Abstract
We consider i.i.d. random variables \(X_1,\ldots , X_n\) with a distribution function F preceding the exponential distribution function V in the convex transform order which means that F has an increasing failure rate. We determine sharp upper bounds on the expectations of order statistics and spacings based on \(X_1,\ldots , X_n\), expressed in the population standard deviation units. We also specify the distributions for which all these bounds are attained. Finally, we indicate some reliability applications.
Highlights
Consider an i.i.d. sample X1, . . . , Xn based on the common cumulative distribution function F with the finite expectation μ = EX1 = F−1(x)d x, A
Let X1:n ≤ · · · ≤ Xn:n denote the order statistics based on X1, . . . , Xn
Goroncy and Rychlik (2015) provided general tools for obtaining sharp upper bounds on the expectations of single order statistics and spacings expressed in terms of the population mean and standard deviation, for the families of all parent distributions preceding various W in the convex transform order
Summary
Rychlik (1998) and Goroncy (2009) established the optimal positive and nonpositive upper bounds on the expectations of L-statistics coming from arbitrary distributions, respectively. Goroncy and Rychlik (2015) provided general tools for obtaining sharp upper bounds on the expectations of single order statistics and spacings expressed in terms of the population mean and standard deviation, for the families of all parent distributions preceding various W in the convex transform order. They characterized the distributions which attain the bounds, and specified the general results for the distributions with increasing density functions. It occurs that in the particular problems we consider below, there is only one α satisfying (1.9), and there is no need for comparing norms of different Pαh
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