The Moments of Log-Weibull Order Statistics

TLDR: In this paper, a number of formulae expressing E(i, n) in terms of M(j), j ≤ n, are developed for order statistics of a random sample of size n.

Abstract: Let X 1n ≤ X 2n ≤,…, ≤ Xnn be the order statistics of a random sample of size n. For any integrable function g(x) define E(i, n) = E(g(X in )) and M(n) = E(1, n) = E(g(X 1n )). A number of formulae expressing E(i, n) in terms of M(j), j ≤ n, are developed. For example, These results are applied to obtain the means and variances of the order statistics of a log-Weibull distribution (F(z) = 1 – exp (− exp x)). Tables of these means and variances are given for 1 ≤ i ≤ n, n = 1 (1) 50 (5) 100. The computations were made using a set of 100 decimal place logarithms of integers. Examples of the use of these tables in obtaining weighted least squares estimates from censored samples from a Weibull distribution are also given.