Testing convexity of the generalised hazard function

Let F, G be a pair of absolutely continuous cumulative distributions, where F is the distribution of interest and G is assumed to be known. The composition $$G^{-1}\circ F$$ G - 1 ∘ F , which is referred to as the generalised hazard function of F with respect to G, provides a flexible framework for statistical inference of F under shape restrictions, determined by G, which enables the generalisation of some well-known models, such as the increasing hazard rate family. This paper is concerned with the problem of testing the null hypothesis $${\mathscr {H}}_0$$ H 0 : “ $$G^{-1}\circ F$$ G - 1 ∘ F is convex”. The test statistic is based on the distance between the empirical distribution function and a corresponding isotonic estimator, which is denoted as the greatest relatively-convex minorant of the empirical distribution with respect to G. Under $${\mathscr {H}}_0$$ H 0 , this estimator converges uniformly to F, giving rise to a rather simple and general procedure for deriving families of consistent tests, without any support restriction. As an application, a goodness-of-fit test for the increasing hazard rate family is provided.