Dependence Structure of Spacings of Order Statistics

Abstract

Let X_1:n ≤ X_2:n ≤···≤ X_n:n denote the order statistics of a sample of n independent random variables X₁, X₂,…, X_n, 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 (X_k1:n, X_k2:n − X_k1:n,…, X_kr:n − X_kr−1:n) is MTP₂ [S-MRR₂] whenever 1 ≤ k₁ < k₂ <···< k_r ≤ 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.

Keywords:

• DLR
• ILR
• MTP₂
• Multivariate likelihood ratio order
• S-MRR₂
• Spacing

Mathematics Subject Classification:

• 60E15
• 62N05

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).