Testing monotonicity of a hazard: asymptotic distribution theory

TL;DR

This paper introduces two new statistical tests for assessing whether a hazard function is increasing over a specified interval, using empirical L1-type distances and establishing their asymptotic normality under certain conditions.

Contribution

The paper develops novel test statistics based on empirical L1 distances between isotonic and empirical estimates, with proven asymptotic normality under mild assumptions.

Findings

Test statistics are asymptotically normal when the hazard is strictly increasing.

The tests measure local non-monotonicity through excursions of empirical estimates.

The asymptotic distribution is derived using a process involving Brownian motion and convex minorants.

Abstract

Two new test statistics are introduced to test the null hypotheses that the sampling distribution has an increasing hazard rate on a specified interval [0,a]. These statistics are empirical L_1-type distances between the isotonic estimates, which use the monotonicity constraint, and either the empirical distribution function or the empirical cumulative hazard. They measure the excursions of the empirical estimates with respect to the isotonic estimates, due to local non-monotonicity. Asymptotic normality of the test statistics, if the hazard is strictly increasing on [0,a], is established under mild conditions. This is done by first approximating the global empirical distance by an distance with respect to the underlying distribution function. The resulting integral is treated as sum of increasingly many local integrals to which a CLT can be applied. The behavior of the local integrals is determined by a canonical process: the difference between the stochastic process x -> W(x)+x^2 where W is standard two-sided Brownian Motion, and its greatest convex minorant.