IID Samples from Shape Restricted Families

This chapter is devoted to evaluations of the expectations of order statistics and L-statistics based on iid samples whose parent distribution functions satisfy some shape restrictions. We mainly focus on the distributions with monotone density, failure rate and reversed failure rate functions. The positive upper bounds presented here were determined with use of the method of projection in Hilbert spaces (see Sect. 2.4.2 ), and they are expressed in the standard deviation units of the baseline distribution. The negative ones were established by means of the norm maximization method (see Sect. 2.4.7 ). They are trivially equal to zero except for ones measured in the mean absolute deviation from the mean units. In Sect. 6.1 we consider samples with decreasing density and failure rate and reversed failure rate functions. We formulate general results for arbitrary L- statistics, and specify them for single order statistics and spacings. In Sect. 6.2, we present bounds for the expectations of order statistics from symmetric unimodal populations. Section 6.3 contains analogs of the results of Sect. 6.1 for order statistics from the populations with increasing density, failure rate and reversed failure rate functions. Parent distributions with monotone density and failure rate on the average functions are treated in Sect. 6.4. Theoretical results of Sects. 6.1–6.4 are illustrated by some numerical evaluations in Sect. 6.5. Section 6.6 contains bounds on the expectations of L-statistics based on iid samples with distributions from restricted families expressed in quantiles of selected orders of the parent distributions.