Testing equality of functions under monotonicity constraints

TL;DR

This paper develops a unified statistical testing framework for assessing the equality of monotone functions across different models, including regression, density, and hazard functions, using $L_1$-based test statistics.

Contribution

It introduces a uniform approach for testing equality of monotone functions across various models with asymptotic normality and bootstrap methods for critical region determination.

Findings

Proposed two $L_1$-distance based test statistics.

Established asymptotic normality of the test statistics.

Validated bootstrap procedures for accurate critical regions.

Abstract

We consider the problem of testing equality of functions $f_j:[0,1]\to \mathbb{R}$ for $j=1,2,...,J$ the basis of $J$ independent samples from possibly different distributions under the assumption that the functions are monotone. We provide a uniform approach that covers testing equality of monotone regression curves, equality of monotone densities and equality of monotone hazards in the random censorship model. Two test statistics are proposed based on $L_1$-distances. We show that both statistics are asymptotically normal and we provide bootstrap implementations, which are shown to have critical regions with asymptotic level $\alpha$.