Least favorability of the uniform distribution for tests of the concavity of a distribution function

TL;DR

This paper demonstrates that for tests of distribution function concavity based on the L^p-norm, the uniform distribution presents the most challenging case, with the limiting distribution stochastically dominating others.

Contribution

It establishes the uniform distribution as the least favorable case for a class of concavity tests based on the L^p-norm of empirical distribution differences.

Findings

Uniform distribution is least favorable for the test.

Limiting distribution under uniformity stochastically dominates others.

Results inform the robustness of concavity testing procedures.

Abstract

A test of the concavity of a distribution function with support contained in the unit interval may be based on a statistic constructed from the $L^p$-norm of the difference between an empirical distribution function and its least concave majorant. It is shown here that the uniform distribution is least favorable for such a test, in the sense that the limiting distribution of the statistic obtained under uniformity stochastically dominates the limiting distribution obtained under any other concave distribution function.