General, Symmetric and Life IID Samples

In this chapter we consider order statistics based on iid samples. In Sect. 4.1 we gathered basic formulae representing the distribution and density functions of order statistics and their expectations. There is also presented the variation diminishing property of the Bernstein polynomials which is very useful for analyzing variability of distribution and density functions of order statistics and their combinations. Section 4.2 contains the most classic bounds for the expecations of L-statistics with monotone vectors of coefficients. Section 4.3 contains analogous results for arbitrary L-statistics. The sharp bounds on properly centered L-statistics measured in the scale units based on central absolute population moments are studied in Sect. 4.4. Nonpositive upper bounds for negative L-statistics and nonnegative lower ones for positive L-statistics are presented in Sect. 4.5. More refined bounds for L-statistics coming from symmetric and life populations are established in Sects. 4.5 and 4.6, respectively. In the latter case, higher moments of order statistics are evaluated as well. Bounds on the expectations of L-statistics gauged in the Gini mean difference units are described in Sect. 4.8. Section 4.9 contains bounds on the expectations of L-statistics expresssed in terms of the mean residual life function. Inequalities for moments of order statistics based on the orthogonal expansions are considered in Sect. 4.10. The last section is devoted to optimal bounds on variances of order and L-statistics. A special case of symmetrically distributed iid samples is treated in particular.