2015/03/03 by Robert W. Chen, Chen, Robert W.
Computer Science · Decision Sciences · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Primary 62E99 #Probability and Risk Models #Secondary 62F99 #Statistical Distribution Estimation and Applications #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1503.01075
openalex publication_date 2015/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X1, X2,…, Xn be n independent and identically distributed random variables, here n ≥ 2. Let X(1), X(2), …, X(n) be the order statistics of X1, X2,..., Xn. In this note we proved that: (I) If X1, X2,..., Xn are exponential random variables with parameter c > 0, then the "correlation coefficient" between X(k) and X(k+t) is strictly increasing in k from 1 to m, and then is strictly decreasing in k from m to n - t, here t is a fixed integer between 1 and n - 3, and m = (n - t)/2 if n - t is even, m = (n - t + 1)/2 if n - t is odd. We also proved that if t = n - 2, then the "correlation coefficient" between X(1) and X(n-1) is greater than the "correlation coefficient" between X(2) andX(n). (II) The "correlation coefficient" between X(k) and X(k+t) for the exponential random variables is always less than the "correlation coefficient" between X(k) and X(k+t) for the uniform random variables for all k and t such that k + t ≤ n. A combinatorial identity is also given as a bi-product. \vs