Probability inequalities for generalized \(L\)-statistics (Q2773618)

Let \(X_n\) be independent random variables which are uniformly distributed on \([0,1]\). The authors study statistics of the type \(\Phi_n=\sum_{i=1}^n h_{ni}(X_{n:i})\), where \(X_{n:1}\leq\cdots\leq X_{n:n}\) are the order statistics based on the sample \(\{X_i;i\leq n\}\) and \(h_{ni}\) are measurable functions. The functionals of this general form are called generalized \(L\)-statistics. In particular, if \(h_{ni}(y)=c_{ni}h(y)\) and the function \(h(y)\) is monotone, then \(\Phi_n\) represents the classical \(L\)-statistics. Every generalized \(L\)-statistics can be represented as follows: NEWLINE\[NEWLINEA_n=\sum_{i=1}^n h_{ni}\bigl(\sqrt{n+1}(X_{n:i}- \mathbf{E}X_{n:i})\bigr).NEWLINE\]NEWLINE Given bounds on the functions \(h_{ni}\) of the type \(|h_{ni}(y)|\leq a_{ni}+b_{ni}|y|^m\), upper bounds for the probability \(\mathbf{P}\{A_n\geq y\}\) are presented. The mean value \(\mathbf{E}|A_n|^r\), \(r\geq 2\), is also estimated from above.