A weighted Weibull detection model for line transect sampling: application on wooden stake perpendicular distance data

Abstract

The line transect survey method is commonly used to estimate the population size. However, recent developments in this field have tended to prefer practical mathematical models for this purpose. In this study, a new model called the weighted Weibull detection model was introduced to specifically address certain criteria related to line transect data. This study thoroughly explored the characteristics of this new model, including its shapes, moments, and probability density functions. The maximum likelihood estimation and Bayesian estimation methods were employed to ensure accurate parameter estimation. To gauge the performance of this model, population size estimates were generated and compared with various existing parametric estimation methods that are commonly used in the field. Through simulations, the resulting estimates were assessed and compared to widely adopted approaches for estimating population size. Additionally, this mathematical model was applied to real-world data involving perpendicular distances, allowing for a direct comparison of its performance against both traditional and contemporary methods using various measures of goodness of fit. Moreover, the study calculates statistical metrics, such as the variance–covariance matrix, parameter confidence intervals, and estimated population size, were obtained using the proposed detection model. These metrics provide valuable insights into the precision and uncertainty associated with estimated parameters and population size estimates. Confidence intervals offer a range of plausible parameter values, whereas the variance–covariance matrix quantifies the relationships and uncertainties between the estimated parameters.

Keywords:

• Weibull distribution
• line transect sampling
• shoulder condition

Reviewing Editor:

• Zude Zhou, Wuhan University of Technology, China

Subjects:

• Applied Mathematics; Statistics & Probability; Material Science

1. Introduction

In place of plot or strip sample techniques, the line transect approach has been employed over the past 50 years to determine population density or abundance. It can take a lot of time to establish a plot and then count every object of interest in it. It can also be difficult to define plots in some habitats, such as those with fast-moving species or at sea. In ecology and wildlife biology, the line transect method is widely used for determining the density or abundance of a specific species in a given area. Using this method, a straight line or transect is drawn through the research area, and the number of individuals or signs of the target species that are seen along that line are counted and recorded in a methodical manner. For instance, see Burnham et al. (1980). A retrospective study of this topic is proposed below, with the main references provided.

• The critical component for estimating population abundance is the detection function, which can be evaluated using nonparametric, semiparametric, or parametric methods.

• Gates et al. (1968) introduced a function characterized by an exponential distribution and a single scaling parameter, while Hemingway (1971) developed a function based on the half-normal distribution with a shoulder function as its foundation.

• Parameter $f(0),$ representing the probability density function (PDF) at distance 0, it also plays a central role in these functions. Consequently, under the shoulder condition, the population abundance, often denoted as D in the literature, was calculated using the various approaches described in the following sections.

• We will cover the parametric estimation methods which are proposed by Burnham and Anderson (1976), Pollock (1978), Ramsey (1979), Karunamuni and Quinn (1995), Buckland (1985), Eberhardt (1978), Eidous (2004), Ababneh and Eidous (2012), Quinn and Gallucci (1980), Eidous and Al-Eibood (2018), Ameeq et al. (2023), Muneeb Hassan et al. (2023), Naz et al. (2023), and Hassan et al. (2023). Al-Hussaini (1991), Fewster et al. (2008), Laake et al. (2008), Porteus et al. (2011), Rodriguez (1977), Schmidt et al. (2012), and Naz et al. (2023).

• Buckland and Turnock (1992) employed primary and secondary observation platforms in a whale survey to challenge the assumption that every whale can be perfectly detected. Mark–recapture distance sampling (MRDS) was developed by Amstrup et al. (2001) represent to estimate the population of polar bears (Ursus maritimus) in northern Alaska. Their methodology also considers population estimation variables such as group size.

• Quang and Becker (1997) introduced an MRDS model using a stratified Lincoln–Petersen estimate and data obtained from a line transect. Becker and Quang (2009) opted for an iteratively reweighted least-squares approach instead of maximum likelihood when fitting the model, and they incorporated a logistic detection technique that considered contour transect factors. They also utilized the Horvitz–Thompson estimate (Horvitz & Thompson, 1952) to determine the size of the brown bear population, sometimes referred to as grizzly bears.

• To estimate the population of harbor porpoises, Borchers et al. (1998) developed an MRDS model with various variables and relied on the Horvitz–Thompson estimate. In their data collection efforts related to harbor porpoises, Amstrup et al. (2001), Becker and Quang (2009), and Borchers et al. (1998) utilized survey designs that involved two observers on a single aerial survey platform.

In order to estimate the unidentified population abundance, represented by D, an observer walks a distance L that is divided into K randomly spaced transects that cover the target region of interest. During the survey, the observer records the number of objects encountered and measures the perpendicular distance x from the centerline to each object’s location. Let’s assume there are n identified objects, and their distances from each other are denoted as $X_1,X_2,\dots ,X_n.$ These distances are used to estimate the population density. The detection function $g(x)$ quantifies the conditional probability of observing an object at a specific perpendicular distance x from the transect line. It provides information about the likelihood of detecting an object at different distances. The detection function $g(x)$ represents the probability of detecting an object given its perpendicular distance x from the line transect. It can be mathematically defined as: $$g(x)=\mathrm{\text{Pr}}(\text{detecting an object}|\text{perpendicular distance}=x).$$

In order to assess the population abundance D of unidentified items in a research area, an observer walks a distance L divided into K randomly distributed transects using the line transect sampling method. The observer records the perpendicular distance x from the centerline to each object found, and the detection function $g(x)$ represents the probability of detecting an object at distance x from the line transect. If an object is found at a perpendicular distance x, there exists a PDF $f(x)$ that has the same shape as $g(x)$ but scaled. For $x\ge 0,$ this PDF can be written as $f(x)=\mathit{\text{kg}}(x),$ where $k=1/{\int }_0^{\infty }g(t)\mathit{\text{dt}}$ is the scaling constant. Since not all objects in a given region are detected using the line transect technique, detection probabilities can be estimated. The likelihood of identifying an object increases with proximity to the center of the transect line. Mathematically, if $x_1$ and $x_2$ are observed perpendicular distances such that $x_1>x_2,$ then $g(x_2)\ge g(x_1).$Considering that the probability of spotting an object at zero distance is one ($g(0)=1$), the population density can be estimated using $\widehat{D}=\frac{n\widehat{f}(0)}{2L},$ where n is the number of identified objects and $\widehat{f}(0)$ is an approximate sample estimate of $f(0).$ The equation $\widehat{N}=A\widehat{D}$ can be used to determine the projected population size N in the target area of area A. Indicating that detection is still certain or virtually definite at a close proximity to the line transect’s center, the detection function $g(x)$ is anticipated to come from a shoulder ($g^{\prime }(0)=0$).

Various references provide in-depth discussions on line transect sampling, including Marques et al. (2001), Barabesi (2000), Eidous (2015), Eidous and Al-Salman (2016), Jang and Loh (2010), Seber (1982), Pollock (1978), Strindberg and Buckland (2004), Eberhardt (1978), Quang and Becker (1997), Drummer and McDonald (1987), Burnham and Anderson (1984), Routledge and Fyfe (1992), Southwell et al. (2008), Brockelman and Srikosamatara (1993), Chen (1996), Melville and Welsh (2001), Anderson et al. (2001), and Naz, Al-Essa, et al. (2023). Line transect sampling can underestimate the true PDF for known-sized populations, as observed in kangaroo populations by Southwell (1994). This underestimation may occur when animals on the centerline become disturbed and move before being observed, violating a fundamental assumption. Mark–recapture experiments and line transect estimation can be combined to directly test the detectability.

The article is organized as follows. The proposed detection function is discussed in Section 2, which also shows that it complies with all requirements for a shoulder function. This section also includes plots of various detection functions for different parametric values. The characteristics of the density function related to the suggested detection function are presented in Section 3, and the rth moments, which can be used to determine the mean and variance, are specifically explored. In Section 4 of the referenced article, the population abundance and PDF at a distance of 0 are calculated using the maximum likelihood method. The simulation is used in Section 5 to evaluate the performance of the calculated parameters. The simulation was performed using Mathematica 10, and plots and graphs were used to provide visual insight. Section 6 describes the application of the proposed function to practical perpendicular distance datasets. The authors demonstrated how the model performs by computing the related measurements for these datasets. Finally, Section 7 concludes the article. This study combines mathematical analysis, simulation, and practical data application to present and evaluate a new detection function based on the Weibull distribution.

2. The suggested detection method

The one-parameter model $g(x)$ exhibits a lack of flexibility, which may result in a rigid shape that does not adequately capture the detection curve. Consequently, the model estimator lacked robustness. However, we propose a weighted Weibull model (WWM) with two parameters and a user-supplied detection function to solve this problem. As a result, the adaptability of WWM was higher than that of the one-parameter model. Therefore, the robustness of the estimator and the ability of the model to conform to the shape of the detection curve can be improved by adding more parameters. Additionally, customization is made possible by the user-supplied detection function, which enables the model to consider particular aspects of the detection process. By combining the Weibull distribution and the detection function, WWM provides a more accurate representation of the detection curve. The Weibull distribution is a commonly used model for survival data, and we modified it by multiplying it by a detection function. This integration ensures that the detection curve aligns more closely with the observed data, resulting in a more reliable estimation. $$g(x;\lambda ,\gamma )=(2-e^{-\lambda x^{\gamma }})e^{-\lambda x^{\gamma }},\hspace{1em}x>0,\hspace{1em}\lambda ,\gamma >0.$$

By utilizing the line transect method, estimate the population density D. Any reliable detection function $g(x;\eta ),$ $\eta =(\lambda ,\gamma )$ should satisfy three main assumptions (Burnham et al. 1980). Those assumptions are justified for the proposed $g(x,\eta )$ as follows:

1. $g(0,\eta )=1,$ that is the objects with perpendicular distance 0 will never be missed.

2. $g^{\prime }(0,\eta )=0,$ where $g^{\prime }(0,\eta )$ is the first derivative of $g(x,\eta )$ at $x=0,$ i.e. the shoulder function of the detection function ought to be close to the origin (Burnham et al., 1980). Specifically, this means that within a relatively small distance of the line transect, the detection probability should continue to be nearly certain.

3. $g^{\prime }(x,\eta )<0$ for all $x>0,$ this shows that while the detection function monotonically decreases in x, the likelihood of detecting an object decreases as its perpendicular distance x increases.

The likelihood of detecting an object given its perpendicular distance on the line transect is one, hence this detection function states that the probability of detection on the line transect center is assured. The detection function’s first derivative with respect to x is, $$g^{\prime }(x;\lambda ,\gamma )=-2\lambda \gamma x^{\gamma -1}{e}^{-2\lambda x^{\gamma }}(1-e^{-\lambda x^{\gamma }}).$$ which gives results $g^{\prime }(0;\lambda ,\gamma )=0$ for $\gamma >1$ and the $g(x;\lambda ,\gamma )$ is uniformly monotonically declining $x>0.$ Various shapes confirm this shoulder condition once more (see Figure 1). The PDF related to $g(x;\lambda ,\beta )$ is possible by averaging the detecting function as $f(x;\eta )=\frac{1}{\mu }g(x;\eta ),$ where $\mu ={\int }_0^{\infty }g(x;\eta )\mathit{\text{dx}}.$ Then the PDF is (2.1) $$f(x;\lambda ,\gamma )=\frac{2^{1/\gamma }{\lambda }^{1/\gamma }}{2^{(1+1/\gamma )}-1\Gamma (1+1/\gamma )}(2-e^{-\lambda x^{\gamma }})e^{-\lambda x^{\gamma }},$$(2.1)

Figure 1. Diagrams displaying the detection function for various parameter values.

In Figure 2, plots of the PDF are given for various parametric values.

Figure 2. Diagrams displaying the PDF for different parametric values.

The shoulder condition assumptions are satisfied since $f(x;\eta )$ shows similar behavior to $g(x;\eta )$ but scaled to be a PDF, and it is monotonically decreasing in x. Also, $f(x,\eta )$ has the same shape as $g(x;\eta )$ with different scales.

Since $g(0;\eta )=1$ for $\lambda >1,$ the parameter $f(0;\eta )$ is (2.2) $$f(0;\eta )=\frac{2^{1/\gamma }{\lambda }^{1/\gamma }}{(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}$$(2.2)

Then, from Equation (2.2)(2.2) $$f(0;\eta )=\frac{2^{1/\gamma }{\lambda }^{1/\gamma }}{(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}$$(2.2) , $\lambda ={\left[\frac{f(0;\eta )(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}{2^{1/\gamma }}\right]}^{\gamma }$ replacing this value with $\lambda$ in Equation (2.1)(2.1) $$f(x;\lambda ,\gamma )=\frac{2^{1/\gamma }{\lambda }^{1/\gamma }}{2^{(1+1/\gamma )}-1\Gamma (1+1/\gamma )}(2-e^{-\lambda x^{\gamma }})e^{-\lambda x^{\gamma }},$$(2.1) , Then, the PDF of the WWM can be stated as $f(0;\eta ),$ such that (2.3) $$\begin{array}{cc}f(x;\lambda ,\gamma )=\hfill & f(0;\eta )\left[2‐\mathrm{\text{exp}}\{-{\left[\frac{f(0;\eta )(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}{2^{1/\gamma }}\right]}^{\gamma }{x}^{\gamma }\}\right]\hfill \end{array}$$(2.3) (2.4) $$\mathrm{\text{exp}}\{-{\left[\frac{f(0;\eta )(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}{2^{1/\gamma }}\right]}^{\gamma }{x}^{\gamma }\}$$(2.4)

This is determined by the two parameters $\gamma$ and $f(0;\eta ).$ The estimation of the parameter’s maximum likelihood will be based on this formula.

With regard to x, the first derivative of the PDF (2.1) is $$f^{\prime }(x;\lambda ,\gamma )=-\frac{2^{1/\gamma +1}\gamma {\lambda }^{1/\gamma +1}{x}^{\gamma -1}{e}^{-2\lambda x^{\gamma }}(e^{\lambda x^{\gamma }}-1)}{(2^{1/\gamma +1}-1)\Gamma (1/\gamma +1)}$$

It is clear from statement $f(x;\eta )=\frac{g(x;\eta )}{{\int }_0^{\infty }g(x;\eta )\mathit{\text{dx}}}$ that $g(x;\eta )$ is immediately connected to $f(x;\eta ).$ As a result, $f(x;\eta )$ and $g(x;\eta )$ share some characteristics, including $f^{\prime }(0;\eta )=0$ and the monotone decreasing shape property. Figures 1 and 2 exhibit these features. The proposed model would then provide a reliable estimate for $f(0;\eta ),$ known as the Shape Criterion (see Burnham et al., 1980). The detection function, $g(x;\eta ),$ which adjusts $f(x;\eta )$ and has parameters $\lambda$ for scale and $\gamma$ for shape, can have a wide range of shapes, as can be seen when carefully examining these pictures. A desirable property for the detection model under consideration is that all plots steadily fade to 0 as $x\to \infty .$ An integrated estimator of $f(0;\eta )=f(0;\lambda ,\gamma )$ corresponds to $\widehat{\lambda }$ and $\widehat{\gamma }$ if they are the estimators of the parameters $\lambda$ and $\gamma ,$ respectively. (2.5) $$f(0;\eta )=f(0;\widehat{\lambda },\widehat{\gamma })=\frac{2^{1/\widehat{\gamma }}{\widehat{\lambda }}^{1/\widehat{\gamma }}}{(2^{(1+1/\widehat{\gamma })}-1)\Gamma (1+1/\widehat{\gamma })}$$(2.5) where the estimate for population density (abundance) $\widehat{D}$ is $$\widehat{D}=\frac{n{2}^{1/\widehat{\gamma }}{\widehat{\lambda }}^{1/\widehat{\gamma }}}{2L(2^{(1+1/\widehat{\gamma })}-1)\Gamma (1+1/\widehat{\gamma })}.$$

To calculate the values of the parameters $\lambda$ and $\gamma ,$ we shall employ the maximum likelihood method. Thus, we calculate $f(0;\lambda ,\gamma )$ and D represents the population abundance. Section 4 will go into further detail.

3. A few statistical characteristics

In this section, we seek immediate outcomes that will aid in the calculation of parameter $f(0;\lambda ;\gamma )$ and, ultimately, population abundance. These instant results offer insightful statistical data regarding the distribution of random variables. These moments can be used to estimate the population abundance parameter, $f(0;\lambda ;\gamma ).$ By calculating and analyzing various moments, such as the mean, variance, and higher-order moments, we gain insight into the central tendency, dispersion, and shape of the distribution. These moment results aid in understanding the underlying characteristics of the population and enable estimation and inference about the parameter of interest. The specific calculations of moments for the PDF given in Equation (2.1)(2.1) $$f(x;\lambda ,\gamma )=\frac{2^{1/\gamma }{\lambda }^{1/\gamma }}{2^{(1+1/\gamma )}-1\Gamma (1+1/\gamma )}(2-e^{-\lambda x^{\gamma }})e^{-\lambda x^{\gamma }},$$(2.1) would require the functional form of $f(x;\lambda ,\gamma ).$ Using this information, the integrals can be solved to obtain the moment results for a given random variable.: $${\mu }_r^{\prime }=E(X^r)={\int }_0^{\infty } x^r \frac{2^{1/\gamma }{\lambda }^{1/\gamma }}{(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}(2-e^{-\lambda x^{\gamma }})e^{-\lambda x^{\gamma }}\mathit{\text{dx}},$$ where E denotes the expectation operator. After some algebra, we have (3.1) $${\mu }_r^{\prime }=E(X^r)=\frac{(2^{\frac{1+r+\gamma }{\gamma }}-1)\Gamma \left(\frac{1+r}{\gamma }\right)}{2^r/\gamma {\lambda }^{r/\gamma }(2^{1/\gamma +1}-1)\Gamma 1/\gamma }.$$(3.1)

By using ${\mu }_1^{\prime }$ and ${\mu }_2^{\prime }$ to determine the mean and variance, it can be calculated that (3.2) $$\mu ={\mu }_1^{\prime }=E(X)=\frac{2^{1/\gamma -1}(2^{\frac{2+\gamma }{2}}-1)\Gamma (1/2+1/\gamma )}{{\lambda }^{1/\gamma }(2^{1/\gamma +1}-1)\sqrt{\pi }},$$(3.2) (3.3) $${\mu }_2^{\prime }=E(X^2)=\frac{(2^{\frac{3+\gamma }{\gamma }}-1)\Gamma (3/\gamma )}{4^{1/\gamma }{\lambda }^{2/\gamma }(2^{1/\gamma +1}-1)\Gamma (1/\gamma )},$$(3.3) and $${\sigma }^2=\mathit{\text{Var}}(X)=\frac{(2^{\frac{3+\gamma }{\gamma }}-1)\Gamma (3/\gamma )}{4^{1/\gamma }{\lambda }^{2/\gamma }(2^{1/\gamma +1}-1)\Gamma (1/\gamma )}-{\left[\frac{2^{1/\gamma -1}(2^{\frac{2+\gamma }{2}}-1)\Gamma (1/2+1/\gamma )}{{\lambda }^{1/\gamma }(2^{1/\gamma +1}-1)\sqrt{\pi }}\right]}^2.$$

Based on Equation (3.1)(3.1) $${\mu }_r^{\prime }=E(X^r)=\frac{(2^{\frac{1+r+\gamma }{\gamma }}-1)\Gamma \left(\frac{1+r}{\gamma }\right)}{2^r/\gamma {\lambda }^{r/\gamma }(2^{1/\gamma +1}-1)\Gamma 1/\gamma }.$$(3.1) , the skewness and kurtosis of X are calculated using the following formulas. $$\mathit{\text{Skewness}}=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }\mu +2{\mu }^3}{{\sigma }^3},\hspace{1em}\mathit{\text{Kurtosis}}=\frac{{\mu }_4^{\prime }-4{\mu }_3^{\prime }\mu +6{\mu }_2^{\prime }{\mu }^2-3{\mu }^4}{{\sigma }^4}.$$

Ghitany et al. (2013) and Wackerly et al. (2014) conducted studies investigating the relationship between the model parameters $\lambda$ and $\gamma$ and their impact on the statistical properties of the random variable X. In Table 1, various parameter options are presented alongside the corresponding mean, variance, skewness, and kurtosis values for X. It is observed that as the parameter values increase, these statistical measures decrease. The skewness and kurtosis values, however, remain constant for fixed $\gamma$ values. The 3D plots shown in Figures 3 and 4 illustrate the relationships between the mean, variance, skewness, kurtosis, and the values of $\lambda$ and $\gamma$ within the model. These plots visually depict how changes in $\lambda$ and $\gamma$ significantly affect the skewness and kurtosis characteristics of the parameter X. The results indicate that adjusting the values of $\lambda$ and $\gamma$ allows for control over and a better understanding of the skewness and kurtosis behaviors of the parameter X, in addition to its mean and variance.

Figure 3. Three-dimensional mean and variance graphs for different parametric values.

Figure 4. Skewness and kurtosis graphs in three dimensions for various parametric variables.

Figure 5. (a) $\beta$=1.0, $w=5.0$ and (b) $\beta$=1.5, $w=3.0$ RME plots for the EP model.

Table 1. For some parameter values, mean, variance, skewness, and kurtosis are present.

Parameters	$\gamma$ = 1
$\lambda$	Mean	Variance	Skewness	Kurtosis
2	0.5833	0.2847	1.7902	16.4277
4	0.2916	0.0711	1.7902	16.4277
6	0.1944	0.0316	1.7902	16.4277
8	0.1458	0.0177	1.7902	16.4277
10	0.1166	0.0113	1.7902	16.4277
12	0.0972	0.0079	1.7902	16.4277
Parameters	$\lambda$ = 2
$\gamma$	Mean	Variance	Skewness	Kurtosis
1	0.5833	0.2847	1.7902	16.4277
2	0.4628	0.1041	0.7808	28.7193
3	0.4556	0.0847	0.4451	38.4799
4	0.4585	0.0795	0.2869	44.1649
5	0.4625	0.0777	0.1999	47.5552
6	0.4663	0.0771	0.1470	49.6907

4. Inference

In this section, we focus on making statistical inferences about the parameter $f(0;\lambda ,\gamma )$ when the values of $\lambda$ and $\gamma$ are unknown. To achieve this, we employed maximum likelihood and Bayesian estimation (BE) methods, which involve determining the values of $\lambda$ and $\gamma$ that maximize the likelihood function. Consider the PDF $f(x;\lambda ,\gamma )$ connected to the suggested detection function. The observed values of a random sample are given by equation $g(x;\lambda ,\gamma ).$

4.1. Maximum likelihood estimator of $f(0;\lambda ,\gamma )$

To simulate an n-dimensional random sample of perpendicular distances, denoted as $x_1,x_2,\dots ,x_n,$ from a random variable $f(x;\lambda ,\gamma )$ defined by Equation (2.5)(2.5) $$f(0;\eta )=f(0;\widehat{\lambda },\widehat{\gamma })=\frac{2^{1/\widehat{\gamma }}{\widehat{\lambda }}^{1/\widehat{\gamma }}}{(2^{(1+1/\widehat{\gamma })}-1)\Gamma (1+1/\widehat{\gamma })}$$(2.5) , we can generate random numbers using the specified PDF with parameter values $\lambda$ and $\gamma .$ The maximum likelihood estimators (MLEs) $\widehat{\gamma }$ and $\widehat{f}(0;\lambda ,\gamma )$ for the parameters $\gamma$ and $f(0;\lambda ,\gamma ),$ respectively, may be estimated once we have the observed sample. The approach outlined in Burnham et al. (1980) can then be used to compute the MLE of $\lambda .$ (4.1) $$\begin{array}{l}\mathit{\text{lnL}}=\sum _{i=1}^n\mathit{\text{lnf}}(x_i;\lambda ,\gamma )=\mathit{\text{nln}}[f(0;\lambda ,\gamma )]+\sum _{i=1}^n\mathit{\text{ln}}\left[2‐\mathrm{\text{exp}}\{-{\left[\frac{f(0;\eta )(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}{2^{1/\gamma }}\right]}^{\gamma }{x}_i^{\gamma }\}\right]\hfill \end{array}$$(4.1) (4.2) $$-{\left[\frac{f(0;\eta )(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}{2^{1/\gamma }}\right]}^{\gamma }\sum _{i=1}^n{x}_i^{\gamma }.$$(4.2)

Differentiating with respect to $\gamma$ and $f(0;\lambda ,\gamma ),$ respectively, we get $$\frac{\partial }{\partial \gamma }\mathit{\text{lnL}}=\frac{n\Phi }{f(0;\lambda ,\gamma )}+\sum _{i=1}^n\frac{\mathrm{\text{exp}}\{-{\Psi }^{\gamma }{x}^{\gamma }\}[{\Psi }^{\gamma }{x}^{\gamma }\mathit{\text{lnx}}+x^{\gamma }{\Psi }^{\gamma }\mathit{\text{lnΨΩ}}]}{2‐\mathrm{\text{exp}}\{-{\Psi }^{\gamma }{x}^{\gamma }\}}-\sum _{i=1}^n[{\Psi }^{\gamma }{x}^{\gamma }\mathit{\text{lnx}}+x^{\gamma }{\Psi }^{\gamma }\mathit{\text{lnΨΩ}}].$$ and $$\begin{array}{ll}\frac{\partial }{\partial f(0;\lambda ,\gamma )}\mathit{\text{lnL}}=\frac{n}{f(0;\lambda ,\gamma )}+\sum _{i=1}^n\frac{\mathrm{\text{exp}}\{-{\Psi }^{\gamma }{x}^{\gamma }\}(\gamma {\Psi }^{\gamma -1}(2^{1/\gamma +1}-1)\Gamma (1/\gamma +1)x^{\gamma })}{2‐\mathrm{\text{exp}}\{-{\Psi }^{\gamma }{x}^{\gamma }\}}\hfill & \hfill \\ -\sum _{i=1}^n(\gamma {\Psi }^{\gamma -1}(2^{1/\gamma +1}-1)\Gamma (1/\gamma +1)x^{\gamma }),\hfill & \hfill \end{array}$$ where $\Phi =\frac{\partial f(0;\lambda ,\gamma )}{\partial \gamma },$ $\Psi =\left[\frac{f(0;\eta )(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}{2^{1/\gamma }}\right]$ and $\Omega =\frac{\partial \left[f(0;\eta )(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )2^{-1/\gamma }\right]}{\partial \beta }.$

The MLEs $\widehat{\gamma }$ and $\widehat{f}(0;\lambda ,\gamma )$ of $\gamma$ and $f(0;\lambda ,\gamma ),$ respectively, are the nonlinear equation’s solutions $\frac{\partial }{\partial \gamma }\mathit{\text{lnL}}=0$ and $\frac{\partial }{\partial f(0;\lambda ,\gamma )}\mathit{\text{lnL}}=0.$ They can be calculated mathematically with the use of any statistical program, such as Mathematica, Python, R, etc. The MLE of $\lambda$ is then obtained by substituting the values of $\widehat{\gamma }$ and $\widehat{f}(0;\lambda ,\gamma )$ in Equation (2.5)(2.5) $$f(0;\eta )=f(0;\widehat{\lambda },\widehat{\gamma })=\frac{2^{1/\widehat{\gamma }}{\widehat{\lambda }}^{1/\widehat{\gamma }}}{(2^{(1+1/\widehat{\gamma })}-1)\Gamma (1+1/\widehat{\gamma })}$$(2.5) , which is where we get the population abundance estimator D. We consider the observed Fisher information matrix for calculating confidence intervals for the model parameters $I=[I_{\mathit{\text{ij}}}],\mathit{\text{ij}}=a,b$ about $(\gamma ,f(0;\lambda ,\gamma ))$ given by $$\begin{array}{cc}I\hfill & =-\left(\begin{array}{cc}{I}_{\mathit{\text{aa}}}& I_{\mathit{\text{ab}}}\\ I_{\mathit{\text{ba}}}& I_{\mathit{\text{bb}}}\end{array}\right)\hfill \end{array}$$ where $I_{\mathit{\text{aa}}}=\frac{{\partial }^2\mathit{\text{lnL}}}{\partial {\gamma }^2},$ $I_{\mathit{\text{bb}}}=\frac{{\partial }^2\mathit{\text{lnL}}}{\partial {(f(0;\lambda ,\gamma ))}^2}$ and $I_{\mathit{\text{ab}}}=I_{\mathit{\text{ba}}}=\frac{{\partial }^2\mathit{\text{lnL}}}{\partial \gamma \partial f(0;\lambda ,\gamma )}.$ So, V, which is the inverse of I, is the variance–covariance matrix connected to $(\widehat{\gamma },f(0;\lambda ,\gamma )).$ On the other hand, it is simple to derive the estimate of $\lambda$ from Equation (2.5)(2.5) $$f(0;\eta )=f(0;\widehat{\lambda },\widehat{\gamma })=\frac{2^{1/\widehat{\gamma }}{\widehat{\lambda }}^{1/\widehat{\gamma }}}{(2^{(1+1/\widehat{\gamma })}-1)\Gamma (1+1/\widehat{\gamma })}$$(2.5) . Furthermore, based on $f(x;\lambda ,\gamma )$ given by Equation (2.1)(2.1) $$f(x;\lambda ,\gamma )=\frac{2^{1/\gamma }{\lambda }^{1/\gamma }}{2^{(1+1/\gamma )}-1\Gamma (1+1/\gamma )}(2-e^{-\lambda x^{\gamma }})e^{-\lambda x^{\gamma }},$$(2.1) , we may calculate the MLEs numerically and then the confidence intervals for $\gamma$ and $\lambda .$

Based on the above discussion, It is simple to calculate the approximate large sample $(1-\alpha )100\%$ confidence interval for $\gamma ,$ $\lambda$ and $f(0;\lambda ,\gamma )$ as $$\widehat{\gamma }\pm Z_{\alpha /2}\sqrt{\widehat{\mathit{\text{Var}}}(\widehat{\gamma })},\hspace{1em}\widehat{\lambda }\pm Z_{\alpha /2}\sqrt{\widehat{\mathit{\text{Var}}}(\widehat{\lambda })},\hspace{1em}\widehat{f}(0;\lambda ,\gamma )\pm Z_{\alpha /2}\sqrt{\widehat{\mathit{\text{Var}}}(\widehat{f}(0;\lambda ,\gamma )},$$ where $\widehat{\mathit{\text{Var}}}(\widehat{\gamma }),$ $\widehat{\mathit{\text{Var}}}(\widehat{\lambda })$ and $\widehat{\mathit{\text{Var}}}(\widehat{f}(0;\lambda ,\gamma )$ are the MLEs of the variances of the estimators behind $\gamma ,$ $\lambda$ and $f(0;\lambda ,\gamma ),$ respectively.

4.2. Bayesian estimator of $f(0;\lambda ,\gamma )$

This section deals with the BE of the unknown parameters $\gamma$ and $f(0;\lambda ,\gamma ),$ respectively. For the Bayesian parameter estimation, loss functions, including squared error loss (SEL) functions, can be taken into consideration by Tolba et al. (2023), Alsadat et al. (2023), Bhat et al. (2023), and Chinedu et al. (2023). We can consider applying independent gamma priors for the variables $\gamma$ and $f(0;\lambda ,\gamma )$ with PDF $f(x;\lambda ,\gamma )$ in the parameter prior detection function. (4.3) $${\pi }_1(\gamma )\propto {\gamma }^{s_1-1}{e}^{-q_1\gamma },\hspace{1em}\gamma >0,s_1>0,q_1>0,$$(4.3) $${\pi }_2(f(0;\lambda ,\gamma ))\propto f{(0;\lambda ,\gamma )}^{s_2-1}{e}^{-q_{2}f(0;\lambda ,\gamma )},\hspace{1em}f(0;\lambda ,\gamma )>0,s_2>0,q_2>0,$$ where the hyper-parameters $s_j,q_j,j=1,2$ are selected to reflect the prior knowledge about the unknown parameters. The joint prior for $\Omega =(\gamma ,f(0;\lambda ,\gamma ))$ is given by (4.4) $$\pi (\Omega )\propto {\pi }_1(\gamma ){\pi }_2(f(0;\lambda ,\gamma ))$$(4.4) $$\pi (\Omega )\propto {\gamma }^{s_1-1}f{(0;\lambda ,\gamma )}^{s_2-1}{e}^{-q_1\gamma -q_{2}f(0;\lambda ,\gamma )}.$$

The corresponding posterior density, given the observed data $x=(x_1,x_2,\dots ,x_n),$ is given by (4.5) $$\pi (\Omega |x)=\frac{\pi (\Omega )\mathcal{l}(\Omega )}{{\int }_{\gamma }{\int }_{f(0;\lambda ,\gamma )}\pi (\Omega )\mathcal{l}(\Omega )\mathit{\text{dγdf}}(0;\lambda ,\gamma )}.$$(4.5)

Consequently, the posterior density function is denoted by (4.6) $$\begin{array}{l}\pi (\Omega |x)\propto {\gamma }^{s_1-1}f{(0;\lambda ,\gamma )}^{s_2-1}{e}^{-q_1\gamma -q_{2}f(0;\lambda ,\gamma )}\\ \times \sum _{i=1}^{n}f(0;\eta )\left[2-\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{-{\left[\frac{f(0;\eta )(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}{2^{1/\gamma }}\right]}^{\gamma }{x}^{\gamma }\right\}\right]\\ \times \hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{-{\left[\frac{f(0;\eta )(2^{(1+1/\gamma )}-1)\Gamma (1+1/\gamma )}{2^{1/\gamma }}\right]}^{\gamma }{x}^{\gamma }\right\}.\end{array}$$(4.6)

Given any function, such as $l(\Omega )$ under the SEL function, the Bayes estimator is given by (4.7) $${\widehat{\Omega }}_{\mathit{\text{BEsel}}}=E[l(\Omega )|x]={\int }_{\Omega }l(\Omega )\pi (\Omega /x)d\Omega .$$(4.7)

They can be calculated mathematically with the use of any statistical program, such as Mathematica, Python, R, etc. The Bayesian estimator of $\lambda$ is then obtained by substituting the values of $\widehat{\gamma }$ and $\widehat{f}(0;\lambda ,\gamma )$ in Equation (2.5)(2.5) $$f(0;\eta )=f(0;\widehat{\lambda },\widehat{\gamma })=\frac{2^{1/\widehat{\gamma }}{\widehat{\lambda }}^{1/\widehat{\gamma }}}{(2^{(1+1/\widehat{\gamma })}-1)\Gamma (1+1/\widehat{\gamma })}$$(2.5) , which is where we get the population abundance estimator D.

5. Simulation research and findings

To assess how well the WWM estimate, $\widehat{f}\mathit{\text{ML}}(0),$ performs in comparison with other available estimates for the function $f(0;\lambda ,\gamma ),$ we perform a simulation exercise. We consider the estimates from the negative exponential model (NEM), half-normal model (HNM), and weighted exponential model (WEM) (see Saeed, 2013). We evaluated two different target detection functions, coupled with observations of $X_1,X_2,\dots ,X_n,$ with the sample sizes of $n=50,$ $n=100,$ and $n=200,$ to duplicate the perpendicular distances. These identification strategies were selected to highlight the range of shapes that might be present in a specific field (see Eidous, 2015). The preferred models for replicating the perpendicular distances are as follows:

Detection of exponential power (EP) (see Pollock, 1978): (5.1) $$f(x)=\frac{e^{-x^{\beta }}}{\Gamma \left(1+\frac{1}{\beta }\right)},\hspace{1em}x>0,\beta >1,$$(5.1) where $\Gamma (x)$ is the standard gamma function defined as $\Gamma (x)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}\mathit{\text{dt}}.$

The beta exponential (BE) detection function is described by Eberhardt (1978): (5.2) $$g(x)=(1+\beta ){(1-x)}^{\beta },\hspace{1em}0<x<1,\beta >0.$$(5.2)

Each model is shortened at specific distances w in order to simulate the data. In contrast to the BE detection function, which is truncated at $w=0.5$ and $w=1,$ the EP detection function is truncated at $w=5,3,2.5,$ and 2. A number of arbitrary $\beta$ values are selected for both models. We consider sample sizes of $n=50,100,200$ and generate 1500 samples of perpendicular distances for each model randomly. Tables 2 and 3 present the relative bias (RB) and the relative mean square error (RME) for the different estimators. Based on these results, we draw the following conclusions:

Table 2. When data from the EP model are simulated, RB and RME are used for the various estimations.

			Estimate
n	$\beta$	W	${\widehat{f}}_{\mathit{\text{ML}}}(0)$		${\widehat{f}}_{1,\mathit{\text{ML}}}(0)$		${\widehat{f}}_{2,\mathit{\text{ML}}}(0)$		${\widehat{f}}_{3,\mathit{\text{ML}}}(0)$
			RB	RME	RB	RME	RB	RME	RB	RME
50	1.0	5.0	−0.1751	0.1401	0.0520	0.1413	0.3751	0.3842	0.1825	0.2134
100	1.0	5.0	−0.1773	0.1101	0.0534	0.1132	0.3785	0.3822	0.1811	0.1972
200	1.0	5.0	−0.1794	0.0422	0.0437	0.0703	0.3847	0.3861	0.1804	0.1962
300	1.0	5.0	−0.1717	0.0413	0.0335	0.0416	0.3863	0.3828	0.1793	0.1920
500	1.0	5.0	−0.1757	0.0323	0.0231	0.0356	0.3843	0.3839	0.1678	0.1860
50	1.5	3.0	−0.1132	0.1291	0.4021	0.4355	0.1332	0.1640	0.0907	0.1578
100	1.5	3.0	−0.1172	0.1281	0.3872	0.4144	0.1378	0.1517	0.0872	0.1251
200	1.5	3.0	−0.1194	0.1261	0.3790	0.3782	0.1421	0.1506	0.0813	0.1046
300	1.5	3.0	−0.1112	0.1246	0.3661	0.3612	0.1325	0.1428	0.0801	0.1005
500	1.5	3.0	−0.1482	0.1317	0.3613	0.3501	0.1211	0.1406	0.0753	0.0241
50	2.0	2.5	−0.1001	0.1041	0.6016	0.6251	0.0232	0.1095	0.2451	0.2790
100	2.0	2.5	−0.1021	0.0632	0.5931	0.6080	0.0171	0.0782	0.2390	0.2601
200	2.0	2.5	−0.1104	0.0401	0.5752	0.5814	0.0052	0.0501	0.2253	0.2342
300	2.0	2.5	−0.1154	0.0210	0.5423	0.5565	0.0026	0.0401	0.2204	0.2123
500	2.0	2.5	−0.1196	0.0142	0.5351	0.5205	0.0013	0.0204	0.2125	0.2103
50	2.5	2.0	−0.0742	0.0746	0.7043	0.7241	0.1165	0.1527	0.3253	0.3512
100	2.5	2.0	−0.0786	0.0732	0.7975	0.7084	0.1288	0.1297	0.3202	0.3340
200	2.5	2.0	−0.0791	0.0712	0.7961	0.7024	0.1261	0.1183	0.3196	0.3272
300	2.5	2.0	−0.0843	0.0681	0.7920	0.7001	0.1220	0.1121	0.3153	0.3216
500	2.5	2.0	−0.0785	0.0630	0.7903	0.6929	0.1200	0.1090	0.3121	0.3173

Table 3. When data are simulating from the BE model, RB and RME for the various estimations.

			Estimate
n	$\beta$	W	${\widehat{f}}_{\mathit{\text{ML}}}(0)$		${\widehat{f}}_{1,\mathit{\text{ML}}}(0)$		${\widehat{f}}_{2,\mathit{\text{ML}}}(0)$		${\widehat{f}}_{3,\mathit{\text{ML}}}(0)$
			RB	RME	RB	RME	RB	RME	RB	RME
50	1.0	0.5	0.0601	0.0912	0.5094	0.5551	0.1913	0.2231	0.2844	0.2560
100	1.0	0.5	0.0521	0.0833	0.4882	0.5184	0.1890	0.2146	0.2431	0.2212
200	1.0	0.5	0.0341	0.0672	0.4710	0.5011	0.1825	0.1801	0.2003	0.2012
300	1.0	0.5	0.0280	0.0451	0.3962	0.5001	0.1732	0.1744	0.1901	0.1902
500	1.0	0.5	0.0092	0.0371	0.3612	0.4926	0.1384	0.1673	0.1801	0.1842
50	1.5	1.0	0.1724	0.1892	0.4110	0.4853	0.1791	0.2025	0.1682	0.2001
100	1.5	1.0	0.1532	0.1670	0.4322	0.4161	0.1472	0.1892	0.1361	0.1866
200	1.5	1.0	0.1413	0.1301	0.3236	0.3913	0.1386	0.1657	0.1172	0.1708
300	1.5	1.0	0.1061	0.1286	0.3202	0.3478	0.1291	0.1424	0.1163	0.1632
500	1.5	1.0	0.1001	0.1123	0.3172	0.3117	0.1031	0.1291	0.1012	0.1587
50	2.0	1.5	0.1251	0.1413	0.3991	0.4320	0.1985	0.2253	0.2101	0.1830
100	2.0	1.5	0.1074	0.1172	0.2846	0.3312	0.1626	0.2113	0.2061	0.1612
200	2.0	1.5	0.0665	0.0753	0.2312	0.2830	0.1357	0.1892	0.1946	0.1542
300	2.0	1.5	0.0517	0.0734	0.2196	0.2655	0.1224	0.1690	0.1912	0.1316
500	2.0	1.5	0.0473	0.0513	0.2167	0.2508	0.1115	0.1523	0.1820	0.1091
50	2.5	2.0	0.1410	0.1675	0.2513	0.3196	0.2498	0.2755	0.1994	0.2182
100	2.5	2.0	0.1302	0.1531	0.2432	0.2936	0.2001	0.2396	0.1701	0.1982
200	2.5	2.0	0.0921	0.1026	0.2312	0.2236	0.1927	0.2278	0.1583	0.1667
300	2.5	2.0	0.0802	0.1017	0.2272	0.2097	0.1812	0.2186	0.1423	0.1528
500	2.5	2.0	0.0801	0.0926	0.2213	0.2008	0.1723	0.2166	0.1382	0.1457

Table 4. Information regarding wooden stakes parallel distances in meters.

2.02	0.45	10.40	3.61	0.92	1.00	3.40	2.90	8.16	6.47	5.66
2.95	3.96	0.09	11.82	14.23	2.44	1.61	6.50	8.27	4.85	1.47
18.60	0.41	0.40	0.20	11.59	3.17	7.10	10.71	3.86	6.05	6.42
3.79	15.24	3.47	3.05	7.93	18.15	10.05	4.41	1.27	13.72	6.25
3.59	9.04	7.68	4.89	9.10	3.25	8.49	6.08	0.40	9.33	0.53
1.23	1.67	4.53	3.12	3.05	6.60	4.40	4.97	3.17	7.67	18.16
4.08

1. As the sample size n increases, the RB and RME of all estimators decrease. This suggests that the estimators are reliable as the sample size grows.

2. The suggested estimate ${\widehat{f}}_{\mathit{\text{ML}}}(0)$ exhibits lesser RB and RME compared to the NEM, HNM, and WEM estimators for all studied target detection functions, which shows that the proposed estimator outperforms the competing models, as shown in Tables 2 and 3.

3. Figures 5–8 depict the RME for $\widehat{f}\mathit{\text{ML}}(0),$ $\widehat{f}1,\mathit{\text{ML}}(0),$ $\widehat{f}2,\mathit{\text{ML}}(0),$ and $\widehat{f}3,\mathit{\text{ML}}(0),$ respectively.

Figure 6. (c) $\beta$=2.0, $w=2.5$ and (d) $\beta$=2.5, $w=2.0$ RME plots for the EP model.

Figure 7. (a) $\beta$=1.0, $w=0.5$ and (b) $\beta$=1.5, $w=1.0$ RME plots for the BE model.

Figure 8. (c) $\beta$=2.0, $w=1.5$ and (d) $\beta$=2.5, $w=2.0$ RME plots for the BE model.

6. Model compatibility and real-world data application

The primary subjects of discussion in this section are model selection and validation. However, choosing these models is the most difficult step in creating a good model. It was chosen after carefully weighing all the available information. The chosen model must be flexible enough to consider the trade-off between model complexity and ease of assessment while accurately reproducing the provided data. Additionally, it takes a lot of effort to simulate behaviors for extreme values of the relevant variable. In this situation, a variety of visualizations and goodness-of-fit checks are required as part of the statistical process step for validating the model.

6.1. Alternative models

Using some goodness-of-fit analyses, the performance of the proposed detection model is compared to a variety of other current detection models. For comparison, the following detection models are utilized:

1. ‘New two-parameter detection model (NDM)’ (Bakouch et al., 2022): $$g(x;\gamma ,\lambda )=(1+\lambda x^{\gamma })e^{-\lambda x^{\gamma }},$$ $$f(x;\gamma ,\lambda )=\frac{{\gamma }^2{\lambda }^{1/\gamma }}{(\gamma +1)\Gamma (1/\gamma )}(1+\lambda x^{\gamma })e^{-\lambda x^{\gamma }},\hspace{1em}\gamma ,\lambda >0.$$

2. ‘Model (2013)’ (Saeed, 2013): $$g(x;\gamma ,\lambda )={(2-e^{-\lambda x})}^{\gamma }{e}^{-\lambda x},$$ $$f(x;\gamma ,\lambda )=\frac{\lambda (1+\gamma )}{2^{\gamma +1}-1}{(2-e^{-\lambda x})}^{\gamma }{e}^{-\lambda x},\hspace{1em}\gamma ,\lambda >0.$$

3. ‘Generalized exponential model (GEM)’ (Quinn & Gallucci, 1980): $$g(x;\gamma ,\lambda )=e^{(1/\gamma ){(x/\lambda )}^{\gamma }},$$ $$f(x;\gamma ,\lambda )=\frac{e^{(1/\gamma ){(x/\lambda )}^{\gamma }}}{\lambda {\gamma }^{1/\gamma }\Gamma (1+1/\gamma )},\hspace{1em}\gamma ,\lambda >0.$$

4. ‘Exponential power series model (EPSM)’ (Pollock, 1978): $$g(x;\gamma ,\lambda )=e^{{(x/\lambda )}^{\gamma }},$$ $$f(x;\gamma ,\lambda )=\frac{e^{{(x/\lambda )}^{\gamma }}}{\lambda \Gamma (1+1/\gamma )},\hspace{1em}\gamma ,\lambda \ge 0.$$

5. ‘Reverse logistic model (RLM)’ (Eberhardt, 1978): $$g(x;\gamma ,\lambda )=\frac{(1+\lambda )e^{-\gamma x}}{1+\lambda e^{-\gamma x}},$$ $$f(x;\gamma ,\lambda )=\frac{\gamma \lambda (1+\lambda )e^{-\gamma x}}{(1+\lambda )(1+\lambda e^{-\gamma x})\mathrm{\text{log}}(1+\lambda )},\hspace{1em}\gamma ,\lambda \ge 0.$$

6. ‘Exponential quadratic model (EQM)’ (Burnham et al., 1980): $$g(x;\gamma ,\lambda )=e^{(-\gamma x-\lambda x^2)},$$ $$f(x;\gamma ,\lambda )=\frac{2\lambda e^{(-\gamma x-\lambda x^2)}}{e^{-{\gamma }^2/4\lambda }\sqrt{\pi }e\mathit{\text{rfc}}(\gamma /2\sqrt{\lambda })},\hspace{1em}\gamma ,\lambda \ge 0.$$

7. ‘Weighted half-normal model (WHNM)’ (Eidous & Al-Salman, 2016): $$g(x;\lambda )=(2-e^{-x^2/2\lambda })e^{-x^2/2\lambda },$$ $$f(x;\lambda )=\frac{2}{2\sqrt{2}-1}(2-e^{-x^2/2\lambda })e^{-x^2/2\lambda },\hspace{1em}\lambda >0.$$

8. ‘Model (2015)’ (Eidous, 2015): $$g(x;\lambda )={(1-0.5e^{-\lambda x/2})}^{2}4e^{-\lambda x},$$ $$f(x;\lambda )=\frac{24\lambda }{11}{(1-0.5e^{-\lambda x/2})}^2{e}^{-\lambda x},\hspace{1em}\lambda >0.$$

9. ‘Weighted exponential model (WEM)’ (Ababneh & Eidous, 2012): $$g(x;\lambda )=(2-e^{-\lambda x})e^{-\lambda x},$$ $$f(x;\lambda )=\frac{2\lambda }{3}(2-e^{-\lambda x})e^{-\lambda x},\hspace{1em}\lambda \ge 0.$$

10. ‘Negative exponential model (NEM)’ (Gates et al., 1968): $$g(x;\lambda )=e^{-\lambda x},$$ $$f(x;\lambda )=\lambda e^{-\lambda x},\hspace{1em}\lambda \ge 0.$$

To compare the MLEs with the data being discussed, we consider the pertinent PDF of each detection model. The effectiveness of the detection model is strongly correlated with its PDF, as shown previously.

6.2. Perpendicular distance datasets

Dataset 1: Information about wooden stakes parallel distances in meters

The dataset comprises perpendicular distances between a sagebrush meadow and wooden stakes. These distances were obtained by walking a 1000 m path and selecting a sample of 67 stakes from a total population of 150 stakes. The true density of the stakes is defined as $D=0.00375$ stakes/meter. The recorded distances, denoted as $x_1,x_2,\dots ,x_n,$ represent the perpendicular distances associated with the selected stakes along the single 1000 m path. The estimated value of $f(0),$ given the sample size n and stake density D, is determined as 0.11029 (refer to Karunamuni & Quinn, 1995; Zhang, 2001 for more details). In addition, the zone of interest’s dimensions, notably the sagebrush meadow east, are calculated as $A=40,000m^2$ using the equation $D=N/A,$ where $N=150$ (see Karunamuni & Quinn, 1995; Zhang, 2001). Burnham et al. (1980) and Barabesi (2000) have both reviewed and reported on the data in great detail. The perpendicular distance value of 31.31 in the accompanying table significantly deviates from the transect path in contrast to other values, as is evident when looking at the wooden stake dataset. The accuracy of the estimation will therefore be increased by removing this outlier (as suggested by Zhang, 2001). As a result, the truncation point is removed.

Table 5 presents the important statistics of the wooden stakes dataset, including the relevant theoretical metrics of the WWM (Wooden Stakes Analysis). Additionally, the accompanying box plot, shown in Figure 9, provides further visual assistance. It is worth noting that the box plots display the minimum score of a dataset, first quartile (lower quartile), median, third quartile (upper quartile), and maximum score as part of the five-number summary.

Figure 9. Box plot and TTT plot for wooden stakes dataset.

Table 5. The wooden stakes dataset’s descriptive statistics and related theoretical WWM metrics.

Sample size (n)	Mean	Median	Standard deviation	Skewness	Kurtosis	$\frac{\mathit{\text{Skewness}}}{\mathit{\text{Kurtosis}}}$
67	5.73201	4.41242	4.55640	1.10830	3.80333	0.29148
67	5.73370	4.47211	4.53042	1.31830	3.60312	0.36582

Table 6. The Hemmingway’s data perpendicular distance data in meters.

0	0	0	0	0	0	0	0	8.72	10.5
22.3	26.0	26.0	30.5	30.5	31.7	34.2	35.1	38.0	41.0
42.1	50.8	55.1	58.5	63.6	64.3	65.0	68.8	71.1	71.8
71.9	72.1	73.1	76.6	77.6	78.1	84.0	84.5	86.0	86.0
87.0	90.0	92.3	94.0	96.4	96.4	106.0	115.0	123.0	123.0
129.0	129.0	143.0	143.0	150.0	151.0	153.0	157.0	161.0	164.0
164.0	164.0	166.0	175.0	188.0	193.0	200.0	200.0	246.0	260.0
272.0	378.0	400.0

Dataset 2: The Hemingway’s perpendicular distance data

The second dataset in Table 6 comes from Hemmingway’s research on several ungulates in Africa (see Burnham et al., 1980). Seventy-three animals were found along a single-line transect that traveled for 60,000 m. Using the sighting method, the perpendicular distances are calculated. Sighted angles and distances. Hemmingway’s dataset does not include the true density D or the area A of the discovered items (not recorded during the survey). However, the TRANSECT software, which adapts the estimate of $f(0)$ and D, which are, respectively, 0.0065 and 0.0396 animal per hectare (or 3.096106 animal per meter), creates the estimation technique. Table 7 presents the important statistics of Hemmingway’s dataset, including the relevant theoretical metrics of the WWM. Additionally, the accompanying box plot is shown in Figure 10.

Figure 10. Box plot and TTT plot for Hemmingway’s dataset.

Table 7. Hemmingway’s dataset descriptive statistics and related theoretical WWM metrics.

Sample size (n)	Mean	Median	Standard deviation	Skewness	Kurtosis	$\frac{\mathit{\text{Skewness}}}{\mathit{\text{Kurtosis}}}$
73	99.2414	84.0012	82.3732	1.36056	5.47687	0.24841
73	99.9404	84.3516	82.3506	1.34128	4.83453	0.27742

The following common criteria are applied in order to compare the fit performance of the models that were taken into consideration for the study of Datasets 1 and 2: the Kolmogorov–Smirnov test p value, the Bayesian information criterion (BIC), the Hannan–Quinn information criterion (HQIC), the Akaike information criterion (AIC), and the corrected AIC (CAIC). Their values are shown in Tables 8–11. It is evident that our model has the lowest information criterion values and the greatest p value for the Kolmogorov–Smirnov test. We can therefore conclude that the suggested model works well in this situation.

Table 8. The MLEs, W*, A*, statistics K–S and p value for wooden stakes dataset.

Detection models	$\widehat{\gamma }$	$\widehat{\lambda }$	$A^{*}$	$W^{*}$	K–S	p Value
WWM	1.3756	0.0694	0.2471	0.0361	0.0819	0.7596
NDM	1.2845	0.1166	0.2641	0.2672	0.0840	0.7314
Model (2013)	2.1208	0.23506	0.2731	0.0385	0.0861	0.7019
GEM	1.7377	6.9115	0.2721	0.0420	0.0885	0.6705
EPSM	1.7377	9.4989	0.2721	0.0420	0.0885	0.6700
RLM	0.2838	6.7168	0.2596	0.0408	0.08907	0.6623
EQM	0.0072	0.0398	0.2792	0.0441	0.0909	0.6361
WHNM	–	42.698	0.8711	0.1512	0.1041	0.4610
Model (2015)	–	0.2369	0.3772	0.0702	0.1136	0.3520
WEM	–	0.2045	0.5024	0.0979	0.1261	0.2366
NEM	–	0.1747	1.0226	0.2065	0.1588	0.0680

Table 9. Information criteria and log-likelihood $(\mathcal{l})$ for wooden stakes dataset.

Detection models	−$\mathcal{l}$	AIC	AICC	BIC	HQIC	CAIC
WWM	181.703	367.008	367.195	371.017	369.053	367.095
NDM	181.662	367.325	367.512	371.734	369.069	367.512
Model (2013)	182.030	368.06	368.248	372.470	369.805	368.248
GEM	181.652	367.303	367.491	371.713	369.048	367.491
EPSM	181.652	367.303	367.491	371.713	369.048	367.491
RLM	181.768	367.536	367.723	371.945	369.281	367.723
EQM	181.656	367.311	367.499	371.720	369.056	367.499
WHNM	181.752	367.503	367.591	371.813	369.058	367.511
Model (2015)	182.165	367.330	367.392	371.535	369.203	367.392
WEM	182.526	367.052	367.213	371.156	369.924	367.113
NEM	183.862	369.723	369.785	371.928	370.596	369.785

Table 10. The MLEs, W*, A*, statistics K–S and p value for Hemmingway’s dataset.

Detection models	$\widehat{\gamma }$	$\widehat{\lambda }$	$A^{*}$	$W^{*}$	K–S	p Value
WWM	1.3181	0.0018	0.0120	0.0850	0.1095	0.3445
NDM	1.2265	0.0041	0.0864	0.0869	0.1095	0.3435
Model (2013)	1.5614	0.0125	0.0981	0.1014	0.1098	0.3131
GEM	1.3125	5.9110	0.0272	0.0420	0.1885	0.2708
EPSM	1.5905	15.965	0.1272	0.0969	0.1095	0.3444
RLM	6.0001	0.0160	0.2626	0.0886	0.1099	0.3440
EQM	0.0072	0.0398	0.2825	0.1068	0.1195	0.2498
WHNM	–	42.698	0.9167	0.1964	0.1223	0.2244
Model (2015)	–	0.0117	0.0772	0.0886	0.1156	0.3012
WEM	–	0.0117	0.0986	0.0886	0.1495	0.0912
NEM	–	0.0100	1.7216	0.2932	0.1443	0.0952

Table 11. Information criteria and log-likelihood $(\mathcal{l})$ for Hemmingway’s dataset.

Detection models	−$\mathcal{l}$	AIC	AICC	BIC	HQIC	CAIC
WWM	406.941	817.883	818.054	822.463	819.708	818.054
NDM	406.980	817.959	818.131	822.449	819.785	818.131
Model (2013)	407.572	819.145	819.316	823.725	820.970	819.316
GEM	407.501	818.303	819.491	822.713	819.748	818.491
EPSM	407.056	818.113	818.285	822.694	819.939	818.285
RLM	407.759	817.919	818.691	822.911	819.945	819.691
EQM	407.009	818.019	818.190	822.600	819.844	818.190
WHNM	408.842	819.685	819.741	822.975	820.598	819.741
Model (2015)	408.291	818.583	818.639	822.873	819.996	818.639
WEM	407.698	817.897	818.954	829.688	819.310	819.454
NEM	408.621	819.299	821.533	822.556	819.299	822.913

The CIs of the PDF at 0 based on the MLEs are obtained in Tables 12 and 13 (with appropriate truncation for the bottom bound of the intervals) based on Datasets 1 and 2 and the proposed model.

Table 12. For a wooden stake, the confidence intervals.

CI	$\gamma$	$\lambda$	$f(0)$
$95\%$	[0.2851, 2.4665]	[0, 1.1603]	[0, 1.2036]
$99\%$	[0, 2.8116]	[0, 1.5054]	[0, 1.5487]

Table 13. For Hemmingway’s dataset, the confidence intervals.

CI	$\gamma$	$\lambda$	$f(0)$
$95\%$	[0, 20.214]	[0, 18.898]	[0, 18.903]
$99\%$	[0, 26.152]	[0, 24.836]	[0, 24.827]

The MLEs of all the models were used to produce the estimated population abundance, denoted as $\widehat{D},$ and the maximum likelihood estimate of the PDF at 0, $\widehat{f}\mathit{\text{ML}}(0),$ for the analyzed data. The theoretical standard deviation of the model is represented by SD, but the standard deviation of the sample is represented by $\widehat{\mathit{\text{SD}}}.$ Tables 14 and 15 display the findings for $\widehat{f}\mathit{\text{ML}}(0)$ and $|\widehat{\mathit{\text{SD}}}-\mathit{\text{SD}}|.$ According to the tables, the WWM exhibits the lowest absolute difference, $|\widehat{\mathit{\text{SD}}}-\mathit{\text{SD}}|,$ indicating a closer match between the estimated and theoretical standard deviations. Furthermore, among the examined detection models, the WWM yields $\widehat{f}\mathit{\text{ML}}(0)$ and $\widehat{D}$ values that closely resemble the genuine values of $f(0)$ and D for the considered data.

Table 14. Estimated population abundance D and $f(0;\lambda ,\gamma )$ for the wooden stakes dataset using the $|\widehat{\mathit{\text{SD}}}-\mathit{\text{SD}}|.$

Detection models	${\widehat{f}}_{\mathit{\text{ML}}}(0)$	$(\widehat{D})$	$|\widehat{\mathit{\text{SD}}}-\mathit{\text{SD}}|$
WWM	0.11271	0.00370	0.02603
NDM	0.11402	0.00384	0.02907
Model (2013)	0.09504	0.00322	0.26611
GEM	0.11820	0.00393	0.03210
EPSM	0.11804	0.00397	0.03206
RLM	0.12006	0.00401	0.03640
EQM	0.12202	0.00401	0.03410
WHNM	0.09401	0.00320	0.33963
Model (2015)	0.12911	0.00430	0.41671
WEM	0.13600	0.00457	0.66913
NEM	0.17401	0.00580	1.17652

Table 15. Estimated population abundance D and $f(0;\lambda ,\gamma )$ for the Hemmingway’s dataset using the $|\widehat{\mathit{\text{SD}}}-\mathit{\text{SD}}|.$

Detection models	${\widehat{f}}_{\mathit{\text{ML}}}(0)$	$(\widehat{D})$	$|\widehat{\mathit{\text{SD}}}-\mathit{\text{SD}}|$
WWM	0.00659	0.00024	0.02201
NDM	0.00657	0.00024	0.02906
Model (2013)	0.00656	0.00023	0.26911
GEM	0.11820	0.00393	0.03120
EPSM	0.00710	0.00025	0.03761
RLM	0.00700	0.00025	0.03649
EQM	0.00764	0.00027	0.03643
WHNM	0.00530	0.00019	0.32190
Model (2015)	0.02550	0.00090	0.31672
WEM	0.00781	0.00028	0.64321
NEM	0.01007	0.00036	1.18136

7. Concluding remarks

This study introduced and examined a novel two-parameter detection model. The proposed model is beneficial for line transect data. This model ensures that it meets the shoulder requirement and that its perpendicular distance monotonically decreases. The proposed WWM is adaptable to suit line transect data because it depends only on two parameters. However, statistical techniques to estimate these parameters, such as MLEs, do not exist in closed form. Consequently, in this context, a numerical method, such as the Newton–Raphson method, is required. According to the simulation results from this study, the estimators obtained under the suggested model are promising for estimating population abundance using line transect approaches. Favorable properties of the proposed estimator, which were obtained using the suggested model, are shown in the simulation results. The performance of our model in comparison with other models shows how the fit can be improved using a valuable collection, such as a table of perpendicular distances and some information on goodness of fit. In addition, investigating the inclusion of a scale parameter in the detection function is a potential extension; however, more work needs to be done on the definition of the normalization constant and the suggested approach needs to be modified.

Disclosure statement

No potential conflict of interest was reported by the authors.