On stochastic orderings between distributions and their sample spacings

Let X-1:n less than or equal to X-2:n less than or equal to ... less than or equal to X-n:n denote the order statistics of a random sample X-1, X-2, ..., X-n from a probability distribution with distribution function F. Similarly, let Y-1:n less than or equal to Y-2:n less than or equal to ... less than or equal to Y-n:n denote the order statistics of an independent random sample Y-1, Y-2, ..., Y-n from G. The corresponding spacings are defined by U-i:n = X-i:n - Xi-1:n and V-i:n - Yi-1:n, for i = 1, 2, ..., n, where X-0:n = Y-0:n = 0. It is proved that if X is smaller than Y in the hazard rate order sense and if either F or G is a DFR (decreasing failure rate) distribution, then the vector of U-t:n's is stochastically smaller than the vector of V-i:n's. If instead, we assume that X is smaller than Y in the likelihood ratio order and if either F or G is DFR, then U-i:n is smaller than V-i:n in the hazard rate sense for 1 less than or equal to i less than or equal to n. Finally, if we make a stronger assumption on the shapes of the distributions that either X or Y has log-convex density, then the random vector of U-i:n's is smaller than the corresponding random vector of V-i:n's in the sense of multivariate likelihood ratio ordering. (C) 1999 Elsevier Science B.V. All rights reserved.