Estimators of a distribution function with increasing failure rate average

Abstract

A nonparametric estimator G n of a distribution function F with increasing failure rate average (IFRA) is constructed. This estimator G n is so constructed that its hazard function C n is the greatest star-shaped minorant of the sample hazard function. The estimator G n is itself IFRA. Denoting the sample distribution function by F n and the sample hazard function by H n , it follows from Millar (1979) that F n is an asymptotically minimax estimator of F if F is IFRA. We prove, under suitable restrictions of F, and for any fixed λ with F( λ) < 1, that n 1 2 sup x⩽λ||G n(x)−F n(x)|| and n 1 2 sup x⩽λ||C n(x)−H n(x)|| both tend to zero in probability. That is, G n and F n are asymptotically n 1 2 - equivalent as estimators of the true distribution functions. Similar results can be obtained for a DFRA distribution.