Deterministic Bounds

In Sect. 3.1 we define order statistics \(X_{i:n}\) , \(i=1,\ldots ,n\) , (n stands here for the sample size) and their linear combinations \(\sum _{i=1}^n c_i X_{i:n}\) , called shortly L-statistics. We describe particular properties of L-statistics useful in statistical inference, and some popular examples of L-statistics. Generally, this chapter is devoted to evaluations of L-statistics with fixed deterministic values \(x_{1:n}, \ldots , x_{n:n}\) of order statistics. In Sect. 3.2 we present simple sharp upper bounds on the values of L-statistics with nondecreasing sequences \(c_1,\ldots ,c_n\) of coefficients, represented in sample standard deviation units. The best reference here is David (Commun Stat Theory Methods 17:2119–2134, 1988). Respective nonnegative upper bounds for arbitrary L-statistics are presented in Sect. 3.3. The results are generalized in Sect. 3.4 to more general scale units constructed on the basis of sample central absolute moments of various orders. These results were described in Rychlik (Commun Stat Theory Methods 22:1053–1068, 1993). Section 3.5 contains sharp upper negative bounds for negative L-statistics, and analogous lower positive bounds for positive L-statistics. The main contribution to the topic is given in Goroncy and Rychlik (Math Inequalities Appl 9:633–647, 2006a). Section 3.6 contains generalizations of the results of Sect. 3.4. They consist in replacing scale units based on sample moments by more general ones.