Optimal bounds on expectations of order statistics and spacings from nonparametric families of distributions generated by convex transform order

Abstract

Assume that \(X_1,\ldots , X_n\) are i.i.d. random variables with a common distribution function \(F\) which precedes a fixed distribution function \(W\) in the convex transform order. In particular, if \(W\) is either uniform or exponential distribution function, then \(F\) has increasing density and failure rate, respectively. We present sharp upper bounds on the expectations of single order statistics and spacings based on \(X_1,\ldots , X_n\), expressed in terms of the population mean and standard deviation, for the family of all parent distributions preceding \(W\) in the convex transform order. We also characterize the distributions which attain the bounds, and specify the general results for the distributions with increasing density function.