 indicates internal population and $\mathit{u}$ is a fixed point in $\mathcal{U}$, the range of $\mathit{U}$. The indicator D can be either random or deterministic. The subscript 1 in ${\mu }_1\left(\mathit{u}\right)$ emphasizes that it is for internal data population (D = 1), which may be different from $\mu \left(\mathit{u}\right)=E(Y\mid \mathit{U}=\mathit{u})$, a mixture of quantities from the internal and external data populations.

When external datasets also have measurements Y and $\mathit{U}$, we may simply combine the internal and external datasets when the populations for internal and external data are identical (homogeneous). However, heterogeneity typically exists among populations for different datasets, especially when there are multiple external datasets collected in different ways and/or different time periods. In Section 2, we propose a method to handle heterogeneity among different populations and derive a kernel regression more efficient than the one using internal data alone. The result is also a crucial building block for the more complicated case in Section 3 where external datasets contain fewer measured covariates as described next.

In applications, it often occurs that an external dataset has measured Y and $\mathit{X}$ from each subject, where $\mathit{X}$ is a part of the vector $\mathit{U}$, i.e., some components of $\mathit{U}$ are not measured due to high measurement cost or the progress of technology and/or scientific relevance. With some unmeasured components of $\mathit{U}$, the external dataset cannot be directly used to estimate ${\mu }_1\left(\mathit{u}\right)$ in (1(1) ${\mu }_1\left(\mathit{u}\right)=E(Y\mid \mathit{U}=\mathit{u},D=1),$(1) ), since conditioning on the entire $\mathit{U}$ is involved. To solve this problem, Dai and Shao (2023) proposes a two-step kernel regression using external information as a constraint to improve kernel regression based on internal data alone, following the idea of using constraints in Chatterjee et al. (2016) and H. Zhang et al. (2020). However, these three cited papers mainly assume that the internal and external datasets share the same population, which may be unrealistic. The challenge in dealing with the heterogeneity among different populations is similar to the difficulty in handling nonignorable missing data if unmeasured components of $\mathit{U}$ is treated as missing data, although in missing data problems we usually want to estimate $\mu \left(\mathit{u}\right)=E(Y\mid \mathit{U}=\mathit{u})\ne {\mu }_1\left(\mathit{u}\right)$ in (1(1) ${\mu }_1\left(\mathit{u}\right)=E(Y\mid \mathit{U}=\mathit{u},D=1),$(1) ).

In Section 3, we develop a methodology to handle population heterogeneity for internal and external datasets, which extends the procedure in Dai and Shao (2023) to heterogeneous populations and greatly widens its application scope.

Under each scenario, we derive asymptotic normality in Section 4 for the proposed kernel estimators and obtain explicitly the asymptotic variances, which is important for large sample inference. Some simulation results are presented in Section 5 to compare finite sample performance of several estimators. Discussions on extensions and handling high dimension covariates are given in Section 6. All technical details are in the Appendix.

Our research fits into a general framework of data integration (Kim et al., 2021; Lohr & Raghunathan, 2017; Merkouris, 2004; Rao, 2021; Yang & Kim, 2020; Y. Zhang et al., 2017).

2. Efficient kernel estimation by combining datasets

The internal dataset contains observations $(Y_i,{\mathit{U}}_i)$, $i=1,\dots ,n$, independent and identically distributed (iid) from ${\mathcal{P}}_1$, the internal population of $(Y,\mathit{U})$, where Y is the response and $\mathit{U}$ is a p-dimensional covariate vector associated with Y. We are interested in the estimation of conditional expectation ${\mu }_1\left(\mathit{u}\right)$ in (1(1) ${\mu }_1\left(\mathit{u}\right)=E(Y\mid \mathit{U}=\mathit{u},D=1),$(1) ). The standard kernel regression estimator of ${\mu }_1\left(\mathit{u}\right)$ based on the internal dataset alone is (2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) where ${\kappa }_b\left(\mathit{a}\right)=b^{-p}\kappa \left(\mathit{a}/b\right)$, $\kappa (\cdot )$ is a given kernel function on $\mathcal{U}$ (the range of $\mathit{u}$), and b>0 is a bandwidth depending on n. We assume that $\mathit{U}$ is standardized so that the same bandwidth b is used for every component of $\mathit{U}$ in kernel regression. Because of the well-known curse of dimensionality for kernel-type methods, we focus on a low dimension p not varying with n. A discussion of handling a large dimensional $\mathit{U}$ is given in Section 6.

We consider the case with one external dataset, independent of the internal dataset. Extension to multiple external datasets is straightforward and discussed in Section 6.

In this section we consider the situation where the external dataset contains iid observations $(Y_i,{\mathit{U}}_i)$, $i=n+1,\dots ,N$, from ${\mathcal{P}}_0$, the external population of $(Y,\mathit{U})$.

2.1. Combing data from homogeneous populations

If we assume that the two populations ${\mathcal{P}}_1$ and ${\mathcal{P}}_0$ are identical, then we can simply combine two datasets to obtain the kernel estimator (3) ${\hat{\mu }}_1^{E1}\left(\mathit{u}\right)=\sum _{i=1}^N{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(3) which is obviously more efficient than ${\hat{\mu }}_1\left(\mathit{u}\right)$ in (2(2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) ) as the sample size is increased to N>n. The estimator ${\hat{\mu }}_1^{E1}\left(\mathit{u}\right)$ in (3(3) ${\hat{\mu }}_1^{E1}\left(\mathit{u}\right)=\sum _{i=1}^N{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(3) ), however, is not correct (i.e., it is biased) when populations ${\mathcal{P}}_1$ and ${\mathcal{P}}_0$ are different, because $E(Y\mid \mathit{U}=\mathit{u},D=0)$ for external population may be different from ${\mu }_1\left(\mathit{u}\right)=E(Y\mid \mathit{U}=\mathit{u},D=1)$ for internal population.

2.2. Combing data from heterogeneous populations

We now derive a kernel estimator using two datasets and is asymptotically correct regardless of whether ${\mathcal{P}}_1$ and ${\mathcal{P}}_0$ are the same or not. Let $f(y|\mathit{u},D)$ be the conditional density of Y given $\mathit{U}=\mathit{u}$ and $D=$ 1 or 0 (for internal or external population). Then (4) ${\mu }_1\left(\mathit{x}\right)=E(Y|\mathit{U}=\mathit{u},D=1)=E\left\{Y\frac{f(Y|\mathit{u},D=1)}{f(Y|\mathit{u},D=0)}\right|\mathit{U}=\mathit{u},D=0\}.$(4) The ratio $f(Y|\mathit{u},D=1)/f(Y|\mathit{u},D=0)$ links internal and external populations so that we can overcome the difficulty in utilizing the external data under heterogeneous populations.

If we can construct an estimator $\hat{f}(y|\mathit{u},D)$ of $f(y|\mathit{u},D)$ for every y, $\mathit{u}$, and D = 0 or 1, then we can modify the estimator in (3(3) ${\hat{\mu }}_1^{E1}\left(\mathit{u}\right)=\sum _{i=1}^N{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(3) ) by replacing every $Y_i$ with i>n by constructed response ${\hat{Y}}_i=Y_i\hat{f}(Y_i|{\mathit{U}}_i,D=1)/\hat{f}(Y_i|{\mathit{U}}_i,D=0)$. The resulting kernel estimator is (5) ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)=\left\{\sum _{i=1}^n{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=n+1}^N{\hat{Y}}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i).$(5) Note that we use internal data $(Y_i,{\mathit{U}}_i)$, $i=1,\dots ,n$, to obtain estimator $\hat{f}(Y_i|{\mathit{U}}_i,D=1)$ and external data $(Y_i,{\mathit{U}}_i)$, $i=n+1,\dots ,N$, to construct estimator $\hat{f}(Y_i|{\mathit{U}}_i,D=0)$. Applying kernel estimation, we obtain that (6) $\begin{array}{rl}\hat{f}(y|\mathit{U}=\mathit{u},=1)& =\sum _{i=1}^n{\tilde{\kappa }}_{\tilde{b}}(y-Y_i,\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\overline{\kappa }}_{\overline{b}}(\mathit{u}-{\mathit{U}}_i),\\ \hat{f}(y|\mathit{U}=\mathit{u},D=0)& =\sum _{i=n+1}^N{\tilde{\kappa }}_{\tilde{b}}(y-Y_i,\mathit{u}-{\mathit{U}}_i)/\sum _{i=n+1}^N{\overline{\kappa }}_{\overline{b}}(\mathit{u}-{\mathit{U}}_i),\end{array}$(6) where $\tilde{\kappa }$ and $\overline{\kappa }$ are kernels with dimensions p + 1 and p and bandwidths $\tilde{b}$ and $\overline{b}$, respectively. The estimator in (5(5) ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)=\left\{\sum _{i=1}^n{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=n+1}^N{\hat{Y}}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i).$(5) ) is asymptotically valid under some regularity conditions for kernel and bandwidth, summarized in Theorem 4.1 of Section 4.

2.3. Combing data from heterogeneous populations with additional information

If additional information exists, then the approach in Section 2.2 can be improved. Assume that the internal and external datasets are formed according to a random binary indicator D such that $(Y_i,{\mathit{U}}_i,D_i)$, $i=1,\dots ,N$, are iid distributed as $(Y,\mathit{U},D)$, where $Y_i$ and ${\mathit{U}}_i$ are observed internal data when $D_i=1$, $Y_i$ and ${\mathit{U}}_i$ are observed external data when $D_i=0$, and N is still the known total sample size for internal and external data. In this situation, the internal and external sample sizes are $n=\sum _{i=1}^N{D}_i$ and N−n, respectively, both of which are random. In most applications, the assumption of random D is not substantial. From the identity (7) $\frac{f(Y|\mathit{u},D=1)}{f(Y|\mathit{u},D=0)}=\frac{P(D=1|\mathit{U}=\mathit{u},Y)}{P(D=0|\mathit{U}=\mathit{u},Y)}\frac{P(D=0|\mathit{U}=\mathit{u})}{P(D=1|\mathit{U}=\mathit{u})},$(7) we just need to estimate $P(D=1|\mathit{U}=\mathit{u},Y)$ and $P(D=1|\mathit{U}=\mathit{u})$ for every $\mathit{u}$, constructed using for example the nonparametric estimators in Fan et al. (1998) for binary response. For each estimator, both internal and external data on $(Y,\mathit{U})$ and the indicator D are used.

A further improvement can be made if the following semi-parametric model holds, (8) $\frac{P(D=0\mid \mathit{U},Y)}{P(D=1\mid \mathit{U},Y)}=\exp \left\{\alpha \right(\mathit{U})+\mathrm{\gamma Y}\},$(8) where $\alpha (\cdot )$ is an unspecified unknown function and γ is an unknown parameter. From (7(7) $\frac{f(Y|\mathit{u},D=1)}{f(Y|\mathit{u},D=0)}=\frac{P(D=1|\mathit{U}=\mathit{u},Y)}{P(D=0|\mathit{U}=\mathit{u},Y)}\frac{P(D=0|\mathit{U}=\mathit{u})}{P(D=1|\mathit{U}=\mathit{u})},$(7) )–(8(8) $\frac{P(D=0\mid \mathit{U},Y)}{P(D=1\mid \mathit{U},Y)}=\exp \left\{\alpha \right(\mathit{U})+\mathrm{\gamma Y}\},$(8) ), (9) $\frac{f(Y|\mathit{u},D=1)}{f(Y|\mathit{u},D=0)}={\mathrm{e}}^{-\mathrm{\gamma Y}}E({\mathrm{e}}^{\mathrm{\gamma Y}}\mid \mathit{U}=\mathit{u},D=1).$(9) If $\gamma =0$, then $f(Y|\mathit{u},D=1)=f(Y|\mathit{u},D=0)$ and the estimator ${\hat{\mu }}_1^{E1}\left(\mathit{u}\right)$ in (3(3) ${\hat{\mu }}_1^{E1}\left(\mathit{u}\right)=\sum _{i=1}^N{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(3) ) is correct. Under (9(9) $\frac{f(Y|\mathit{u},D=1)}{f(Y|\mathit{u},D=0)}={\mathrm{e}}^{-\mathrm{\gamma Y}}E({\mathrm{e}}^{\mathrm{\gamma Y}}\mid \mathit{U}=\mathit{u},D=1).$(9) ) with $\gamma \ne 0$, we just need to derive an estimator $\hat{\gamma }$ of γ and apply kernel estimation to estimate $E({\mathrm{e}}^{\hat{\gamma }Y}\mid \mathit{U}=\mathit{u},D=1)$ as a function of $\mathit{u}$. Note that we do not need to estimate the unspecified function $\alpha (\cdot )$ in (8(8) $\frac{P(D=0\mid \mathit{U},Y)}{P(D=1\mid \mathit{U},Y)}=\exp \left\{\alpha \right(\mathit{U})+\mathrm{\gamma Y}\},$(8) ), which is a nice feature of semi-parametric model (8(8) $\frac{P(D=0\mid \mathit{U},Y)}{P(D=1\mid \mathit{U},Y)}=\exp \left\{\alpha \right(\mathit{U})+\mathrm{\gamma Y}\},$(8) ).

We now derive an estimator $\hat{\gamma }$. Applying (7(7) $\frac{f(Y|\mathit{u},D=1)}{f(Y|\mathit{u},D=0)}=\frac{P(D=1|\mathit{U}=\mathit{u},Y)}{P(D=0|\mathit{U}=\mathit{u},Y)}\frac{P(D=0|\mathit{U}=\mathit{u})}{P(D=1|\mathit{U}=\mathit{u})},$(7) )–(8(8) $\frac{P(D=0\mid \mathit{U},Y)}{P(D=1\mid \mathit{U},Y)}=\exp \left\{\alpha \right(\mathit{U})+\mathrm{\gamma Y}\},$(8) ) to (4(4) ${\mu }_1\left(\mathit{x}\right)=E(Y|\mathit{U}=\mathit{u},D=1)=E\left\{Y\frac{f(Y|\mathit{u},D=1)}{f(Y|\mathit{u},D=0)}\right|\mathit{U}=\mathit{u},D=0\}.$(4) ), we obtain that $\begin{array}{rl}{\mu }_1\left(\mathit{u}\right)& =E\left\{Y\frac{P(D=1|\mathit{U}=\mathit{u},Y)}{P(D=0|\mathit{U}=\mathit{u},Y)}\right|\mathit{U}=\mathit{u},D=0\}\frac{P(D=0|\mathit{U}=\mathit{u})}{P(D=1|\mathit{U}=\mathit{u})}\\ & =E(Y{\mathrm{e}}^{-\alpha \left(\mathit{u}\right)-\mathrm{\gamma Y}}|\mathit{U}=\mathit{u},D=0)\frac{E\left\{P\right(D=0|\mathit{U}=\mathit{u},Y)|\mathit{U}=\mathit{u}\}}{P(D=1|\mathit{U}=\mathit{u})}\\ & ={\mathrm{e}}^{-\alpha \left(\mathit{u}\right)}E(Y{\mathrm{e}}^{-\mathrm{\gamma Y}}|\mathit{U}=\mathit{u},D=0)\frac{E\left\{{\mathrm{e}}^{\alpha \left(\mathit{u}\right)+\mathrm{\gamma Y}}P\right(D=1|\mathit{U}=\mathit{u},Y)|\mathit{U}=\mathit{u}\}}{P(D=1|\mathit{U}=\mathit{u})}\\ & =E(Y{\mathrm{e}}^{-\mathrm{\gamma Y}}|\mathit{U}=\mathit{u},D=0)\frac{E\left\{{\mathrm{e}}^{\mathrm{\gamma Y}}E\right(D|\mathit{U}=\mathit{u},Y)|\mathit{U}=\mathit{u}\}}{P(D=1|\mathit{U}=\mathit{u})}\\ & =E(Y{\mathrm{e}}^{-\mathrm{\gamma Y}}|\mathit{U}=\mathit{u},D=0)\frac{E({\mathrm{e}}^{\mathrm{\gamma Y}}D|\mathit{U}=\mathit{u})}{P(D=1|\mathit{U}=\mathit{u})}\\ & =E(Y{\mathrm{e}}^{-\mathrm{\gamma Y}}|\mathit{U}=\mathit{u},D=0)E({\mathrm{e}}^{\mathrm{\gamma Y}}|\mathit{U}=\mathit{u},D=1),\end{array}$where the second and third equalities follow from (8(8) $\frac{P(D=0\mid \mathit{U},Y)}{P(D=1\mid \mathit{U},Y)}=\exp \left\{\alpha \right(\mathit{U})+\mathrm{\gamma Y}\},$(8) ) and the last equation follows from $\begin{array}{rl}E({\mathrm{e}}^{\mathrm{\gamma Y}}D|\mathit{U}=\mathit{u})& =E({\mathrm{e}}^{\mathrm{\gamma Y}}D|\mathit{U}=\mathit{u},D=1)P(D=1|\mathit{U}=\mathit{u})\\ & +E({\mathrm{e}}^{\mathrm{\gamma Y}}D|\mathit{U}=\mathit{u},D=0)P(D=0|\mathit{U}=\mathit{u})\\ & =E({\mathrm{e}}^{\mathrm{\gamma Y}}|\mathit{U}=\mathit{u},D=1)P(D=1|\mathit{U}=\mathit{u}),\end{array}$as $E({\mathrm{e}}^{\mathrm{\gamma Y}}D|\mathit{U}=\mathit{u},D=0)=0$. For every real number t, define $h(\mathit{u},t)=E(Y{\mathrm{e}}^{-\mathrm{tY}}|\mathit{U}=\mathit{u},D=0)E({\mathrm{e}}^{\mathrm{tY}}|\mathit{U}=\mathit{u},D=1).$Its estimator by kernel regression is (10) $\hat{h}(\mathit{u},t)=\frac{\sum _{i=1}^N(1-D_i){\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i)Y_i{\mathrm{e}}^{-t{Y}_i}}{\sum _{i=1}^N(1-D_i){\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i)}\frac{\sum _{i=1}^N{D}_i{\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i){\mathrm{e}}^{t{Y}_i}}{\sum _{i=1}^N{D}_i{\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i)}$(10) where $\check{\kappa }$ is a kernel and $\check{b}$ is a bandwidth. Then, we estimate γ by (11) $\hat{\gamma }=\arg \underset{t}{min}\frac{1}{N}\sum _{i=1}^N{D}_i\{Y_i-\hat{h}({\mathit{U}}_i,t){\}}^2,$(11) motivated by the fact that the objective function for minimization in (11(11) $\hat{\gamma }=\arg \underset{t}{min}\frac{1}{N}\sum _{i=1}^N{D}_i\{Y_i-\hat{h}({\mathit{U}}_i,t){\}}^2,$(11) ) approximates $E\left[D\right\{Y-h(\mathit{U},t){\}}^2|D=1]$ and, for any t, $E\left[D\right\{Y-h(\mathit{U},\gamma ){\}}^2|D=1]\le E[D\{Y-h(\mathit{U},t){\}}^2|D=1]$because $h(\mathit{u},\gamma )={\mu }_1\left(\mathit{u}\right)$.

Once $\hat{\gamma }$ is obtained, our estimator of ${\mu }_1\left(\mathit{u}\right)$ is (12) ${\hat{\mu }}_1^{E3}\left(\mathit{u}\right)=\left\{\sum _{i=1}^N{D}_i{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=1}^N(1-D_i\left){\hat{Y}}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i)$(12) with ${\hat{Y}}_i=Y_i{\mathrm{e}}^{-\hat{\gamma }{Y}_i}\sum _{j=1}^n{\mathrm{e}}^{\hat{\gamma }{Y}_j}{\check{\kappa }}_{\check{b}}({\mathit{U}}_i-{\mathit{U}}_j)/\sum _{j=1}^n{\check{\kappa }}_{\check{b}}({\mathit{U}}_i-{\mathit{U}}_j),$in view of (9(9) $\frac{f(Y|\mathit{u},D=1)}{f(Y|\mathit{u},D=0)}={\mathrm{e}}^{-\mathrm{\gamma Y}}E({\mathrm{e}}^{\mathrm{\gamma Y}}\mid \mathit{U}=\mathit{u},D=1).$(9) ).

In applications, we need to choose bandwidths with given sample sizes n and N−n. We can apply the k-fold cross-validation as described in Györfi et al. (2002). Requirements on the rates of bandwidths are described in theorems in Section 3.

3. Constrained kernel regression with unmeasured covariates

We still consider the case with one external dataset, independent of the internal dataset. In this section, the external dataset contains iid observations $(Y_i,{\mathit{X}}_i)$, $i=n+1,\dots ,N$, from the external population ${\mathcal{P}}_0$, where $\mathit{X}$ is a q-dimensional sub-vector of $\mathit{U}$ with q<p.

Since the external dataset has only $\mathit{X}$, not the entire $\mathit{U}$, we cannot apply the method in Section 2 when q<p. Instead, we consider kernel regression using external information in a constraint. First, we consider the estimation of the n-dimensional vector ${\mathit{\mu }}_1=\left({\mu }_1\right({\mathit{U}}_1),\dots ,{\mu }_1({\mathit{U}}_n){)}^{\mathrm{\top }}$, where ${\mathit{A}}^{\mathrm{\top }}$ denotes the transpose of vector or matrix $\mathit{A}$ throughout. Note that the standard kernel regression (2(2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) ) estimates ${\mathit{\mu }}_1$ as ${\hat{\mathit{\mu }}}_1={\left(\sum _{i=1}^n{Y}_i{\kappa }_b\right({\mathit{U}}_1-{\mathit{U}}_i\left)/\sum _{i=1}^n{\kappa }_b({\mathit{U}}_1-{\mathit{U}}_i),\dots ,\sum _{i=1}^n{Y}_i{\kappa }_b({\mathit{U}}_n-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b\right({\mathit{U}}_n-{\mathit{U}}_i\left)\right)}^{\mathrm{\top }}.$Taking partial derivatives with respect to ${\mu }_i$'s, we obtain that (13) ${\hat{\mathit{\mu }}}_1=\arg \underset{{\mu }_1,\dots ,{\mu }_n}{min}\sum _{i=1}^n\sum _{j=1}^n{\kappa }_b({\mathit{U}}_i-{\mathit{U}}_j)(Y_j-{\mu }_i{)}^2/\sum _{k=1}^n{\kappa }_b({\mathit{U}}_i-{\mathit{U}}_k).$(13) We improve ${\hat{\mathit{\mu }}}_1$ by the following constrained minimization, (14) $\begin{array}{rl}{\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}& =\arg \underset{{\mu }_1,\dots ,{\mu }_n}{min}\sum _{i=1}^n\sum _{j=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j)(Y_j-{\mu }_i{)}^2/\sum _{k=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_k)\end{array}$(14) (15) $\begin{array}{rl}& \text{subject to}\sum _{i=1}^n\{{\mu }_i-{\hat{h}}_1^{\mathrm{Ej}}({\mathit{X}}_i\left)\right\}\mathit{g}({\mathit{X}}_i{)}^{\mathrm{\top }}=0,\end{array}$(15) where $\mathit{g}(\mathit{x}{)}^{\mathrm{\top }}=(1,{\mathit{x}}^{\mathrm{\top }})$, l in (14(14) $\begin{array}{rl}{\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}& =\arg \underset{{\mu }_1,\dots ,{\mu }_n}{min}\sum _{i=1}^n\sum _{j=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j)(Y_j-{\mu }_i{)}^2/\sum _{k=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_k)\end{array}$(14) ) is a bandwidth that may be different from b in (2(2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) ) or (13(13) ${\hat{\mathit{\mu }}}_1=\arg \underset{{\mu }_1,\dots ,{\mu }_n}{min}\sum _{i=1}^n\sum _{j=1}^n{\kappa }_b({\mathit{U}}_i-{\mathit{U}}_j)(Y_j-{\mu }_i{)}^2/\sum _{k=1}^n{\kappa }_b({\mathit{U}}_i-{\mathit{U}}_k).$(13) ), and ${\hat{h}}_1^{\mathrm{Ej}}\left(\mathit{x}\right)$ is the kernel estimator of $h_1\left(\mathit{x}\right)=E(Y\mid \mathit{X}=\mathit{x},D=1)$ using the jth of the three methods described in Section 2, j = 1, 2, 3. Specifically, ${\hat{h}}_1^{E1}\left(\mathit{x}\right)$ is given by (3(3) ${\hat{\mu }}_1^{E1}\left(\mathit{u}\right)=\sum _{i=1}^N{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(3) ), ${\hat{h}}_1^{E2}\left(\mathit{x}\right)$ is given by (5(5) ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)=\left\{\sum _{i=1}^n{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=n+1}^N{\hat{Y}}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i).$(5) ), and ${\hat{h}}_1^{E3}\left(\mathit{x}\right)$ is given by (12(12) ${\hat{\mu }}_1^{E3}\left(\mathit{u}\right)=\left\{\sum _{i=1}^N{D}_i{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=1}^N(1-D_i\left){\hat{Y}}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i)$(12) ), with $\mathit{u}$ and $\mathit{U}$ replaced by $\mathit{x}$ and $\mathit{X}$, respectively, and kernels and bandwidths suitably adjusted as dimensions of $\mathit{U}$ and $\mathit{X}$ are different. Note that ${\hat{h}}_1^{\mathrm{Ej}}$ can be computed as both internal and external datasets have measured ${\mathit{X}}_i$'s.

It turns out that ${\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}$ in (14(14) $\begin{array}{rl}{\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}& =\arg \underset{{\mu }_1,\dots ,{\mu }_n}{min}\sum _{i=1}^n\sum _{j=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j)(Y_j-{\mu }_i{)}^2/\sum _{k=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_k)\end{array}$(14) ) has an explicit form ${\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}={\hat{\mathit{\mu }}}_1+\mathit{G}\left({\mathit{G}}^{\mathrm{\top }}\mathit{G}{)}^{-1}{\mathit{G}}^{\mathrm{\top }}\right({\hat{\mathit{h}}}_1^{\mathrm{Ej}}-{\hat{\mathit{\mu }}}_1),$ where $\mathit{G}$ is the $n\times n$ matrix whose ith row is $\mathit{g}({\mathit{X}}_i{)}^{\mathrm{\top }}$ and ${\hat{\mathit{h}}}_1^{\mathrm{Ej}}$ is the n-dimensional vector whose ith component is ${\hat{h}}_1^{\mathrm{Ej}}\left({\mathit{X}}_i\right)$. Constraint (15(15) $\begin{array}{rl}& \text{subject to}\sum _{i=1}^n\{{\mu }_i-{\hat{h}}_1^{\mathrm{Ej}}({\mathit{X}}_i\left)\right\}\mathit{g}({\mathit{X}}_i{)}^{\mathrm{\top }}=0,\end{array}$(15) ) is an empirical analog of the theoretical constraint $E\left[\left\{{\mu }_1\right(\mathit{U})-h_1(\mathit{X}\left)\right\}\mathit{g}\right(\mathit{X}{)}^{\mathrm{\top }}\mid D=1]=0$(based on internal data), as $E\left\{E\right(Y\mid \mathit{U},D=1)\mid \mathit{X},D=1\}=E(Y\mid \mathit{X},D=1)=h_1\left(\mathit{X}\right)$. Thus, if ${\hat{h}}_1^{\mathrm{Ej}}(\cdot )$ is a good estimator of $h_1(\cdot )$, then ${\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}$ in (14(14) $\begin{array}{rl}{\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}& =\arg \underset{{\mu }_1,\dots ,{\mu }_n}{min}\sum _{i=1}^n\sum _{j=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j)(Y_j-{\mu }_i{)}^2/\sum _{k=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_k)\end{array}$(14) ) is more accurate than the unconstrained ${\hat{\mathit{\mu }}}_1$ in (13(13) ${\hat{\mathit{\mu }}}_1=\arg \underset{{\mu }_1,\dots ,{\mu }_n}{min}\sum _{i=1}^n\sum _{j=1}^n{\kappa }_b({\mathit{U}}_i-{\mathit{U}}_j)(Y_j-{\mu }_i{)}^2/\sum _{k=1}^n{\kappa }_b({\mathit{U}}_i-{\mathit{U}}_k).$(13) ).

To obtain an improved estimator of the entire regression function ${\mu }_1(\cdot )$ in (1(1) ${\mu }_1\left(\mathit{u}\right)=E(Y\mid \mathit{U}=\mathit{u},D=1),$(1) ), not just the function at $\mathit{u}={\mathit{U}}_i$, $i=1,\dots ,n$, we apply the standard kernel regression with response vector $(Y_1,\dots ,Y_n{)}^{\mathrm{\top }}$ replaced by ${\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}$ in (14(14) $\begin{array}{rl}{\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}& =\arg \underset{{\mu }_1,\dots ,{\mu }_n}{min}\sum _{i=1}^n\sum _{j=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j)(Y_j-{\mu }_i{)}^2/\sum _{k=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_k)\end{array}$(14) ), which results in the following three estimators of ${\mu }_1\left(\mathit{u}\right)$: (16) ${\hat{\mu }}_1^{\mathrm{Cj}}\left(\mathit{u}\right)=\sum _{i=1}^n{\hat{\mu }}_i^{\mathrm{Cj}}{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),j=1,2,3,$(16) where ${\hat{\mu }}_i^{\mathrm{Cj}}$ is the ith component of ${\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}$ in (14(14) $\begin{array}{rl}{\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}& =\arg \underset{{\mu }_1,\dots ,{\mu }_n}{min}\sum _{i=1}^n\sum _{j=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j)(Y_j-{\mu }_i{)}^2/\sum _{k=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_k)\end{array}$(14) ) and b is the same bandwidth in (2(2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) ). The first estimator ${\hat{\mu }}_i^{C1}$ is simple, but can be incorrect when populations ${\mathcal{P}}_1$ and ${\mathcal{P}}_0$ are different. The asymptotic validity of ${\hat{\mathit{\mu }}}_1^{C2}$ and ${\hat{\mathit{\mu }}}_1^{C3}$ are established in the next section.

4. Asymptotic normality

We now establish the asymptotic normality of ${\hat{\mu }}_1^{\mathrm{Ej}}\left(\mathit{u}\right)$ and ${\hat{\mu }}_1^{\mathrm{Cj}}\left(\mathit{u}\right)$ for a fixed $\mathit{u}$, as the sample size of the internal dataset increases to infinity. All technical proofs are given in the Appendix.

The first result is about ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)$ in (5(5) ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)=\left\{\sum _{i=1}^n{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=n+1}^N{\hat{Y}}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i).$(5) ). The result is also applicable to ${\hat{\mu }}_1^{E1}\left(\mathit{u}\right)$ in (3(3) ${\hat{\mu }}_1^{E1}\left(\mathit{u}\right)=\sum _{i=1}^N{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(3) ) with an added condition that ${\mathcal{P}}_1={\mathcal{P}}_0$.

Theorem 4.1

Assume the following conditions.

(B1)	The densities $f_1\left(\mathit{u}\right)$ and $f_0\left(\mathit{u}\right)$ for $\mathit{U}$, respectively under internal and external populations have continuous and bounded first- and second-order partial derivatives.
(B2)	${\mu }_1^2\left(\mathit{u}\right)f_k\left(\mathit{u}\right)$, ${\sigma }_k^2\left(\mathit{u}\right)f_k\left(\mathit{u}\right)$, and the first- and second-order partial derivatives of ${\mu }_1\left(\mathit{u}\right)f_k\left(\mathit{u}\right)$ are continuous and bounded, where ${\sigma }_1^2\left(\mathit{u}\right)=E\left[\right\{Y-{\mu }_1\left(\mathit{U}\right){\}}^2\mid \mathit{U}=\mathit{u},D=1]$, ${\sigma }_0^2\left(\mathit{u}\right)=E\left[\right\{\tilde{Y}-{\mu }_1\left(\mathit{U}\right){\}}^2\mid \mathit{U}=\mathit{u},D=0]$, and $\tilde{Y}=\mathrm{Yf}(Y|\mathit{U},D=1)/f(Y|\mathit{U},D=0)$. Also, $E(|Y{|}^s|\mathit{U}=\mathit{u},D=1)f_1\left(\mathit{u}\right)$ and $E(|\tilde{Y}{|}^s|\mathit{U}=\mathit{u},D=0)f_0\left(\mathit{u}\right)$ are bounded for a constant s>2.
(B3)	The kernel κ is second order, i.e., $\int \mathit{u}\kappa \left(\mathit{u}\right)\mathrm{d}\mathit{u}=0$ and $0<\int {\mathit{u}}^{\mathrm{\top }}\mathit{u}\kappa \left(\mathit{u}\right)\mathrm{d}\mathit{u}<\mathrm{\infty }$.
(B4)	The bandwidth b satisfies $b\to 0$ and $(a+1)n{b}^{p+4}\to c\in [0,\mathrm{\infty })$, where $a=\underset{n\to \mathrm{\infty }}{lim}(N-n)/n$ (assumed to exist without loss of generality).
(B5)	The kernels $\tilde{\kappa }$ and $\overline{\kappa }$ in (6(6) $\begin{array}{rl}\hat{f}(y|\mathit{U}=\mathit{u},=1)& =\sum _{i=1}^n{\tilde{\kappa }}_{\tilde{b}}(y-Y_i,\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\overline{\kappa }}_{\overline{b}}(\mathit{u}-{\mathit{U}}_i),\\ \hat{f}(y|\mathit{U}=\mathit{u},D=0)& =\sum _{i=n+1}^N{\tilde{\kappa }}_{\tilde{b}}(y-Y_i,\mathit{u}-{\mathit{U}}_i)/\sum _{i=n+1}^N{\overline{\kappa }}_{\overline{b}}(\mathit{u}-{\mathit{U}}_i),\end{array}$(6) ) have bounded supports and orders $\tilde{m}>2+2/p$ and $\overline{m}>2$, respectively, as defined by Bierens (1987), $f(y,\mathit{u}|D=1)$, $f(y,\mathit{u}|D=0)$ are $\tilde{m}$th-order continuously differentiable with bounded partial derivatives, and $f_1\left(\mathit{u}\right)$ and $f_0\left(\mathit{u}\right)$ are $\overline{m}$th-order continuously differentiable with bounded partial derivatives. Functions $f(y,\mathit{u}|D=0)$ and $f_1\left(\mathit{u}\right)$ are bounded away from zero. The bandwidths $\tilde{b}$ and $\overline{b}$ satisfy $n{\tilde{b}}^{p+1}/\log \left(n\right)\to \mathrm{\infty }$ and $n{\overline{b}}^p/\log \left(n\right)\to \mathrm{\infty }$.

Then, for any fixed $\mathit{u}$ with $f_0\left(\mathit{u}\right)>0$ and $f_1\left(\mathit{u}\right)>0$ and ${\hat{\mu }}_1^{E2}$ in (5(5) ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)=\left\{\sum _{i=1}^n{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=n+1}^N{\hat{Y}}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i).$(5) ), (17) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{E2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_a\right(\mathit{u}),V_a(\mathit{u}\left)\right),$(17) where $⟹d$ denotes convergence in distribution as $n\to \mathrm{\infty }$, $\begin{array}{rl}{B}_a\left(\mathit{u}\right)& =\frac{c^{1/2}\left\{f_1\right(\mathit{u}\left)A_1\right(\mathit{u})+a{f}_0(\mathit{u}\left)A_0\right(\mathit{u}\left)\right\}}{(a+1{)}^{1/2}\{f_1\left(\mathit{u}\right)+a{f}_0\left(\mathit{u}\right)\}},\\ A_1\left(\mathit{u}\right)& =\int \kappa \left(\mathit{v}\right)\left\{\frac{1}{2}{\mathit{v}}^{\mathrm{\top }}{\mathrm{\nabla }}^2{\mu }_1\right(\mathit{u})\mathit{v}+{\mathit{v}}^{\mathrm{\top }}\mathrm{\nabla log}{f}_1(\mathit{u}\left)\mathrm{\nabla }{\mu }_1\right(\mathit{u}{)}^{\mathrm{\top }}\mathit{v}\}\mathrm{d}\mathit{v},\\ A_0\left(\mathit{u}\right)& =\int \kappa \left(\mathit{v}\right)\left\{\frac{1}{2}{\mathit{v}}^{\mathrm{\top }}{\mathrm{\nabla }}^2{\mu }_1\right(\mathit{u})\mathit{v}+{\mathit{v}}^{\mathrm{\top }}\mathrm{\nabla log}{f}_0(\mathit{u}\left)\mathrm{\nabla }{\mu }_1\right(\mathit{u}{)}^{\mathrm{\top }}\mathit{v}\}\mathrm{d}\mathit{v},\\ V_a\left(\mathit{u}\right)& =\frac{f_1\left(\mathit{u}\right){\sigma }_1^2\left(\mathit{u}\right)+a{f}_0\left(\mathit{u}\right){\sigma }_0^2\left(\mathit{u}\right)}{\left\{f_1\right(\mathit{u})+a{f}_0(\mathit{u}){\}}^2}\int \kappa (\mathit{v}{)}^2\mathrm{d}\mathit{v}.\end{array}$

Conditions (B1)–(B4) are typically assumed for kernel estimation (Bierens, 1987). Condition (B5) is a sufficient condition for (18) $\underset{i=n+1,\dots ,N}{max}|\frac{\hat{f}(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{\hat{f}(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}-\frac{f(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{f(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}|=\frac{o_p\left(1\right)}{\sqrt{n{b}^p}}$(18) (Lemma 8.10 in Newey & McFadden, 1994), where $o_p\left(1\right)$ denotes a term tending to 0 in probability. Result (18(18) $\underset{i=n+1,\dots ,N}{max}|\frac{\hat{f}(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{\hat{f}(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}-\frac{f(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{f(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}|=\frac{o_p\left(1\right)}{\sqrt{n{b}^p}}$(18) ) implies that the estimation of ratio $f(Y|\mathit{U},D=1)/f(Y|\mathit{U},D=0)$ does not affect the asymptotic distribution of ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)$ in (5(5) ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)=\left\{\sum _{i=1}^n{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=n+1}^N{\hat{Y}}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i).$(5) ).

Note that both the squared bias $B_a^2\left(\mathit{u}\right)$ and variance $V_a\left(\mathit{u}\right)$ in (17(17) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{E2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_a\right(\mathit{u}),V_a(\mathit{u}\left)\right),$(17) ) are decreasing in the limit $a=\underset{n\to \mathrm{\infty }}{lim}(N-n)/n$, a quantity reflecting how many external data we have. In the extreme case of a = 0, i.e., the size of the external dataset is negligible compared with the size of the internal dataset, result (17(17) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{E2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_a\right(\mathit{u}),V_a(\mathit{u}\left)\right),$(17) ) reduces to the well-known asymptotic normality for the standard kernel estimator ${\hat{\mu }}_1\left(\mathit{u}\right)$ in (2(2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) ) (Bierens, 1987). In the other extreme case of $a=\mathrm{\infty }$, on the other hand, $B_a\left(\mathit{u}\right)=V_a\left(\mathit{u}\right)=0$ and, hence, ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)$ has a convergence rate tending to 0 faster than $1/\sqrt{n{b}^p}$, the convergence rate of the standard kernel estimator ${\hat{\mu }}_1\left(\mathit{u}\right)$.

The next result is about ${\hat{\mu }}_1^{C2}\left(\mathit{u}\right)$ in (16(16) ${\hat{\mu }}_1^{\mathrm{Cj}}\left(\mathit{u}\right)=\sum _{i=1}^n{\hat{\mu }}_i^{\mathrm{Cj}}{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),j=1,2,3,$(16) ) as described in Section 3.

Theorem 4.2

Assume (B1)–(B5) with $\mathit{U}$ and p replaced by $\mathit{X}$ and q, respectively, and the following conditions, where $f_k\left(\mathit{u}\right)$ and ${\sigma }_k^2\left(\mathit{u}\right)$, k = 0, 1, are defined in (B1)–(B2).

(C1)	The range $\mathcal{U}$ of $\mathit{U}$ is a compact set in the p-dimensional Euclidean space and $f_1\left(\mathit{u}\right)$ is bounded away from infinity and zero on $\mathcal{U}$; $f_1\left(\mathit{u}\right)$ and $f_0\left(\mathit{u}\right)$ have continuous and bounded first- and second-order partial derivatives.
(C2)	Functions ${\mu }_1\left(\mathit{u}\right)=E(Y|\mathit{U}=\mathit{u})$ and ${\sigma }_1^2\left(\mathit{u}\right)$ are Lipschitz continuous; ${\mu }_1\left(\mathit{u}\right)$ has bounded third-order partial derivatives; $h_1\left(\mathit{x}\right)=E(Y\mid \mathit{X}=\mathit{x},D=1)$ has bounded first- and second-order partial derivatives; and $E(|Y{|}^s|\mathit{U}=\mathit{u},D=1)$ is bounded with $s>2+p/2$.
(C3)	All kernel functions are positive, bounded, and Lipschitz continuous with mean zero and finite sixth moments.
(C4)	$a=\underset{n\to \mathrm{\infty }}{lim}(N-n)/n>0$ and the bandwidths b in (2(2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) ) and l in (14(14) $\begin{array}{rl}{\hat{\mathit{\mu }}}_1^{\mathrm{Cj}}& =\arg \underset{{\mu }_1,\dots ,{\mu }_n}{min}\sum _{i=1}^n\sum _{j=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j)(Y_j-{\mu }_i{)}^2/\sum _{k=1}^n{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_k)\end{array}$(14) ) satisfy $b\to 0$, $l\to 0$, $l/b\to r\in (0,\mathrm{\infty })$, $n{b}^p\to \mathrm{\infty }$, and $n{b}^{4+p}\to c\in [0,\mathrm{\infty })$, as $n\to \mathrm{\infty }$.
(C5)	The densities $f_{X0}\left(\mathit{x}\right)$ and $f_{X1}\left(\mathit{x}\right)$ for $\mathit{X}$, respectively under internal and external populations are bounded away from zero. There exists a constant s>4 such that $E(|Y{|}^s\mid D=1)$ and $E(|\tilde{Y}{|}^s\mid D=0)$ are finite, $E(|Y{|}^s\mid \mathit{X}=\mathit{x},D=1)f_{X1}\left(\mathit{x}\right)$ and $E(|\tilde{Y}{|}^s\mid \mathit{X}=\mathit{x},D=0)f_{X0}\left(\mathit{x}\right)$ are bounded, and the bandwidth $b_h$ for ${\hat{h}}_1$ satisfies $n^{1-2/s}{b}_h^q/\log \left(n\right)\to \mathrm{\infty }.$

Then, for any fixed $\mathit{u}\in \mathcal{U}$ and ${\hat{\mu }}_1^{C2}\left(\mathit{u}\right)$ in (16(16) ${\hat{\mu }}_1^{\mathrm{Cj}}\left(\mathit{u}\right)=\sum _{i=1}^n{\hat{\mu }}_i^{\mathrm{Cj}}{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),j=1,2,3,$(16) ), (19) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{C2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_r\right(\mathit{u}),V_r(\mathit{u}\left)\right),$(19) where $\begin{array}{rl}{B}_r\left(\mathit{u}\right)& =c^{1/2}\left[\right(1+r^2\left)A_1\right(\mathit{u})-r^2\mathit{g}(\mathit{x}{)}^{\mathrm{\top }}{\mathbf{\Sigma }}_g^{-1}E\left\{\mathit{g}\right(\mathit{X}\left)A_1\right(\mathit{U})|D=1\}],\\ A_1\left(\mathit{u}\right)& =\int \kappa \left(\mathit{v}\right)\left\{\frac{1}{2}{\mathit{v}}^{\mathrm{\top }}{\mathrm{\nabla }}^2{\mu }_1\right(\mathit{u})\mathit{v}+{\mathit{v}}^{\mathrm{\top }}\mathrm{\nabla log}{f}_1(\mathit{u}\left)\mathrm{\nabla }{\mu }_1\right(\mathit{u}{)}^{\mathrm{\top }}\mathit{v}\}\mathrm{d}\mathit{v},\\ V_r\left(\mathit{u}\right)& =\frac{{\sigma }_1^2\left(\mathit{u}\right)}{f_1\left(\mathit{u}\right)}\int {\{\int \kappa (\mathit{v}-r\mathit{w}\left)\kappa \right(\mathit{w}\left)d\mathit{w}\right\}}^2\mathrm{d}\mathit{v},\end{array}$

and $\mathbf{\Sigma }=E\left\{\mathit{g}\right(\mathit{X}\left)\mathit{g}\right(\mathit{X}{)}^{\mathrm{\top }}\mid D=1\}$ is assumed to be positive definite without loss of generality.

The next result is about $\hat{\gamma }$ in (11(11) $\hat{\gamma }=\arg \underset{t}{min}\frac{1}{N}\sum _{i=1}^N{D}_i\{Y_i-\hat{h}({\mathit{U}}_i,t){\}}^2,$(11) ).

Theorem 4.3

Suppose that (8(8) $\frac{P(D=0\mid \mathit{U},Y)}{P(D=1\mid \mathit{U},Y)}=\exp \left\{\alpha \right(\mathit{U})+\mathrm{\gamma Y}\},$(8) ) holds for binary random D indicating internal and external data. Assume also the following conditions.

(D1)	The kernel $\check{\kappa }$ in (10(10) $\hat{h}(\mathit{u},t)=\frac{\sum _{i=1}^N(1-D_i){\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i)Y_i{\mathrm{e}}^{-t{Y}_i}}{\sum _{i=1}^N(1-D_i){\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i)}\frac{\sum _{i=1}^N{D}_i{\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i){\mathrm{e}}^{t{Y}_i}}{\sum _{i=1}^N{D}_i{\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i)}$(10) ) is Lipschitz continuous, satisfies $\int \check{\kappa }\left(\mathit{u}\right)\mathrm{d}\mathit{u}=1$, has a bounded support, and has order $d>max\left\{\right(p+4)/2,p\}$.
(D2)	The bandwidth $\check{b}$ in (10(10) $\hat{h}(\mathit{u},t)=\frac{\sum _{i=1}^N(1-D_i){\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i)Y_i{\mathrm{e}}^{-t{Y}_i}}{\sum _{i=1}^N(1-D_i){\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i)}\frac{\sum _{i=1}^N{D}_i{\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i){\mathrm{e}}^{t{Y}_i}}{\sum _{i=1}^N{D}_i{\check{\kappa }}_{\check{b}}(\mathit{u}-{\mathit{U}}_i)}$(10) ) satisfies $N{\check{b}}^{2q}/(\log N{)}^2\to \mathrm{\infty }$ and $N{\check{b}}^{2d}\to 0$ as the total sample size of internal and external datasets $N\to \mathrm{\infty }$, where d is given in (D1).
(D3)	γ in (8(8) $\frac{P(D=0\mid \mathit{U},Y)}{P(D=1\mid \mathit{U},Y)}=\exp \left\{\alpha \right(\mathit{U})+\mathrm{\gamma Y}\},$(8) ) is an interior point of a compact domain Γ and it is the unique solution to $h_1(\cdot )=h(\cdot ,t)$, $t\in \mathrm{\Gamma }$. For any $\mathit{u}$, $h(\mathit{u},t)$ is second-order continuously differentiable in t, and h, ${\mathrm{\nabla }}_{t}h$, ${\mathrm{\nabla }}_t^{2}h$ are bounded over t and $\mathit{u}$. As $t\to \gamma$, $h(\cdot ,t)$, ${\mathrm{\nabla }}_{t}h(\cdot ,t)$, and ${\mathrm{\nabla }}_t^{2}h(\cdot ,t)$ converge uniformly.
(D4)	$\underset{t\in \mathrm{\Gamma }}{sup}E\Vert {\mathit{W}}_t{\Vert }^4<\mathrm{\infty }$ and $\underset{t\in \mathrm{\Gamma }}{sup}E[\Vert {\mathit{W}}_t{\Vert }^4|\mathit{U}]f_U\left(\mathit{U}\right)$ is bounded, where $\Vert \mathit{a}{\Vert }^2={\mathit{a}}^{\mathrm{\top }}\mathit{a}$, ${\mathit{W}}_t=(D{\mathrm{e}}^{\mathrm{tY}},(1-D)Y{\mathrm{e}}^{-\mathrm{tY}},D,(1-D),\mathrm{DY}{\mathrm{e}}^{\mathrm{tY}},(1-D)Y^2{\mathrm{e}}^{-\mathrm{tY}},$ $D{Y}^2{\mathrm{e}}^{\mathrm{tY}},(1-D)Y^3{\mathrm{e}}^{-\mathrm{tY}}{)}^{\mathrm{\top }}$, and $f_U$ is the density of $\mathit{U}$. Furthermore, there is a function $\tau (Y,D)$ with $E\left\{\tau \right(Y,D\left)\right\}<\mathrm{\infty }$ such that $\Vert {\mathit{W}}_t-{\mathit{W}}_{t^{\prime }}\Vert <\tau (Y,D)|t-t^{\prime }|$.
(D5)	The function ${\mathit{\omega }}_t\left(\mathit{u}\right)=E({\mathit{W}}_t|\mathit{U}=\mathit{u})f_U\left(\mathit{u}\right)$ is bounded away from zero, and it is dth-order continuously differentiable with bounded partial derivatives on an open set containing the support of $\mathit{U}$. There is a functional $G(Y,D,\mathit{\omega })$ linear in $\mathit{\omega }$ such that $|G(Y,D,\mathit{\omega })|\le \iota (Y,D)\Vert \mathit{\omega }{\Vert }_{\mathrm{\infty }}$ and, for small enough $\Vert \mathit{\omega }-{\mathit{\omega }}_{\gamma }{\Vert }_{\mathrm{\infty }}$, $|\psi (Y,D,\mathit{\omega })-\psi (Y,D,{\mathit{\omega }}_{\gamma })-G(Y,D,\mathit{\omega }-{\mathit{\omega }}_{\gamma })|\le$ $\iota (Y,D)\Vert \mathit{\omega }-{\mathit{\omega }}_{\gamma }{\Vert }_{\mathrm{\infty }}^2$, where $\iota (Y,D)$ is a function with $E\left\{\iota \right(Y,D\left)\right\}<\mathrm{\infty }$, $\psi (Y,D,\mathit{\omega })$ $=-2D(Y-\frac{{\omega }_1{\omega }_2}{{\omega }_3{\omega }_4})\left(\frac{{\omega }_2{\omega }_5-{\omega }_1{\omega }_6}{{\omega }_3{\omega }_4}\right)$, ${\omega }_j$ is the jth component of $\mathit{\omega }$, $\Vert \mathit{\omega }{\Vert }_{\mathrm{\infty }}=\underset{\mathit{x}\in \mathcal{U}}{sup}\Vert \mathit{\omega }\left(\mathit{u}\right)\Vert$, $\Vert \mathit{\omega }-{\mathit{\omega }}_{\gamma }{\Vert }_{\mathrm{\infty }}=\underset{\mathit{x}\in \mathcal{U}}{sup}\Vert \mathit{\omega }\left(\mathit{u}\right)-{\mathit{\omega }}_{\gamma }\left(\mathit{u}\right)\Vert$, and $\mathcal{U}$ is the range of $\mathit{U}$. Also, there exists an almost everywhere continuous 8-dimensional function $\mathit{\nu }\left(\mathit{U}\right)$ with $\int \Vert \mathit{\nu }\left(\mathit{u}\right)\Vert \mathrm{d}\mathit{u}<\mathrm{\infty }$ and $E\{\underset{\Vert \mathit{\delta }\Vert \le ϵ}{sup}\Vert \mathit{\nu }(\mathit{U}+\mathit{\delta }\left){\Vert }^4\right\}<\mathrm{\infty }$ for some $ϵ>0$ such that $E\left\{G\right(Y,D,\mathit{\omega }\left)\right\}=\int \mathit{\nu }\left(\mathit{u}{)}^{\mathrm{\top }}\mathit{\omega }\right(\mathit{u})\mathrm{d}\mathit{u}$ for all $\Vert \mathit{\omega }{\Vert }_{\mathrm{\infty }}<\mathrm{\infty }$.

Then, as the total sample size of internal and external datasets $N\to \mathrm{\infty }$, (20) $\sqrt{N}(\hat{\gamma }-\gamma )⟹dN(0,{\sigma }_{\gamma }^2),$(20) where ${\sigma }_{\gamma }^2=\left[2E\right\{D{\mathrm{\nabla }}_{\gamma }h(\mathit{U},\gamma ){\}}^2{]}^{-1}\mathrm{Var}\left[\psi \right(Y,D,{\mathit{\omega }}_{\gamma })+\mathit{\nu }(\mathit{U}{)}^{\mathrm{\top }}{\mathit{W}}_{\gamma }-E\left\{\mathit{\nu }\right(\mathit{U}{)}^{\mathrm{\top }}{\mathit{W}}_{\gamma }\left\}\right]$.

Conditions (D1)–(D5) are technical assumptions discussed in Lemmas 8.11 and 8.12 in Newey and McFadden (1994). As discussed by Newey and McFadden (1994), the condition that $\check{\kappa }$ has a bounded support can be relaxed, as it is imposed for a simple proof.

Combining Theorems 4.1–4.3, we obtain the following result for ${\hat{\mu }}_1^{E3}\left(\mathit{u}\right)$ in (12(12) ${\hat{\mu }}_1^{E3}\left(\mathit{u}\right)=\left\{\sum _{i=1}^N{D}_i{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=1}^N(1-D_i\left){\hat{Y}}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i)$(12) ) or ${\hat{\mu }}_1^{C3}\left(\mathit{u}\right)$ in (16(16) ${\hat{\mu }}_1^{\mathrm{Cj}}\left(\mathit{u}\right)=\sum _{i=1}^n{\hat{\mu }}_i^{\mathrm{Cj}}{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),j=1,2,3,$(16) ).

Corollary 4.1

Suppose that (8(8) $\frac{P(D=0\mid \mathit{U},Y)}{P(D=1\mid \mathit{U},Y)}=\exp \left\{\alpha \right(\mathit{U})+\mathrm{\gamma Y}\},$(8) ) holds for the binary random D indicating internal and external data.

(i)	Under (B1)–(B4) and (D1)–(D5), result (17(17) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{E2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_a\right(\mathit{u}),V_a(\mathit{u}\left)\right),$(17) ) holds with ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)$ replaced by ${\hat{\mu }}_1^{E3}\left(\mathit{u}\right)$.
(ii)	Under (C1)–(C4) and (D1)–(D5) with $\mathit{U}$ and p replaced by $\mathit{X}$ and q, respectively, result (19(19) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{C2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_r\right(\mathit{u}),V_r(\mathit{u}\left)\right),$(19) ) holds with ${\hat{\mu }}_1^{C2}\left(\mathit{u}\right)$ replaced by ${\hat{\mu }}_1^{C3}\left(\mathit{u}\right)$.

5. Simulation results

5.1. The performance of ${\hat{\mu }}_1^{\mathrm{Cj}}$ given by (16)

We first present simulation results to examine and compare the performance of the standard kernel estimator ${\hat{\mu }}_1$ in (2(2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) ) without using external information and our proposed estimator (16(16) ${\hat{\mu }}_1^{\mathrm{Cj}}\left(\mathit{u}\right)=\sum _{i=1}^n{\hat{\mu }}_i^{\mathrm{Cj}}{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),j=1,2,3,$(16) ) with three variations, ${\hat{\mu }}_1^{C1}$, ${\hat{\mu }}_1^{C2}$, and ${\hat{\mu }}_1^{C3}$, as described in the end of Section 3. We consider $\mathit{U}=(X,Z{)}^{\mathrm{\top }}$ with univariate covariates X and Z, where Z is unmeasured in the external dataset (p = 2 and q = 1). The covariates are generated in two ways:

1. normal covariates: $(X,Z{)}^{\mathrm{\top }}$ is bivariate normal with means 0, variances 1, and correlation 0.5;

2. bounded covariates: $X=B{W}_1+(1-B)W_2$ and $Z=B{W}_1+(1-B)W_3$, where $W_1,$ $W_2$, and $W_3$ are identically distributed as uniform on $[-1,1]$, B is uniform on $[0,1]$, and $W_1,$ $W_2$, $W_3$, and B are independent.

Conditioned on $(X,Z{)}^{\mathrm{\top }}$, the response Y is normal with mean $\mu (X,Z)$ and variance 1, where $\mu (X,Z)$ follows one of the following four models:

(M1)	$\mu (X,Z)=X/2-Z^2/4$;
(M2)	$\mu (X,Z)=\cos \left(2X\right)/2+\sin \left(Z\right)$;
(M3)	$\mu (X,Z)=\cos \left(2\mathrm{XZ}\right)/2+\sin \left(Z\right)$;
(M4)	$\mu (X,Z)=X/2-Z^2/4+\cos \left(\mathrm{XZ}\right)/4$.

Note that all four models are nonlinear in $(X,Z{)}^{\mathrm{\top }}$; (M1)-(M2) are additive models, while (M3)-(M4) are non-additive.

A total of N = 1, 200 data are generated from the population of $(Y,X,Z)$ as previously described. A data point is treated as internal or external according to a random binary D with conditional probability $P(D=1\mid Y,X,Z)=1/\exp ({\gamma }_0+2|X|+\mathrm{\gamma Y})$, where $\gamma =0$ or 1/2, and ${\gamma }_0=1$ or $-1.5$. Under the setting ${\gamma }_0=1$ or $-1.5$, the unconditional $P(D=1)\approx n/N$ is around 13% or 50%.

The simulation studies performance of kernel estimators in terms of mean integrated square error (MISE). The following measure is calculated by simulation with S replications: (21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) where $\{{\mathit{U}}_{s,t}:t=1,\dots ,T\}$ are test data for each simulation replication s, the simulation is repeated independently for $s=1,\dots ,S$, and ${\hat{\mu }}_1^{\ast }$ is one of ${\hat{\mu }}_1$, ${\hat{\mu }}_1^{C1}$, ${\hat{\mu }}_1^{C2}$, and ${\hat{\mu }}_1^{C3}$, independent of test data. We consider two ways of generating test data ${\mathit{U}}_{s,t}$'s. The first one is to use T = 121 fixed grid points on $[-1,1]\times [-1,1]$ with equal space. The second one is to take a random sample of T = 121 without replacement from the covariate $\mathit{U}$'s of the internal dataset, for each fixed $s=1,\dots ,S$ and independently across s.

To show the benefit of using external information, we calculate the improvement in efficiency defined as follows: (22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) where the minimum is over ${\hat{\mu }}_1^{\ast }=$ one of ${\hat{\mu }}_1$, ${\hat{\mu }}_1^{C1}$, ${\hat{\mu }}_1^{C2}$, and ${\hat{\mu }}_1^{C3}$.

In all cases, we use the Gaussian kernel. The bandwidths b and l affect the performance of kernel methods. We consider two types of bandwidths in the simulation. The first one is ‘the best bandwidth’; for each method, we evaluate MISE in a pool of bandwidths and display the one that has the minimal MISE. This shows the best we can achieve in terms of bandwidth, but it cannot be used in applications. The second one is to select bandwidth from a pool of bandwidths via 10-fold cross validation (Györfi et al., 2002), which produces a decent bandwidth that can be applied to real data.

The simulated MISE values based on S = 200 replications are shown in Tables 1–4.

Table 1. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains only X, with S = 200 under $\gamma =0$, $n/N\approx 13\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{C1}$	${\hat{\mu }}_1^{C2}$	${\hat{\mu }}_1^{C3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.040	0.017	0.020	0.022	56.96	−0.003
			CV	0.057	0.043	0.042	0.051	27.16	−0.007
		Grid	Best	0.026	0.009	0.013	0.014	66.49	−0.012
			CV	0.037	0.022	0.022	0.028	40.53	−0.003
	M2	Sample	Best	0.042	0.021	0.027	0.026	50.58	−0.042
			CV	0.074	0.056	0.061	0.060	23.52	−0.039
		Grid	Best	0.030	0.015	0.025	0.023	50.36	−0.026
			CV	0.051	0.033	0.040	0.036	35.62	−0.035
	M3	Sample	Best	0.039	0.020	0.025	0.023	48.55	−0.023
			CV	0.063	0.050	0.053	0.055	21.16	−0.020
		Grid	Best	0.028	0.014	0.020	0.021	51.73	−0.010
			CV	0.046	0.029	0.033	0.033	36.33	−0.024
	M4	Sample	Best	0.041	0.018	0.020	0.023	56.28	−0.007
			CV	0.064	0.050	0.047	0.056	25.83	−0.017
		Grid	Best	0.026	0.009	0.014	0.015	65.54	−0.006
			CV	0.036	0.021	0.020	0.026	44.18	−0.007
Bounded	M1	Sample	Best	0.010	0.002	0.006	0.007	79.58	−0.007
			CV	0.016	0.006	0.007	0.010	63.65	0.001
		Grid	Best	0.013	0.002	0.006	0.007	82.93	−0.000
			CV	0.041	0.012	0.013	0.017	70.86	0.009
	M2	Sample	Best	0.011	0.003	0.007	0.008	74.88	−0.031
			CV	0.037	0.008	0.011	0.013	77.46	−0.018
		Grid	Best	0.016	0.004	0.009	0.010	74.00	−0.023
			CV	0.086	0.018	0.019	0.021	79.20	−0.019
	M3	Sample	Best	0.011	0.003	0.006	0.007	73.85	−0.003
			CV	0.037	0.009	0.011	0.014	76.48	−0.008
		Grid	Best	0.016	0.004	0.008	0.009	73.94	−0.004
			CV	0.083	0.018	0.020	0.023	78.46	0.004
	M4	Sample	Best	0.009	0.002	0.005	0.006	77.43	−0.007
			CV	0.018	0.006	0.008	0.011	68.73	0.002
		Grid	Best	0.013	0.002	0.006	0.007	82.23	−0.007
			CV	0.036	0.010	0.011	0.015	71.59	0.003

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.005 and 0.006.

Table 2. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains only X, with S = 200 under $\gamma =0$, $n/N\approx 50\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{C1}$	${\hat{\mu }}_1^{C2}$	${\hat{\mu }}_1^{C3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.012	0.007	0.006	0.007	44.75	−0.017
			CV	0.033	0.027	0.021	0.028	36.37	−0.005
		Grid	Best	0.006	0.004	0.005	0.005	34.29	−0.009
			CV	0.017	0.012	0.008	0.012	53.01	−0.007
	M2	Sample	Best	0.013	0.008	0.010	0.009	36.08	−0.057
			CV	0.052	0.027	0.027	0.026	48.95	−0.062
		Grid	Best	0.010	0.005	0.007	0.006	45.83	−0.058
			CV	0.036	0.016	0.018	0.016	57.27	−0.063
	M3	Sample	Best	0.014	0.009	0.010	0.010	37.36	−0.032
			CV	0.050	0.027	0.027	0.027	46.48	−0.041
		Grid	Best	0.008	0.005	0.006	0.005	43.13	−0.027
			CV	0.030	0.013	0.015	0.014	55.91	−0.031
	M4	Sample	Best	0.014	0.009	0.008	0.009	44.00	−0.020
			CV	0.041	0.033	0.026	0.033	36.63	−0.014
		Grid	Best	0.006	0.004	0.005	0.005	37.33	−0.008
			CV	0.017	0.012	0.008	0.013	53.05	−0.023
Bounded	M1	Sample	Best	0.002	0.001	0.001	0.002	40.67	−0.008
			CV	0.009	0.005	0.004	0.005	52.83	0.009
		Grid	Best	0.004	0.002	0.002	0.002	55.17	−0.010
			CV	0.021	0.008	0.007	0.008	67.74	−0.000
	M2	Sample	Best	0.004	0.002	0.002	0.002	58.55	−0.035
			CV	0.027	0.007	0.006	0.007	77.51	−0.039
		Grid	Best	0.006	0.002	0.003	0.003	64.61	−0.033
			CV	0.056	0.011	0.011	0.011	80.27	−0.036
	M3	Sample	Best	0.004	0.002	0.002	0.002	60.82	−0.006
			CV	0.024	0.006	0.006	0.006	76.60	−0.008
		Grid	Best	0.006	0.002	0.002	0.003	66.02	−0.011
			CV	0.050	0.011	0.011	0.011	78.56	−0.010
	M4	Sample	Best	0.002	0.001	0.001	0.002	43.16	−0.011
			CV	0.010	0.005	0.005	0.005	54.67	−0.003
		Grid	Best	0.004	0.002	0.002	0.002	53.45	−0.007
			CV	0.024	0.009	0.008	0.009	68.53	−0.002

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.004 and 0.005.

Table 3. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains only X, with S = 200 under $\gamma =0.5$, $n/N\approx 13\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{C1}$	${\hat{\mu }}_1^{C2}$	${\hat{\mu }}_1^{C3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.042	0.181	0.033	0.028	32.15	0.449
			CV	0.055	0.193	0.061	0.055	−0.91	0.449
		Grid	Best	0.022	0.164	0.019	0.013	40.36	0.453
			CV	0.030	0.193	0.048	0.030	2.13	0.435
	M2	Sample	Best	0.041	0.295	0.047	0.026	37.01	0.429
			CV	0.077	0.324	0.103	0.066	15.26	0.424
		Grid	Best	0.030	0.279	0.040	0.022	27.11	0.428
			CV	0.059	0.315	0.087	0.043	28.20	0.421
	M3	Sample	Best	0.048	0.335	0.047	0.028	40.81	0.452
			CV	0.076	0.370	0.104	0.066	12.50	0.448
		Grid	Best	0.031	0.300	0.035	0.021	31.59	0.441
			CV	0.053	0.355	0.082	0.041	23.25	0.447
	M4	Sample	Best	0.048	0.193	0.035	0.030	37.27	0.452
			CV	0.067	0.205	0.067	0.061	8.73	0.433
		Grid	Best	0.023	0.177	0.023	0.014	39.42	0.437
			CV	0.036	0.204	0.049	0.031	15.80	0.452
Bounded	M1	Sample	Best	0.009	0.146	0.011	0.007	23.90	0.482
			CV	0.017	0.150	0.026	0.011	35.66	0.483
		Grid	Best	0.013	0.152	0.016	0.007	44.41	0.470
			CV	0.037	0.159	0.030	0.017	52.24	0.485
	M2	Sample	Best	0.012	0.196	0.018	0.008	33.06	0.468
			CV	0.044	0.206	0.036	0.017	60.76	0.468
		Grid	Best	0.016	0.205	0.022	0.010	39.95	0.463
			CV	0.088	0.214	0.043	0.024	72.45	0.469
	M3	Sample	Best	0.011	0.209	0.016	0.007	34.46	0.479
			CV	0.040	0.216	0.034	0.016	60.92	0.475
		Grid	Best	0.018	0.218	0.019	0.011	39.73	0.486
			CV	0.089	0.228	0.042	0.026	70.77	0.483
	M4	Sample	Best	0.010	0.152	0.012	0.007	28.69	0.484
			CV	0.019	0.157	0.025	0.012	36.41	0.485
		Grid	Best	0.014	0.164	0.015	0.007	46.99	0.485
			CV	0.040	0.170	0.029	0.019	52.63	0.492

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.005 and 0.006.

Table 4. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains only X, with S = 200 under $\gamma =0.5$, $n/N\approx 50\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{C1}$	${\hat{\mu }}_1^{C2}$	${\hat{\mu }}_1^{C3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.013	0.035	0.008	0.009	39.89	0.431
			CV	0.037	0.055	0.027	0.034	25.63	0.434
		Grid	Best	0.007	0.032	0.005	0.004	34.86	0.439
			CV	0.014	0.046	0.014	0.012	14.98	0.437
	M2	Sample	Best	0.015	0.061	0.014	0.010	33.07	0.408
			CV	0.063	0.086	0.041	0.030	52.47	0.406
		Grid	Best	0.009	0.061	0.011	0.007	28.26	0.406
			CV	0.043	0.078	0.032	0.018	59.17	0.400
	M3	Sample	Best	0.015	0.070	0.014	0.010	33.24	0.441
			CV	0.056	0.093	0.037	0.029	48.27	0.440
		Grid	Best	0.008	0.070	0.011	0.006	30.34	0.439
			CV	0.034	0.095	0.032	0.017	51.53	0.438
	M4	Sample	Best	0.015	0.041	0.010	0.010	35.19	0.423
			CV	0.048	0.063	0.033	0.040	31.52	0.414
		Grid	Best	0.006	0.033	0.005	0.005	21.67	0.422
			CV	0.017	0.054	0.016	0.014	17.89	0.423
Bounded	M1	Sample	Best	0.003	0.022	0.003	0.002	20.26	0.477
			CV	0.010	0.027	0.008	0.006	39.55	0.474
		Grid	Best	0.004	0.024	0.003	0.002	37.63	0.480
			CV	0.022	0.029	0.009	0.009	57.30	0.480
	M2	Sample	Best	0.004	0.034	0.004	0.002	45.43	0.462
			CV	0.030	0.042	0.011	0.008	72.70	0.462
		Grid	Best	0.007	0.037	0.006	0.003	56.85	0.457
			CV	0.057	0.041	0.014	0.012	78.52	0.468
	M3	Sample	Best	0.004	0.037	0.004	0.002	47.13	0.474
			CV	0.026	0.045	0.010	0.007	73.40	0.487
		Grid	Best	0.006	0.043	0.005	0.002	56.60	0.476
			CV	0.050	0.046	0.014	0.011	77.10	0.483
	M4	Sample	Best	0.003	0.024	0.002	0.002	26.50	0.483
			CV	0.010	0.030	0.007	0.006	43.47	0.475
		Grid	Best	0.004	0.027	0.003	0.002	45.69	0.480
			CV	0.023	0.032	0.009	0.009	61.42	0.485

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.004 and 0.005.

Consider first the results in Tables 1–2. Since $\gamma =0$, all three estimators, ${\hat{\mu }}_1^{C1}$, ${\hat{\mu }}_1^{C2}$, and ${\hat{\mu }}_1^{C3}$, are correct and more efficient than the standard estimator ${\hat{\mu }}_1$ in (2(2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) ) without using external information. The estimator ${\hat{\mu }}_1^{C1}$ is the best, as it uses the correct information that populations are homogeneous ($\gamma =0$) and is simpler than ${\hat{\mu }}_1^{C2}$ and ${\hat{\mu }}_1^{C3}$.

Next, the results in Tables 3–4 for $\gamma =1/2$ indicate that the estimator ${\hat{\mu }}_1^{C2}$ or ${\hat{\mu }}_1^{C3}$ using a correct constraint is better than the estimator ${\hat{\mu }}_1^{C1}$ using an incorrect constraint or the estimator ${\hat{\mu }}_1$ without using external information. Since ${\hat{\mu }}_1^{C3}$ uses more information, it is in general better than ${\hat{\mu }}_1^{C2}$. Furthermore, with an incorrect constraint, ${\hat{\mu }}_1^{C1}$ can be much worse than ${\hat{\mu }}_1$ without using external information.

5.2. The performance of ${\hat{\mu }}_1^{\mathrm{Ej}}$ given by (3), (5), or (12)

Under the same simulation setting as described in Section 5.1 but with covariate Z measured in both internal and external datasets, we compare the performance of three estimators, ${\hat{\mu }}_1^{E1}$, ${\hat{\mu }}_1^{E2}$, and ${\hat{\mu }}_1^{E3}$ given by (3(3) ${\hat{\mu }}_1^{E1}\left(\mathit{u}\right)=\sum _{i=1}^N{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(3) ), (5(5) ${\hat{\mu }}_1^{E2}\left(\mathit{u}\right)=\left\{\sum _{i=1}^n{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=n+1}^N{\hat{Y}}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i).$(5) ), and (12(12) ${\hat{\mu }}_1^{E3}\left(\mathit{u}\right)=\left\{\sum _{i=1}^N{D}_i{Y}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i)+\sum _{i=1}^N(1-D_i\left){\hat{Y}}_i{\kappa }_b\right(\mathit{u}-{\mathit{U}}_i\left)\right\}/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i)$(12) ), respectively, with the standard kernel estimator ${\hat{\mu }}_1$ in (2(2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) ) without using external information. The mean integrated squared error (MISE) and improvement (IMP) are calculated using formulas (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ), respectively, with ${\hat{\mu }}_1^{\ast }=$ one of ${\hat{\mu }}_1$, ${\hat{\mu }}_1^{E1}$, ${\hat{\mu }}_1^{E2}$, and ${\hat{\mu }}_1^{E3}$.

Tables 5–9 present the simulation results. The relative performance of ${\hat{\mu }}_1^{E1}$, ${\hat{\mu }}_1^{E2}$, ${\hat{\mu }}_1^{E3}$, and ${\hat{\mu }}_1$ follows the same pattern as ${\hat{\mu }}_1^{C1}$, ${\hat{\mu }}_1^{C2}$, ${\hat{\mu }}_1^{C3}$, and ${\hat{\mu }}_1$ in Section 5.1.

Table 5. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains both X and Z, with S = 200 under $\gamma =0$, $n/N\approx 13\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{E1}$	${\hat{\mu }}_1^{E2}$	${\hat{\mu }}_1^{E3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.055	0.020	0.035	0.025	64.46	0.020
			CV	0.072	0.032	0.052	0.036	55.51	0.015
		Grid	Best	0.047	0.010	0.027	0.017	78.44	0.013
			CV	0.060	0.019	0.034	0.024	68.49	0.014
	M2	Sample	Best	0.066	0.026	0.044	0.032	60.99	−0.054
			CV	0.088	0.035	0.052	0.040	60.73	−0.058
		Grid	Best	0.072	0.019	0.032	0.025	74.04	−0.051
			CV	0.091	0.027	0.041	0.034	70.63	−0.058
	M3	Sample	Best	0.065	0.027	0.047	0.033	58.73	−0.014
			CV	0.078	0.035	0.053	0.041	54.98	−0.009
		Grid	Best	0.055	0.016	0.031	0.023	70.04	−0.004
			CV	0.067	0.026	0.038	0.033	60.79	0.002
	M4	Sample	Best	0.055	0.020	0.037	0.025	63.83	0.018
			CV	0.075	0.034	0.053	0.040	54.39	0.012
		Grid	Best	0.045	0.010	0.026	0.019	77.21	0.015
			CV	0.057	0.017	0.031	0.023	71.13	0.012
Bounded	M1	Sample	Best	0.016	0.005	0.009	0.009	69.52	0.003
			CV	0.023	0.011	0.015	0.015	53.87	−0.003
		Grid	Best	0.036	0.017	0.023	0.022	54.15	0.009
			CV	0.075	0.043	0.050	0.050	42.12	0.005
	M2	Sample	Best	0.026	0.006	0.011	0.010	78.31	−0.024
			CV	0.030	0.015	0.020	0.018	49.71	−0.027
		Grid	Best	0.084	0.018	0.028	0.025	77.95	−0.021
			CV	0.100	0.055	0.064	0.059	44.75	−0.016
	M3	Sample	Best	0.025	0.006	0.011	0.011	78.40	0.000
			CV	0.037	0.014	0.022	0.019	63.38	0.010
		Grid	Best	0.073	0.017	0.024	0.023	76.96	0.006
			CV	0.107	0.048	0.060	0.061	55.27	0.004
	M4	Sample	Best	0.016	0.005	0.009	0.010	69.74	0.002
			CV	0.027	0.011	0.017	0.015	59.30	0.001
		Grid	Best	0.038	0.016	0.021	0.021	59.45	0.011
			CV	0.063	0.042	0.048	0.044	33.36	0.011

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.005 and 0.007.

Table 6. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains both X and Z, with S = 200 under $\gamma =0$, $n/N\approx 50\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{E1}$	${\hat{\mu }}_1^{E2}$	${\hat{\mu }}_1^{E3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.027	0.017	0.026	0.017	36.71	0.024
			CV	0.033	0.020	0.024	0.020	39.59	0.023
		Grid	Best	0.017	0.007	0.013	0.008	57.00	0.021
			CV	0.019	0.010	0.014	0.011	46.34	0.023
	M2	Sample	Best	0.041	0.022	0.025	0.023	46.91	−0.089
			CV	0.043	0.023	0.025	0.024	46.51	−0.093
		Grid	Best	0.022	0.013	0.016	0.014	42.47	−0.095
			CV	0.027	0.013	0.016	0.015	49.76	−0.090
	M3	Sample	Best	0.039	0.022	0.025	0.023	43.14	−0.034
			CV	0.042	0.022	0.025	0.023	48.01	−0.026
		Grid	Best	0.023	0.013	0.016	0.014	43.63	−0.032
			CV	0.024	0.013	0.015	0.014	44.94	−0.028
	M4	Sample	Best	0.029	0.017	0.026	0.018	40.57	0.014
			CV	0.039	0.021	0.026	0.022	45.54	0.013
		Grid	Best	0.018	0.008	0.013	0.009	54.08	0.005
			CV	0.020	0.011	0.014	0.011	46.79	0.015
Bounded	M1	Sample	Best	0.006	0.004	0.005	0.005	31.24	0.007
			CV	0.008	0.005	0.006	0.006	28.27	0.008
		Grid	Best	0.019	0.012	0.012	0.012	39.18	−0.003
			CV	0.025	0.017	0.018	0.017	30.05	0.004
	M2	Sample	Best	0.007	0.005	0.006	0.006	32.97	−0.035
			CV	0.012	0.012	0.013	0.012	2.38	−0.040
		Grid	Best	0.021	0.013	0.015	0.014	38.04	−0.036
			CV	0.042	0.036	0.040	0.036	14.28	−0.030
	M3	Sample	Best	0.007	0.005	0.006	0.005	31.15	−0.006
			CV	0.010	0.010	0.011	0.010	6.03	−0.010
		Grid	Best	0.022	0.013	0.015	0.015	37.83	−0.007
			CV	0.032	0.027	0.030	0.029	13.81	−0.002
	M4	Sample	Best	0.007	0.004	0.005	0.005	32.99	−0.004
			CV	0.008	0.006	0.007	0.006	26.81	−0.004
		Grid	Best	0.019	0.012	0.012	0.013	36.25	0.007
			CV	0.025	0.017	0.019	0.018	32.72	0.001

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.004 and 0.006.

Table 7. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains both X and Z, with S = 200 under $\gamma =0.5$, $n/N\approx 13\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{E1}$	${\hat{\mu }}_1^{E2}$	${\hat{\mu }}_1^{E3}$	IMP %	Mean of $\hat{\gamma }$
	M1	Sample	Best	0.055	0.180	0.060	0.034	37.57	0.436
			CV	0.064	0.184	0.064	0.039	38.64	0.434
		Grid	Best	0.040	0.177	0.035	0.018	54.10	0.431
			CV	0.053	0.196	0.045	0.022	59.51	0.432
	M2	Sample	Best	0.076	0.170	0.064	0.041	46.31	0.380
			CV	0.092	0.177	0.071	0.045	51.02	0.377
		Grid	Best	0.071	0.179	0.050	0.032	54.78	0.386
			CV	0.082	0.190	0.056	0.038	53.31	0.385
	M3	Sample	Best	0.072	0.188	0.064	0.043	39.63	0.425
			CV	0.091	0.192	0.067	0.049	45.96	0.421
		Grid	Best	0.053	0.191	0.042	0.033	38.08	0.436
			CV	0.074	0.202	0.048	0.035	52.56	0.425
	M4	Sample	Best	0.060	0.187	0.066	0.033	44.90	0.435
			CV	0.070	0.197	0.070	0.040	42.84	0.427
		Grid	Best	0.041	0.180	0.037	0.018	57.34	0.421
			CV	0.059	0.197	0.041	0.021	64.72	0.430
Bounded	M1	Sample	Best	0.016	0.155	0.019	0.010	39.49	0.478
			CV	0.022	0.160	0.022	0.013	41.00	0.494
		Grid	Best	0.037	0.193	0.036	0.021	44.14	0.478
			CV	0.061	0.212	0.047	0.037	38.34	0.489
	M2	Sample	Best	0.030	0.162	0.022	0.015	48.90	0.460
			CV	0.036	0.164	0.031	0.022	37.67	0.464
		Grid	Best	0.087	0.202	0.040	0.049	53.27	0.473
			CV	0.105	0.228	0.086	0.083	20.55	0.471
	M3	Sample	Best	0.029	0.171	0.020	0.015	48.81	0.485
			CV	0.039	0.174	0.029	0.022	43.49	0.492
		Grid	Best	0.076	0.205	0.036	0.045	52.75	0.490
			CV	0.107	0.240	0.076	0.073	31.97	0.492
	M4	Sample	Best	0.016	0.163	0.019	0.009	39.27	0.479
			CV	0.025	0.169	0.025	0.015	37.49	0.479
		Grid	Best	0.039	0.193	0.034	0.023	41.60	0.484
			CV	0.072	0.230	0.062	0.052	27.74	0.483

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.005 and 0.007.

Table 8. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains both X and Z, with S = 200 under $\gamma =0.5$, $n/N\approx 50\mathrm{\%}$.

				Estimator
Covariate	Model	Test data	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{E1}$	${\hat{\mu }}_1^{E2}$	${\hat{\mu }}_1^{E3}$	IMP %	Mean of $\hat{\gamma }$
Normal	M1	Sample	Best	0.027	0.061	0.032	0.019	29.74	0.430
			CV	0.034	0.061	0.033	0.022	36.98	0.422
		Grid	Best	0.015	0.041	0.015	0.008	47.14	0.423
			CV	0.018	0.049	0.018	0.010	45.47	0.417
	M2	Sample	Best	0.043	0.060	0.033	0.027	36.46	0.353
			CV	0.044	0.062	0.035	0.030	33.60	0.360
		Grid	Best	0.023	0.053	0.022	0.016	27.23	0.367
			CV	0.024	0.052	0.022	0.016	31.83	0.353
	M3	Sample	Best	0.043	0.068	0.031	0.029	32.86	0.417
			CV	0.042	0.067	0.030	0.029	30.68	0.424
		Grid	Best	0.022	0.058	0.018	0.015	30.01	0.416
			CV	0.023	0.059	0.019	0.017	27.17	0.417
	M4	Sample	Best	0.033	0.062	0.034	0.022	34.75	0.413
			CV	0.039	0.064	0.033	0.023	40.31	0.414
		Grid	Best	0.017	0.046	0.017	0.010	42.46	0.409
			CV	0.020	0.056	0.018	0.011	44.86	0.421
Bounded	M1	Sample	Best	0.006	0.028	0.007	0.006	8.96	0.481
			CV	0.007	0.029	0.008	0.006	13.22	0.482
		Grid	Best	0.018	0.047	0.016	0.017	13.08	0.482
			CV	0.023	0.054	0.023	0.021	8.56	0.478
	M2	Sample	Best	0.008	0.033	0.008	0.006	20.59	0.465
			CV	0.015	0.040	0.017	0.013	14.82	0.465
		Grid	Best	0.023	0.059	0.021	0.020	12.38	0.460
			CV	0.045	0.081	0.050	0.048	-6.84	0.473
	M3	Sample	Best	0.008	0.037	0.008	0.006	16.16	0.493
			CV	0.010	0.040	0.012	0.009	2.05	0.497
		Grid	Best	0.022	0.060	0.018	0.020	17.75	0.495
			CV	0.033	0.078	0.041	0.040	-21.18	0.498
	M4	Sample	Best	0.007	0.032	0.007	0.005	21.96	0.476
			CV	0.008	0.032	0.008	0.006	21.82	0.489
		Grid	Best	0.019	0.052	0.016	0.019	14.56	0.481
			CV	0.023	0.059	0.022	0.023	2.61	0.485

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.004 and 0.006.

The only difference between the results here and those in Section 5.1 is that the use of more external data (a smaller n/N) results in a better performance of ${\hat{\mu }}_1^{E2}$ or ${\hat{\mu }}_1^{E3}$ (or ${\hat{\mu }}_1^{E1}$ when it is correct). This is actually consistent with our theoretical result Theorem 4.1 in Section 4, which shows that both the squared bias $B_a^2\left(\mathit{u}\right)$ and variance $V_a\left(\mathit{u}\right)$ in (17(17) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{E2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_a\right(\mathit{u}),V_a(\mathit{u}\left)\right),$(17) ) are decreasing in the limit $a=\underset{n\to \mathrm{\infty }}{lim}(N-n)/n$. On the other hand, the simulation results in Section 5.1 and Theorem 4.2 in Section 4 do not show a clear indication of using more external data produces better estimators. The main reason for this is that, when Z is not observed in the external dataset, the estimator ${\hat{\mu }}_1^{\mathrm{Cj}}$ relies more on internal data to recover the loss of Z from external dataset in a complicated way.

5.3. The performance of ${\hat{\mu }}_1^{\mathrm{Cj}}$ given by (16) with q = 2

We re-consider the simulation in Section 5.1 but with the dimension of $\mathit{X}$ to be q = 2, i.e., $\mathit{U}=(X_1,X_2,Z{)}^{\mathrm{\top }}$. We only consider normally distributed covariates with means 0, variances 1, and the correlations in $(X_1,Z)$, $(X_2,Z)$, and $(X_1,X_2)$ being 0.5, 0.5, and 0.25, respectively. Given $\mathit{U}$, the response variable Y is normally distributed with mean $\mu (X_1,X_2,Z)=X_1/2+X_2/4-Z^2/4$ and variance 1. Moreover, $P(D=1|Y,\mathit{X},Z)=1/\exp ({\gamma }_0+2|X_1|+\mathrm{\gamma Y})$, while the remaining settings are the same as in Section 5.1. In calculating MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ), we only a random ${\mathit{U}}_{s,t}$ with T = 121, not fixed grid points. Also, we consider only evaluating the performance of estimators ${\hat{\mu }}_1^{\mathrm{Cj}}$, since estimators ${\hat{\mu }}_1^{\mathrm{Ej}}$ are simpler.

The results are shown in Table 9. Compared with results in Tables 1–4 for the case of q = 1, the MISEs in this case are larger due to the fact of having more covariates (q = 2). But the relative performances of estimators are the same as those shown in Tables 1–4.

Table 9. Simulated MISE (21(21) $\mathrm{MISE}\left({\hat{\mu }}_1^{\ast }\right)=\frac{1}{S}\sum _{s=1}^S\frac{1}{T}\sum _{t=1}^T\left\{{\hat{\mu }}_1^{\ast }\right({\mathit{U}}_{s,t})-{\mu }_1({\mathit{U}}_{s,t}){\}}^2,$(21) ) and IMP (22(22) $\text{IMP}=1-\frac{min\left\{\mathrm{MISE}\right({\hat{\mu }}_1^{\ast }\left)\right\}}{\mathrm{MISE}\left({\hat{\mu }}_1\right)},$(22) ) when the external dataset contains only normally distributed $(X_1,X_2)$, with S = 200.

				Estimator
Test data	γ	n/N	b, l	${\hat{\mu }}_1$	${\hat{\mu }}_1^{C1}$	${\hat{\mu }}_1^{C2}$	${\hat{\mu }}_1^{C3}$	IMP %	Mean of $\hat{\gamma }$
Sample	0	13%	Best	0.090	0.046	0.054	0.051	49.496	−0.006
			CV	0.080	0.056	0.063	0.062	29.115	0.007
		50%	Best	0.027	0.020	0.023	0.020	25.701	−0.006
			CV	0.031	0.025	0.027	0.026	20.040	−0.008
	0.5	13%	Best	0.085	0.198	0.067	0.055	35.178	0.421
			CV	0.070	0.190	0.072	0.062	11.429	0.406
		50%	Best	0.025	0.049	0.025	0.021	17.000	0.389
			CV	0.031	0.052	0.031	0.026	15.798	0.391

Note: Simulation standard deviations of $\hat{\gamma }$ for all cases are between 0.004 and 0.006.

6. Discussion

Curse of dimensionality is a well-known problem for nonparametric methods. Thus, the proposed method in Section 2 is intended for low dimensional covariate $\mathit{U}$, i.e., p is small. If p is not small, then we should reduce the dimension of $\mathit{U}$ prior to applying the CK, or any kernel methods. For example, consider a single index model assumption (K.-C. Li, 1991), i.e., ${\mu }_1\left(\mathit{U}\right)$ in (1(1) ${\mu }_1\left(\mathit{u}\right)=E(Y\mid \mathit{U}=\mathit{u},D=1),$(1) ) is assumed to be (23) ${\mu }_1\left(\mathit{U}\right)={\mu }_1\left({\mathit{\eta }}^{\mathrm{\top }}\mathit{U}\right),$(23) where $\mathit{\eta }$ is an unknown p-dimensional vector. The well-known SIR technique (K.-C. Li, 1991) can be applied to obtain a consistent and asymptotically normal estimator $\hat{\mathit{\eta }}$ of $\mathit{\eta }$ in (23(23) ${\mu }_1\left(\mathit{U}\right)={\mu }_1\left({\mathit{\eta }}^{\mathrm{\top }}\mathit{U}\right),$(23) ). Once $\mathit{\eta }$ is replaced by $\hat{\mathit{\eta }}$, the kernel method can be applied with $\mathit{U}$ replaced by the one-dimensional ‘covariate’ ${\hat{\mathit{\eta }}}^{\mathrm{\top }}\mathit{U}$. We can also apply other dimension reduction techniques developed under assumptions weaker than (23(23) ${\mu }_1\left(\mathit{U}\right)={\mu }_1\left({\mathit{\eta }}^{\mathrm{\top }}\mathit{U}\right),$(23) ) (Cook & Weisberg, 1991; B. Li & Wang, 2007; Ma & Zhu, 2012; Y. Shao et al., 2007; Xia et al., 2002).

We turn to the dimension of $\mathit{X}$ in the external dataset. When the dimension of $\mathit{X}$ is high, we may consider the following approach. Instead of using constraint (15(15) $\begin{array}{rl}& \text{subject to}\sum _{i=1}^n\{{\mu }_i-{\hat{h}}_1^{\mathrm{Ej}}({\mathit{X}}_i\left)\right\}\mathit{g}({\mathit{X}}_i{)}^{\mathrm{\top }}=0,\end{array}$(15) ), we use component-wise constraints (24) $\sum _{i=1}^n\{{\mu }_i-{\hat{h}}_1^{\left(k\right)}(X_i^{\left(k\right)}\left)\right\}{\mathit{g}}_k(X_i^{\left(k\right)}{)}^{\mathrm{\top }}=0,k=1,\dots ,q,$(24) where $X_i^{\left(k\right)}$ is the kth component of ${\mathit{X}}_i$, ${\mathit{g}}_k\left(X^{\left(k\right)}\right)=(1,X^{\left(k\right)}{)}^{\mathrm{\top }}$, and ${\hat{h}}_1^{\left(k\right)}\left(X_i^{\left(k\right)}\right)$ is an estimator of $h_1^{\left(k\right)}\left(X^{\left(k\right)}\right)=E(Y\mid X^{\left(k\right)},D=1)$ using methods described in Section 2. More constraints are involved in (24(24) $\sum _{i=1}^n\{{\mu }_i-{\hat{h}}_1^{\left(k\right)}(X_i^{\left(k\right)}\left)\right\}{\mathit{g}}_k(X_i^{\left(k\right)}{)}^{\mathrm{\top }}=0,k=1,\dots ,q,$(24) ), but estimation only involves one dimensional $X^{\left(k\right)}$, $k=1,\dots ,q$.

The kernel κ we adopted in (2(2) ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{Y}_i{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),$(2) ) and (16(16) ${\hat{\mu }}_1^{\mathrm{Cj}}\left(\mathit{u}\right)=\sum _{i=1}^n{\hat{\mu }}_i^{\mathrm{Cj}}{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i),j=1,2,3,$(16) ) is the second-order kernel so that the convergence rate of ${\hat{\mu }}_1^E\left(\mathit{u}\right)-{\mu }_1\left(\mathit{u}\right)$ is $n^{-2/(4+p)}$. An mth-order kernel with m>2 as defined by Bierens (1987) may be used to achieve convergence rate $n^{-m/(2m+p)}$. Alternatively, we may apply other nonparametric smoothing techniques such as the local polynomial (Fan et al., 1997) to achieve convergence rate $n^{-m/(2m+p)}$ with $m\ge 2$.

Our results can be extended to the scenarios where several external datasets are available. Since each external source may provide different covariate variables, we may need to apply component-wise constraints (24(24) $\sum _{i=1}^n\{{\mu }_i-{\hat{h}}_1^{\left(k\right)}(X_i^{\left(k\right)}\left)\right\}{\mathit{g}}_k(X_i^{\left(k\right)}{)}^{\mathrm{\top }}=0,k=1,\dots ,q,$(24) ) by estimating ${\hat{h}}_1^{\left(k\right)}$ via combining all the external sources that collects covariate $X^{\left(k\right)}$. If populations of external datasets are different, then we may have to apply a combination of the methods described in Section 2.

Acknowledgments

The authors would like to thank two anonymous referees for helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Jun Shao's research was partially supported by the National Natural Science Foundation of China [Grant Number 11831008] and the U.S. National Science Foundation [Grant Number DMS-1914411].

Appendix

Proof of Theorem 4.1.

Proof of Theorem 4.1.

Let ${\tilde{\mu }}_1\left(\mathit{u}\right)=\hat{p}\left(\mathit{u}\right){\hat{\mu }}_1\left(\mathit{u}\right)+\{1-\hat{p}(\mathit{u}\left)\right\}{\hat{\mu }}_0\left(\mathit{u}\right),$ where ${\hat{\mu }}_1\left(\mathit{u}\right)=\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i)Y_i/\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i)$, ${\hat{\mu }}_0\left(\mathit{u}\right)=\sum _{i=n+1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i){\tilde{Y}}_i/\sum _{i=n+1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i)$, and $\hat{p}\left(\mathit{u}\right)=\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/\sum _{i=1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i).$ Under (B3)–(B4), Theorem 2 in Nadaraya (1964) shows that $\hat{p}\left(\mathit{u}\right)$ converges to $P(D=1|\mathit{U}=\mathit{u})$ in probability. Under (B1)–(B4), $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_1\right(\mathit{u}),V_1(\mathit{u}\left)\right),$ $B_1\left(\mathit{u}\right)=c^{1/2}{A}_1\left(\mathit{u}\right)$, $V_1\left(\mathit{u}\right)=\frac{{\sigma }_1^2\left(\mathit{u}\right)}{f_1\left(\mathit{u}\right)}\int \kappa (\mathit{v}{)}^2\mathrm{d}\mathit{v},$ and $\sqrt{n/(N-n)}\sqrt{(N-n)b^p}\left\{{\hat{\mu }}_0\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_0\right(\mathit{u}),V_0(\mathit{u}\left)\right),$ $B_0\left(\mathit{u}\right)=c^{1/2}{A}_0\left(\mathit{u}\right)$, $V_0\left(\mathit{u}\right)=\frac{{\sigma }_0^2\left(\mathit{u}\right)}{a{f}_0\left(\mathit{u}\right)}\int \kappa (\mathit{v}{)}^2\mathrm{d}\mathit{v}$. Then (17(17) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{E2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_a\right(\mathit{u}),V_a(\mathit{u}\left)\right),$(17) ) holds for ${\tilde{\mu }}_1\left(\mathit{u}\right)$, by Slutsky's theorem, the independence between ${\hat{\mu }}_1$ and ${\hat{\mu }}_0$, and the definition of a. The desired result (17(17) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{E2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_a\right(\mathit{u}),V_a(\mathit{u}\left)\right),$(17) ) follows from the fact that $|{\hat{\mu }}_1^{E2}\left(\mathit{u}\right)-{\tilde{\mu }}_1\left(\mathit{u}\right)|$ is bounded by (A1) $\{1-\hat{p}(\mathit{u}\left)\right\}\underset{i>n}{max}|\frac{\hat{f}(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{\hat{f}(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}-\frac{f(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{f(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}|\frac{\sum _{i=n+1}^N|Y_i|{\kappa }_b(\mathit{u}-{\mathit{U}}_i)}{\sum _{i=n+1}^N{\kappa }_b(\mathit{u}-{\mathit{U}}_i)},$(A1) which is $o_p\left(1/\sqrt{n{b}^p}\right)$ by result (18(18) $\underset{i=n+1,\dots ,N}{max}|\frac{\hat{f}(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{\hat{f}(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}-\frac{f(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{f(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}|=\frac{o_p\left(1\right)}{\sqrt{n{b}^p}}$(18) ) under condition (B5).

Proof of Theorem 4.2.

Proof of Theorem 4.2.

Write (A2) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{C2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}=T_1+\cdots +T_6,$(A2) where $T_1=n^{-1/2}{b}^{p/2}{\mathit{\delta }}_b\left(\mathit{u}{)}^{\mathrm{\top }}\right({\mathit{I}}_n-\mathit{P}\left){\mathit{B}}_l^{-1}{\mathbf{\Delta }}_l\mathit{ϵ}/{\hat{f}}_b\right(\mathit{u})$, $T_2=n^{-1/2}{b}^{p/2}{\mathit{\delta }}_b\left(\mathit{u}{)}^{\mathrm{\top }}\right\{{\mathit{\mu }}_1-{\mu }_1\left(\mathit{u}\right){\mathbf{1}}_n\left\}/{\hat{f}}_b\right(\mathit{u})$, $T_3=n^{-1/2}{b}^{p/2}{\mathit{\delta }}_b\left(\mathit{u}{)}^{\mathrm{\top }}\right({\mathit{B}}_l^{-1}{\mathbf{\Delta }}_l{\mathit{\mu }}_1-{\mathit{\mu }}_1\left)/{\hat{f}}_b\right(\mathit{u})$, $T_4=-n^{-1/2}{b}^{p/2}{\mathit{\delta }}_b\left(\mathit{u}{)}^{\mathrm{\top }}\mathit{P}\right({\mathit{B}}_l^{-1}{\mathbf{\Delta }}_l{\mathit{\mu }}_1-{\mathit{\mu }}_1\left)/{\hat{f}}_b\right(\mathit{u})$, $T_5=n^{-1/2}{b}^{p/2}{\mathit{\delta }}_b\left(\mathit{u}{)}^{\mathrm{\top }}\mathit{P}\right({\hat{\mathit{h}}}_1-{\mathit{h}}_1\left)/{\hat{f}}_b\right(\mathit{u})$, $T_6=n^{-1/2}{b}^{p/2}{\mathit{\delta }}_b\left(\mathit{u}{)}^{\mathrm{\top }}\mathit{P}\right({\mathit{h}}_1-{\mathit{\mu }}_1\left)/{\hat{f}}_b\right(\mathit{u})$, ${\hat{f}}_b\left(\mathit{u}\right)=\sum _{i=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_i)/n$, ${\mathit{\delta }}_b\left(\mathit{u}\right)=\left({\kappa }_b\right(\mathit{u}-{\mathit{U}}_1),\dots ,{\kappa }_b(\mathit{u}-{\mathit{U}}_n){)}^{\mathrm{\top }}$, ${\mathit{I}}_n$ is the identity matrix of order n, ${\mathbf{1}}_n$ is the n-vector with all components being 1, ${\mathit{B}}_l$ is the $n\times n$ diagonal matrix whose ith diagonal element is ${\hat{f}}_l\left({\mathit{U}}_i\right)$, ${\mathbf{\Delta }}_l$ is the $n\times n$ matrix whose $(i,j)$th entry is ${\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j)/n$, $\mathit{ϵ}=({ϵ}_1,\dots ,{ϵ}_n{)}^{\mathrm{\top }}$ with ${ϵ}_i=Y_i-{\mu }_1\left({\mathit{U}}_i\right)$, ${\mathit{h}}_1$ is the n-dimensional vector whose ith component is $h_1\left({\mathit{X}}_i\right)$, $\mathit{P}=\mathit{G}({\mathit{G}}^{\mathrm{\top }}\mathit{G}{)}^{-1}{\mathit{G}}^{\mathrm{\top }}$, and $\mathit{G}$, ${\hat{\mathit{h}}}_1$, and ${\mathit{\mu }}_1$ are defined in Section 2.

We first show that $T_1$ in (A2(A2) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{C2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}=T_1+\cdots +T_6,$(A2) ) is asymptotically normal with mean 0 and variance $V_r\left(\mathit{u}\right)$ defined in Theorem 4.2. Consider a further decomposition $T_1=\sqrt{n}V+T_{11}+T_{12}+T_{13}$, where $V=\frac{1}{n^2}\sum _{j=1}^n\sum _{i=1}^{n}S({\mathit{U}}_i,{ϵ}_i,{\mathit{U}}_j,{ϵ}_j)$is a V-statistic with $\begin{array}{rl}S({\mathit{U}}_i,{ϵ}_i,{\mathit{U}}_j,{ϵ}_j)& =\frac{b^{p/2}}{2f_1\left(\mathit{u}\right)}\{\frac{{\kappa }_b(\mathit{u}-{\mathit{U}}_i){\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j){ϵ}_j}{f_1\left({\mathit{U}}_i\right)}+\frac{{\kappa }_b(\mathit{u}-{\mathit{U}}_j){\kappa }_l({\mathit{U}}_j-{\mathit{U}}_i){ϵ}_i}{f_1\left({\mathit{U}}_j\right)}\},\\ T_{11}& =\frac{b^{p/2}}{n^{3/2}}\sum _{i=1}^n\frac{{\kappa }_b(\mathit{u}-{\mathit{U}}_i){\kappa }_l\left(0\right){ϵ}_i}{f_1\left(\mathit{u}\right)f_1\left({\mathit{U}}_i\right)},\\ T_{12}& =\frac{b^{p/2}}{n^{3/2}}\sum _{j=1}^n\sum _{i=1}^n\frac{{\kappa }_b(\mathit{u}-{\mathit{U}}_i){\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j)}{f_1\left(\mathit{u}\right)f_1\left({\mathit{U}}_i\right)}\{\frac{f_1\left(\mathit{u}\right)f_1\left({\mathit{U}}_i\right)}{{\hat{f}}_b\left(\mathit{u}\right){\hat{f}}_l\left({\mathit{U}}_i\right)}-1\}{ϵ}_j,\end{array}$and $T_{13}=-n^{-1/2}{b}^{p/2}{\mathit{\delta }}_b\left(\mathit{u}{)}^{\mathrm{\top }}\mathit{P}{\mathit{B}}_l^{-1}{\mathbf{\Delta }}_l\mathit{ϵ}/{\hat{f}}_b\right(\mathit{u}).$ Note that $S_1({\mathit{U}}_1,{ϵ}_1)=E\left\{S\right({\mathit{U}}_1,{ϵ}_1,{\mathit{U}}_2,{ϵ}_2)\mid {\mathit{U}}_1,{ϵ}_1\}=\frac{b^{p/2}}{2f_1\left(\mathit{u}\right)}\{\int {\kappa }_l({\mathit{u}}_2-{\mathit{U}}_1\left){\kappa }_b\right(\mathit{u}-{\mathit{u}}_2\left)\mathrm{d}{\mathit{u}}_2\right\}{ϵ}_1$having variance $\begin{array}{rl}\mathrm{Var}\left\{S_1\right({\mathit{U}}_1,{ϵ}_1\left)\right\}& =\frac{b^{p/2}}{4f_1^2\left(\mathit{u}\right)}\int f_1\left({\mathit{u}}_1\right){\sigma }_1^2\left({\mathit{u}}_1\right){\{\int {\kappa }_l({\mathit{u}}_2-{\mathit{u}}_1\left){\kappa }_b\right(\mathit{u}-{\mathit{u}}_2\left)\mathrm{d}{\mathit{u}}_2\right\}}^2\mathrm{d}{\mathit{u}}_1\\ & =\frac{b^{p/2}}{4f_1^2\left(\mathit{u}\right)}\int f_1\left({\mathit{u}}_1\right){\sigma }_1^2\left({\mathit{u}}_1\right){\{\int {\kappa }_l(\mathit{v}\left){\kappa }_b\right(\mathit{u}-{\mathit{u}}_1-l\mathit{\nu }\left)\mathrm{d}\mathit{\nu }\right\}}^2\mathrm{d}{\mathit{u}}_1\\ & =\frac{1}{4f_1^2\left(\mathit{u}\right)}\int f_1(\mathit{u}-b\mathit{w}){\sigma }_1^2(\mathit{u}-b\mathit{w}){\{\int \kappa (\mathit{v}\left)\kappa (\mathit{w}-\mathit{\nu }\frac{l}{b})\mathrm{d}\mathit{\nu }\right\}}^2\mathrm{d}\mathit{w},\end{array}$where ${\sigma }_1^2(\cdot )$ is given in condition (C2), the second and third equalities follow from changing variables ${\mathit{u}}_2-{\mathit{u}}_1=l\mathit{\nu }$ and $\mathit{u}-{\mathit{u}}_1=b\mathit{w}$, respectively. From the continuity of $f_1(\cdot )$ and ${\sigma }_1^2(\cdot )$, $\mathrm{Var}\left\{S_1\right({\mathit{u}}_1,{ϵ}_1\left)\right\}$ converges to $V_r\left(\mathit{u}\right).$ Therefore, by the theory for asymptotic normality of V-statistics (e.g., Theorem 3.16 in J. Shao, 2003), $\sqrt{n}V⟹dN(0,V_r(\mathit{u}\left)\right)$.

Conditioned on ${\mathit{U}}_1,\dots ,{\mathit{U}}_n$, $T_{11}$ has mean 0 and variance $\begin{array}{rl}\mathrm{Var}(T_{11}|{\mathit{U}}_1,\dots ,{\mathit{U}}_n)& =\frac{b^p}{4f_1^2\left(\mathit{u}\right)n^3}\sum _{i=1}^n\frac{{\kappa }_b(\mathit{u}-{\mathit{U}}_i{)}^2{\kappa }_l(0{)}^2{\sigma }_1^2\left({\mathit{U}}_i\right)}{f_1\left({\mathit{U}}_i\right)}\\ & \le \frac{\underset{\mathit{u}\in \mathcal{U}}{sup}\kappa (\mathit{u}{)}^3}{4f_1^2\left(\mathit{u}\right)n^3{b}^{2p}}\sum _{i=1}^n\frac{{\kappa }_b(\mathit{u}-{\mathit{U}}_i){\sigma }_1^2\left({\mathit{U}}_i\right)}{f_1\left({\mathit{U}}_i\right)}=o_p\left(1\right).\end{array}$This proves that $T_{11}=o_p\left(1\right)$. Note that $E(T_{12}\mid {\mathit{U}}_1,\dots ,{\mathit{U}}_n)=0$ and $\mathrm{Var}(T_{12}\mid {\mathit{U}}_1,\dots ,{\mathit{U}}_n)$ is bounded by $max\{\frac{1}{f_1^2\left(\mathit{u}\right)},\frac{1}{{\hat{f}}_b^2\left(\mathit{u}\right)}\}\underset{i=1,\dots ,n}{max}{|\frac{f_1\left({\mathit{U}}_i\right)}{{\hat{f}}_l\left({\mathit{U}}_i\right)}-1|}^2\mathrm{Var}(\sqrt{n}V+T_{11}|{\mathit{U}}_1,\dots ,{\mathit{U}}_n).$Therefore, under the assumed condition that $f_1$ is bounded away from zero, Lemma 3 in Dai and Shao (2023) implies $T_{12}=o_p\left(1\right)$. Note that $T_{13}=\frac{b^{p/2}}{n^{1/2}}\sum _{j=1}^n{W}_j\left(\mathit{u}\right){ϵ}_j,W_j\left(\mathit{u}\right)=\frac{1}{n}\sum _{i=1}^n\frac{{\kappa }_b(\mathit{u}-{\mathit{U}}_i)\mathit{g}({\mathit{X}}_i{)}^{\mathrm{\top }}}{{\hat{f}}_b\left(\mathit{u}\right)}({\mathit{G}}^{\mathrm{\top }}\mathit{G}{)}^{-1}\sum _{i=1}^n\frac{{\kappa }_l({\mathit{U}}_i-{\mathit{U}}_j)\mathit{g}\left({\mathit{X}}_i\right)}{{\hat{f}}_l\left({\mathit{U}}_i\right)}.$Conditioned on ${\mathit{U}}_1,\dots ,{\mathit{U}}_n$, $T_{13}$ has mean 0 and variance $\mathrm{Var}(T_{13}\mid {\mathit{U}}_1,\dots ,{\mathit{U}}_n)=\frac{b^p}{n}\sum _{j=1}^n{W}_j^2\left(\mathit{u}\right){\sigma }_1^2\left({\mathit{U}}_j\right)=O_p\left(b^p\right)=o_p\left(1\right),$because, under the assumed condition that $f_1$ is bounded away from zero, Lemma 3 in Dai and Shao (2023) implies $\underset{j=1,\dots ,n}{max}|W_j\left(\mathit{u}\right)-\mathit{g}\left(\mathit{u}{)}^{\mathrm{\top }}{\mathbf{\Sigma }}_g^{-1}\mathit{g}\right({\mathit{X}}_j)|=o_p(1).$ Thus, $T_{13}=o_p\left(1\right).$ Consequently, $T_1$ has the same asymptotic distribution as $\sqrt{n}V$, the claimed result.

From Lemma 4 in Dai and Shao (2023) and (C4), $T_2=\sqrt{c}{A}_1\left(\mathit{u}\right)\{1+o_p(1\left)\right\}.$ Note that $\begin{array}{rl}{T}_3& =\frac{\sqrt{n{b}^p}{l}^2}{n{\hat{f}}_b\left(\mathit{u}\right)}\sum _{j=1}^n{\kappa }_b(\mathit{u}-{\mathit{U}}_j)\left[\frac{1}{n{l}^2{\hat{f}}_b\left({\mathit{U}}_j\right)}\sum _{i=1}^n{\kappa }_l\right(\mathit{u}-{\mathit{U}}_i\left)\right\{{\mu }_1\left({\mathit{U}}_i\right)-{\mu }_1\left({\mathit{U}}_j\right)\left\}\right]\\ & =\left\{\frac{\sqrt{c}{r}^2}{n{\hat{f}}_b\left(\mathit{u}\right)}\sum _{j=1}^n{\kappa }_b\right(\mathit{u}-{\mathit{U}}_j\left)A_1\right({\mathit{U}}_j\left)\right\}\{1+o_p(1\left)\right\}=\sqrt{c}{r}^2{A}_1\left(\mathit{u}\right)\{1+o_p(1\left)\right\},\end{array}$where the second equality follows from (A4) and Lemmas 3–4 in Dai and Shao (2023), and the last equality follows from Lemma 2 in Dai and Shao (2023) and continuity of $A_1(\cdot )$. Also, $\begin{array}{rl}-\frac{n^{1/2}{T}_4}{b^{p/2}}& =\frac{1}{n}\sum _{i=1}^n\frac{{\kappa }_b(\mathit{u}-{\mathit{U}}_i)\mathit{g}({\mathit{X}}_i{)}^{\mathrm{\top }}}{{\hat{f}}_b\left(\mathit{u}\right)}\left({\mathit{G}}^{\mathrm{\top }}\mathit{G}{)}^{-1}\sum _{j=1}^n\frac{\mathit{g}\left({\mathit{X}}_j\right)}{n{\hat{f}}_b\left({\mathit{U}}_j\right)}\sum _{i=1}^n{\kappa }_l\right(\mathit{u}-{\mathit{U}}_i\left)\right\{{\mu }_1\left({\mathit{U}}_i\right)-{\mu }_1\left({\mathit{U}}_j\right)\}\\ & =\left\{\mathit{g}\right(\mathit{x}{)}^{\mathrm{\top }}{\mathbf{\Sigma }}_g^{-1}\frac{1}{n}\sum _{j=1}^n\frac{\mathit{g}\left({\mathit{X}}_j\right)}{n{\hat{f}}_b\left({\mathit{U}}_j\right)}\sum _{i=1}^n{\kappa }_l(\mathit{u}-{\mathit{U}}_i)\left\{{\mu }_1\right({\mathit{U}}_i)-{\mu }_1({\mathit{U}}_j\left)\right\}\}\{1+o_p(1\left)\right\}\\ & =\left\{\mathit{g}\right(\mathit{x}{)}^{\mathrm{\top }}{\mathbf{\Sigma }}_g^{-1}\frac{l^{2/p}}{n}\sum _{j=1}^n\mathit{g}\left({\mathit{X}}_j\right)A_1\left({\mathit{U}}_j\right)\}\{1+o_p(1\left)\right\}\\ & =\sqrt{c}{r}^2\mathit{g}\left(\mathit{x}{)}^{\mathrm{\top }}{\mathbf{\Sigma }}_g^{-1}E\right\{\mathit{g}\left(\mathit{X}\right)A_1\left(\mathit{U}\right)\left\}\right\{1+o_p\left(1\right)\},\end{array}$where the first equality follows from Lemma 3 in Dai and Shao (2023) and the law of large numbers, the second equality follows from Lemma 4 in Dai and Shao (2023), and the last equality follows from the law of large numbers. Similarly, $\begin{array}{rl}\frac{n^{1/2}{T}_5}{b^{p/2}}& =\frac{1}{n}\sum _{i=1}^n\frac{{\kappa }_b(\mathit{u}-{\mathit{U}}_i)\mathit{g}({\mathit{X}}_i{)}^{\mathrm{\top }}}{{\hat{f}}_b\left(\mathit{u}\right)}\left({\mathit{G}}^{\mathrm{\top }}\mathit{G}{)}^{-1}\sum _{i=1}^n\mathit{g}\right({\mathit{X}}_i\left)\right\{{\hat{h}}_1\left({\mathit{X}}_i\right)-h_1\left({\mathit{X}}_i\right)\}\\ & =\left[\mathit{g}\right(\mathit{x}{)}^{\mathrm{\top }}{\mathbf{\Sigma }}_g^{-1}\frac{1}{n}\sum _{i=1}^n\mathit{g}\left({\mathit{X}}_i\right)\left\{{\hat{h}}_1\right({\mathit{X}}_i)-h_1({\mathit{X}}_i\left)\right\}]\{1+o_p(1\left)\right\}\\ & \le \{1+o_p(1\left)\right\}{O}_p\left(1\right)\underset{j=1,\dots ,n}{max}|{\hat{h}}_1\left({\mathit{X}}_j\right)-h_1\left({\mathit{X}}_j\right)|,\end{array}$where the second equality follows from Lemma 3 in Dai and Shao (2023). Under (B1)–(B5) with $\mathit{U}$ and p replaced by $\mathit{X}$ and q, and (C5), Lemma 8.10 in Newey and McFadden (1994) implies that (A3) $max|{\hat{h}}_1\left({\mathit{X}}_i\right)-h_1\left({\mathit{X}}_1\right)|=O_p\left(\sqrt{\log \left(n\right)}{n}^{-2/(q+4)}\right),$(A3) which is $o_p\left(1/\sqrt{n{b}^p}\right)=o_p\left(n^{-2/(p+4)}\right)$ and, hence, $T_5=o_p\left(1\right).$ From Lemma 3 in Dai and Shao (2023) and the Central Limit Theorem, $T_6=\frac{b^{p/2}}{n^{1/2}}\sum _{i=1}^n\frac{{\kappa }_b(\mathit{u}-{\mathit{U}}_i)\mathit{g}({\mathit{X}}_i{)}^{\mathrm{\top }}}{{\hat{f}}_b\left(\mathit{u}\right)}\left({\mathit{G}}^{\mathrm{\top }}\mathit{G}{)}^{-1}\sum _{i=1}^n\mathit{g}\right({\mathit{X}}_i\left)\right\{h_1\left({\mathit{X}}_i\right)-{\mu }_1\left({\mathit{U}}_i\right)\}=O_p(b^{p/2})=o_p(1).$Combining these results, we obtain that $T_2+\cdots +T_6=B_r\left(\mathit{u}\right)+o_p\left(1\right)$. This completes the proof.

Proof of Theorem 4.3.

Proof of Theorem 4.3.

Define $\begin{array}{rl}{\hat{\omega }}_{t1}& =\frac{1}{N}\sum _{i=1}^N{R}_i{\check{\kappa }}_b(\mathit{u}-{\mathit{U}}_i){\mathrm{e}}^{t{Y}_i},{\hat{\omega }}_{t2}=\frac{1}{N}\sum _{i=1}^N(1-R_i){\check{\kappa }}_b(\mathit{u}-{\mathit{U}}_i)Y_i{\mathrm{e}}^{-t{Y}_i},\\ {\hat{\omega }}_{t3}& =\frac{1}{N}\sum _{i=1}^N{R}_i{\check{\kappa }}_b(\mathit{u}-{\mathit{U}}_i),{\hat{\omega }}_{t4}=\frac{1}{N}\sum _{i=1}^N(1-R_i){\check{\kappa }}_b(\mathit{u}-{\mathit{U}}_i),\\ {\hat{\omega }}_{t5}& =\frac{1}{N}\sum _{i=1}^N{R}_i{\check{\kappa }}_b(\mathit{u}-{\mathit{U}}_i)Y_i{\mathrm{e}}^{t{Y}_i},{\hat{\omega }}_{t6}=\frac{1}{N}\sum _{i=1}^N(1-R_i){\check{\kappa }}_b(\mathit{u}-{\mathit{U}}_i)Y_i^2{\mathrm{e}}^{-t{Y}_i},\\ {\hat{\omega }}_{t7}& =\frac{1}{N}\sum _{i=1}^N{R}_i{\check{\kappa }}_b(\mathit{u}-{\mathit{U}}_i)Y_i^2{\mathrm{e}}^{t{Y}_i},{\hat{\omega }}_{t8}=\frac{1}{N}\sum _{i=1}^N(1-R_i){\check{\kappa }}_b(\mathit{u}-{\mathit{U}}_i)Y_i^3{\mathrm{e}}^{-t{Y}_i}.\end{array}$Then, $\hat{h}(\mathit{u},t)={\hat{\omega }}_{t1}{\hat{\omega }}_{t2}/{\hat{\omega }}_{t3}{\hat{\omega }}_{t4}$, ${\mathrm{\nabla }}_t\hat{h}(\mathit{u},t)=({\hat{\omega }}_{t2}{\hat{\omega }}_{t5}-{\hat{\omega }}_{t1}{\hat{\omega }}_{t6})/{\hat{\omega }}_{t3}{\hat{\omega }}_{t4}$, and ${\mathrm{\nabla }}_t^2\hat{h}(\mathit{u},t)=({\hat{\omega }}_{t1}{\hat{\omega }}_{t8}-2{\hat{\omega }}_{t5}{\hat{\omega }}_{t6}+{\hat{\omega }}_{t2}{\hat{\omega }}_{t7})/{\hat{\omega }}_{t3}{\hat{\omega }}_{t4}.$ Let $L\left(t\right)=E\left[R\right\{Y-h(\mathit{U},t){\}}^2]$, ${\hat{L}}_n\left(t\right)=N^{-1}\sum _{i=1}^N{R}_i\{Y_i-\hat{h}({\mathit{U}}_i,t){\}}^2$, and $L_n\left(t\right)=N^{-1}\sum _{i=1}^N{R}_i\{Y_i-h({\mathit{U}}_i,t){\}}^2.$ Taking derivatives with respect to t, we obtain $\begin{array}{rl}{\mathrm{\nabla }}_t{\hat{L}}_n\left(t\right)& =\frac{1}{N}\sum _{i=1}^N-2R_i\{Y_i-\hat{h}({\mathit{U}}_i,t\left)\right\}{\mathrm{\nabla }}_t\hat{h}({\mathit{U}}_i,t)=\frac{1}{N}\sum _{i=1}^N\psi \{Y_i,R_i,{\hat{\mathit{\omega }}}_t({\mathit{U}}_i\left)\right\},\\ {\mathrm{\nabla }}_t{L}_n\left(t\right)& =\frac{1}{N}\sum _{i=1}^N-2R_i\{Y_i-h({\mathit{U}}_i,t\left)\right\}{\mathrm{\nabla }}_{t}h({\mathit{U}}_i,t)=\frac{1}{N}\sum _{i=1}^N\psi \{Y_i,R_i,{\mathit{\omega }}_t({\mathit{U}}_i\left)\right\},\end{array}$and ${\mathrm{\nabla }}_{t}L\left(t\right)=-2E\left[R\right\{Y-h(\mathit{u},t)\left\}{\mathrm{\nabla }}_{t}h\right(\mathit{u},t\left)\right]=\mathrm{E}\left[\psi \right\{Y,R,{\mathit{\omega }}_t\left(\mathit{U}\right)\left\}\right],$where ψ is given in (D5). Note that ${\mathrm{\nabla }}_{t}L\left(\gamma \right)=0$ and ${\mathrm{\nabla }}_t^{2}L\left(\gamma \right)=2E\left[\right\{{\mathrm{\nabla }}_{t}h(\mathit{U},\gamma ){\}}^{2}R]={\nu }_{\gamma }\ge 0$. We establish the asymptotic normality of $\hat{\gamma }$ in the following four steps.

Step 1: Since γ is the unique minimizer of $L\left(t\right)$, from Theorem 2.1 in Newey and McFadden (1994), it suffices to prove that $\underset{t\in \mathrm{\Gamma }}{sup}|{\mathrm{\nabla }}_t{\hat{L}}_n\left(t\right)-{\mathrm{\nabla }}_{t}L\left(t\right)|⟹p0.$ Note that $\begin{array}{rl}\underset{t\in \mathrm{\Gamma }}{sup}|{\mathrm{\nabla }}_t{\hat{L}}_n\left(t\right)-{\mathrm{\nabla }}_{t}L\left(t\right)|& \le \underset{t\in \mathrm{\Gamma }}{sup}|{\mathrm{\nabla }}_t{L}_n\left(t\right)-{\mathrm{\nabla }}_{t}L\left(t\right)|+\underset{t\in \mathrm{\Gamma }}{sup}|{\mathrm{\nabla }}_t{\hat{L}}_n\left(t\right)-{\mathrm{\nabla }}_t{L}_n\left(t\right)|\\ & \le \underset{t\in \mathrm{\Gamma }}{sup}|{\mathrm{\nabla }}_t{L}_n\left(t\right)-{\mathrm{\nabla }}_{t}L\left(t\right)|\\ & +\frac{2}{n}\sum _{i=1}^n{R}_i|Y_i|\left\{\underset{t\in \mathrm{\Gamma },\mathit{x}\in \mathcal{U}}{sup}|{\mathrm{\nabla }}_t\hat{h}\right(\mathit{u},t)-{\mathrm{\nabla }}_{t}h(\mathit{u},t)|\\ & +\underset{t\in \mathrm{\Gamma },\mathit{u}\in \mathcal{U}}{sup}|\hat{h}(\mathit{u},t){\mathrm{\nabla }}_t\hat{h}(\mathit{u},t)-h(\mathit{u},t){\mathrm{\nabla }}_{t}h(\mathit{u},t)|\}\end{array}$From (D3), $|2R\{Y-h(\mathit{u},t\left)\right\}{\mathrm{\nabla }}_{t}h(\mathit{U},t)|$ is bounded by $c|Y|$ for a constant c and hence Lemma 2.4 in Newey and McFadden (1994) implies that $\underset{t\in \mathrm{\Gamma }}{sup}|{\mathrm{\nabla }}_t{L}_n\left(t\right)-{\mathrm{\nabla }}_{t}L\left(t\right)|=o_p\left(1\right).$ Based on Lemma B.3 in Newey (1994), conditions (D1)–(D4) imply that $\underset{\mathit{u}\in \mathcal{U}}{sup}|{\hat{\mathit{\omega }}}_t\left(\mathit{u}\right)-{\mathit{\omega }}_t\left(\mathit{u}\right)|\to 0$ for all $t\in \mathrm{\Gamma }$. As a result, by a similar argument of the proof of Lemma B.3 in Newey (1994), we obtain that $\underset{t\in \mathrm{\Gamma },\mathit{u}\in \mathcal{U}}{sup}|{\hat{\mathit{\omega }}}_t\left(\mathit{u}\right)-{\mathit{\omega }}_t\left(\mathit{u}\right)|⟹p0.$ Since ${\mathit{\omega }}_t$ is bounded away from zero and $h(\cdot ,t)$ and ${\mathrm{\nabla }}_{t}h(\cdot ,t)$ are Lipschitz continuous functions with respect to ${\mathit{\omega }}_t$, $\underset{t\in \mathrm{\Gamma },\mathit{u}\in \mathcal{U}}{sup}|\hat{h}(\mathit{u},t)-h(\mathit{u},t)|⟹p0$ and $\underset{t\in \mathrm{\Gamma },\mathit{u}\in \mathcal{U}}{sup}|{\mathrm{\nabla }}_t\hat{h}(\mathit{u},t)-{\mathrm{\nabla }}_{t}h(\mathit{u},t)|⟹p0.$ These results together with the previous inequality implies that $\hat{\gamma }⟹p\gamma .$

Step 2: Conditions (D1)–(D5) ensure that Lemma 8.11 in Newey and McFadden (1994) holds and hence $\sqrt{N}{\mathrm{\nabla }}_t{\hat{L}}_n\left(\gamma \right)⟹dN(0,{\sigma }_L^2)$ with ${\sigma }_L^2=\mathrm{Var}\left\{m\right(Y,R,\mathit{U},{\mathit{\omega }}_{\gamma })+\tau (Y,R,\mathit{U},\gamma \left)\right\}$.

Step 3: Note that ${\mathrm{\nabla }}_t^2{L}_n\left(t\right)=N^{-1}\sum _{i=1}^N-2R_i\{Y_i-h({\mathit{U}}_i,t\left)\right\}{\mathrm{\nabla }}_t^{2}h({\mathit{U}}_i,t)+2R_i\left\{{\mathrm{\nabla }}_{t}h\right({\mathit{U}}_i,t){\}}^2$ and $\underset{|t-\gamma |\le |\hat{\gamma }-\gamma |}{sup}|{\mathrm{\nabla }}_t^2{\hat{L}}_n\left(t\right)-{\mathrm{\nabla }}_t^{2}L\left(\gamma \right)|\le A_1+A_2+A_3,$ where $A_1=|{\mathrm{\nabla }}_t^2{L}_n\left(\gamma \right)-{\mathrm{\nabla }}_t^{2}L\left(\gamma \right)|$, $A_2=\underset{t\in \mathrm{\Gamma }}{sup}|{\mathrm{\nabla }}_t^2{\hat{L}}_n\left(t\right)-{\mathrm{\nabla }}_t^2{L}_n\left(t\right)|$, and the last term $A_3=\underset{|t-\gamma |\le |\hat{\gamma }-\gamma |}{sup}|{\mathrm{\nabla }}_t^2{L}_n\left(t\right)-{\mathrm{\nabla }}_t^2{L}_n\left(\gamma \right)|.$ The law of large numbers guarantees that $A_1=o_p\left(1\right)$. A similar argument in Step 1 shows that $A_2=o_p\left(1\right)$. For $A_3$, we have $\begin{array}{rl}|{\mathrm{\nabla }}_t^2{L}_n\left(t\right)-{\mathrm{\nabla }}_t^2{L}_n\left(\gamma \right)|& \le \frac{2}{N}\sum _{i=1}^N|\left\{{\mathrm{\nabla }}_{t}h\right({\mathit{U}}_i,t){\}}^2-\{{\mathrm{\nabla }}_{t}h({\mathit{U}}_i,\gamma ){\}}^2|\\ & +\frac{2}{N}\sum _{i=1}^N|Y_i||{\mathrm{\nabla }}_t^{2}h({\mathit{U}}_i,t)-{\mathrm{\nabla }}_t^{2}h({\mathit{U}}_i,\gamma )|\\ & +\frac{2}{N}\sum _{i=1}^N|h({\mathit{U}}_i,t){\mathrm{\nabla }}_t^{2}h({\mathit{U}}_i,t)-h({\mathit{U}}_i,\gamma ){\mathrm{\nabla }}_t^{2}h({\mathit{U}}_i,\gamma )|.\end{array}$Under (D3), $h(\cdot ,t)$, $\mathrm{\nabla h}(\cdot ,t)$, and $\mathrm{\nabla h}(\cdot ,t)$ converge uniformly for all $\mathit{x}$ as $t\to \gamma$ and, thus, the $A_3=o_p\left(1\right)$ because $\hat{\gamma }⟹p\gamma$. This shows that $\underset{|t-\gamma |\le |\hat{\gamma }-\gamma |}{sup}|{\mathrm{\nabla }}_t^2{\hat{L}}_n\left(t\right)-{\mathrm{\nabla }}_t^{2}L\left(\gamma \right)|⟹p0.$

Step 4: By Taylor's expansion, ${\hat{L}}_n\left(\hat{\gamma }\right)-{\hat{L}}_n\left(\gamma \right)=0-{\hat{L}}_n\left(\gamma \right)={\mathrm{\nabla }}_t{\hat{L}}_n\left(\xi \right)(\hat{\gamma }-\gamma )$ for some $\xi \in (\gamma ,\hat{\gamma })$. From the results in Steps 1-3, $\sqrt{N}(\hat{\gamma }-\gamma )⟹dN(0,[2E\left\{R{\mathrm{\nabla }}_{\gamma }h\right(\mathit{U},\gamma \left){\}}^2{]}^{-1}{\sigma }_L^2\right).$ This completes the proof of (20(20) $\sqrt{N}(\hat{\gamma }-\gamma )⟹dN(0,{\sigma }_{\gamma }^2),$(20) ).

Proof of Corollary 4.1.

Proof of Corollary 4.1.

1. From Theorem 4.3, (20(20) $\sqrt{N}(\hat{\gamma }-\gamma )⟹dN(0,{\sigma }_{\gamma }^2),$(20) ) shows that $\hat{\gamma }-\gamma =O_p\left(1/\sqrt{N}\right)$. Furthermore, Lemma 8.10 in Newey and McFadden (1994) shows that (A4) $\underset{i}{max}|\frac{{\mathrm{e}}^{-\gamma Y_i}\sum _{j=1}^n{\mathrm{e}}^{\gamma Y_j}{\check{\kappa }}_{\check{b}}({\mathit{U}}_i-{\mathit{U}}_j)}{\sum _{j=1}^n{\check{\kappa }}_{\check{b}}({\mathit{U}}_i-{\mathit{U}}_j)}-\frac{f(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{f(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}|=O_p(\sqrt{\frac{\log N}{N{\check{b}}^p}}+{\check{b}}^d),$(A4) which is $o_p\left(N^{-2/(p+4)}\right)=o_p\left(1\right)/\sqrt{n{b}^p}$ under the assumed condition $d>max\left\{\right(p+4)/2,p\}$ and $N{\check{b}}^{2d}\to 0$. Since $\hat{\gamma }-\gamma$ converges faster than (A4(A4) $\underset{i}{max}|\frac{{\mathrm{e}}^{-\gamma Y_i}\sum _{j=1}^n{\mathrm{e}}^{\gamma Y_j}{\check{\kappa }}_{\check{b}}({\mathit{U}}_i-{\mathit{U}}_j)}{\sum _{j=1}^n{\check{\kappa }}_{\check{b}}({\mathit{U}}_i-{\mathit{U}}_j)}-\frac{f(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{f(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}|=O_p(\sqrt{\frac{\log N}{N{\check{b}}^p}}+{\check{b}}^d),$(A4) ), (18(18) $\underset{i=n+1,\dots ,N}{max}|\frac{\hat{f}(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{\hat{f}(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}-\frac{f(Y_i|\mathit{U}={\mathit{U}}_i,D=1)}{f(Y_i|\mathit{U}={\mathit{U}}_i,D=0)}|=\frac{o_p\left(1\right)}{\sqrt{n{b}^p}}$(18) ) holds. As a result, (17(17) $\sqrt{n{b}^p}\left\{{\hat{\mu }}_1^{E2}\right(\mathit{u})-{\mu }_1(\mathit{u}\left)\right\}⟹dN\left(B_a\right(\mathit{u}),V_a(\mathit{u}\left)\right),$(17) ) holds with ${\mu }_1^{E2}\left(\mathit{u}\right)$ replaced by ${\mu }_1^{E3}\left(\mathit{u}\right)$ under (B1)–(B4) and (D1)–(D5).

2. Under (D1)–(D5) with $\mathit{U}$ replaced by $\mathit{X}$ and p replaced by q, Lemma 8.10 in Newey and McFadden (1994) implies that $\underset{\mathit{x}\in \mathbb{X}}{sup}|\hat{h}(\mathit{x},\gamma )-h_1\left(\mathit{x}\right)|=O_p\left(\right(\log N{)}^{1/2}(N{\check{b}}^q{)}^{-1/2}+{\check{b}}^d)=o_p\left(n^{-2/(p+4)}\right).$From the asymptotic normality of $\hat{\gamma }$, $\hat{\gamma }-\gamma =O_p\left(1/\sqrt{N}\right)$, which converges to 0 faster than $\underset{\mathit{x}\in \mathcal{X}}{sup}|\hat{h}(\mathit{x},\gamma )-h_1\left(\mathit{x}\right)|\to 0$. Hence (A3(A3) $max|{\hat{h}}_1\left({\mathit{X}}_i\right)-h_1\left({\mathit{X}}_1\right)|=O_p\left(\sqrt{\log \left(n\right)}{n}^{-2/(q+4)}\right),$(A3) ) holds while ${\hat{h}}_1$ is estimated by $\hat{h}(\mathit{X},\hat{\gamma })$. Then, the rest of proof of the second claims follows the argument in the proof of Theorem 4.2.