Nonparametric two-sample tests for increasing convex order

TL;DR

This paper introduces two bootstrap-based nonparametric tests for comparing two non-negative distributions in increasing convex order, demonstrating their consistency and asymptotic size accuracy.

Contribution

It proposes novel tests based on empirical stop-loss transforms for the increasing convex order hypothesis, including size switching adjustments for bootstrap resampling.

Findings

Tests are consistent against all alternatives.

Tests are asymptotically of the specified size.

Comparison of test behavior inside the hypothesis.

Abstract

Given two independent samples of non-negative random variables with unknown distribution functions $F$ and $G$, respectively, we introduce and discuss two tests for the hypothesis that $F$ is less than or equal to $G$ in increasing convex order. The test statistics are based on the empirical stop-loss transform, critical values are obtained by a bootstrap procedure. It turns out that for the resampling a size switching is necessary. We show that the resulting tests are consistent against all alternatives and that they are asymptotically of the given size $\alpha$. A specific feature of the problem is the behavior of the tests `inside' the hypothesis, where $F\not=G$. We also investigate and compare this aspect for the two tests.