Abstract
Let X1:n ≤ X2:n ≤···≤ Xn:n denote the order statistics of a sample of n independent random variables X1, X2,…, Xn, all identically distributed as some X. It is shown that if X has a log-convex [log-concave] density function, then the general spacing vector (Xk1:n, Xk2:n − Xk1:n,…, Xkr:n − Xkr−1:n) is MTP2 [S-MRR2] whenever 1 ≤ k1 < k2 <···< kr ≤ n and 1 ≤ r ≤ n. Multivariate likelihood ratio ordering of such general spacing vectors corresponding to two random samples is also considered. These extend some of the results in the literature for usual spacing vectors.
Acknowledgments
We thank the referees and associate editor for comments on a previous draft of the article.
The fourth author is supported by National Natural Science Foundation of China (No.: 10771204), and by the Knowledge Innovation Program of the Chinese Academy of Sciences (No.: KJCX3-SYW-S02).