Abstract
Shape constraints yield flexible middle grounds between fully nonparametric and fully parametric approaches to modeling distributions of data. The specific assumption of log-concavity is motivated by applications across economics, survival modeling, and reliability theory. However, there do not currently exist valid tests for whether the underlying density of given data is log-concave. The recent universal inference methodology provides a valid test. The universal test relies on maximum likelihood estimation (MLE), and efficient methods already exist for finding the log-concave MLE. This yields the first test of log-concavity that is provably valid in finite samples in any dimension, for which we also establish asymptotic consistency results. Empirically, we find that a random projections approach that converts the d-dimensional testing problem into many one-dimensional problems can yield high power, leading to a simple procedure that is statistically and computationally efficient.
Supplementary Materials
Appendix: The appendix contains proofs of all theoretical results (Appendix A), additional simulations and visualizations for the two-component normal mixture setting (Appendix B), discussions on the relative power of full-dimensional and projection tests (Appendix C), simulations to test log-concavity when data arise from a Beta distribution (Appendix D), and additional details on the permutation test and trace test for log-concavity (Appendix E). (pdf file)
R code: The R code to reproduce the simulations and figures is available at https://github.com/RobinMDunn/LogConcaveUniv.
Acknowledgments
This work made extensive use of the R statistical software (R Core Team Citation2021), as well as the clustermq (Schubert Citation2019), data.table (Dowle and Srinivasan Citation2021), fitdistrplus (Delignette-Muller and Dutang Citation2015), kde1d (Nagler and Vatter Citation2020), ks (Duong Citation2021), LogConcDEAD (Cule, Gramacy, and Samworth Citation2009), logcondens (Dümbgen and Rufibach Citation2011), MASS (Venables and Ripley Citation2002), mclust (Scrucca et al. Citation2016), mvtnorm (Genz et al. Citation2021; Genz and Bretz Citation2009), and tidyverse (Wickham et al. Citation2019) packages.
Disclosure Statement
The authors report that there are no competing interests to declare.