Testing monotonicity of a hazard: Asymptotic distribution theory

Abstract

Two 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, owing 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 a distance with respect to the underlying distribution function. The resulting integral is treated as sum of increasingly many local integrals to which a central limit theorem can be applied. The behavior of the local integrals is determined by a canonical process, the difference between the stochastic process $x\mapsto W(x)+x^{2}$, where $W$ is standard two-sided Brownian motion, and its greatest convex minorant.