Estimating a distribution function subject to a stochastic order restriction: a comparative study

Abstract

In this article, we compare four nonparametric estimators of a distribution function (DF), estimated under a stochastic order restriction. The estimators are compared by simulation using four criteria: (1) the estimation of cumulative DFs; (2) the estimation of quantiles; (3) the estimation of moments and other functionals; and (4) as tools for testing for stochastic order. Our simulation study shows that estimators based on the pointwise maximum-likelihood estimator (p-MLE) outperform all other estimators when the underlying distributions are 'close' to each other. The gain in efficiency may be as high as 25%. If the DFs are far apart then the p-MLE may not be the best. However, the efficiency loss using the p-MLE relative to the best estimator in each case is generally low (about 5%). We also find that the test based on the p-MLE is the most powerful in the majority of cases although the gain in power relative to other tests is generally small.

Isotonic Distributional Regression

“…In distributional regression, it may not be immediately clear what is meant by isotonicity, and the literature typically considers one ordinal covariate only (e.g. Davidov & Iliopoulos, 2012;El Barmi & Mukerjee, 2005;Hogg, 1965;Rojo & El Barmi, 2003), with a notable exception being the work of Mösching & Dümbgen (2020b), whose considerations allow for a real-valued covariate. In the general case of a partially ordered covariate space, which we consider here, it is unclear whether semi-or non-parametric techniques might be capable of handling monotonicity contraints, and suitable notions of isotonicity remain to be developed.…”

Isotonic Distributional Regression

“…In distributional regression it may not be immediately clear what is meant by isotonicity, and the literature typically considers one ordinal covariate only (e.g., Hogg, 1965;Rojo and El Barmi, 2003;El Barmi and Mukerjee, 2005;Davidov and Iliopoulos, 2012) with a notable exception being the work of Mösching and Dümbgen (2019), whose considerations allow for a real-valued covariate. In the general case of a partially ordered covariate space, which we consider here, it is unclear whether semi-or nonparametric techniques might be capable of handling monotonicity contraints, and suitable notions of isotonicity remain to be developed.…”