Stochastic orderings between distributions and their sample spacings - II Article

Khaledi, BE, Kochar, S. (1999). Stochastic orderings between distributions and their sample spacings - II . STATISTICS & PROBABILITY LETTERS, 44(2), 161-166. 10.1016/s0167-7152(99)00004-8

cited authors

  • Khaledi, BE; Kochar, S

authors

abstract

  • Let X1:n≤X2:n≤≤Xn:n denote the order statistics of a random sample of size n from a probability distribution with distribution function F. Similarly, let Y1:m≤Y2:m≤≤Ym:m denote the order statistics of an independent random sample of size m from another distribution with distribution function G. We assume that F and G are absolutely continuous with common support (0,∞). The corresponding normalized spacings are defined by Ui:n≡(n-i+1)(Xi:n-Xi-1:n) and Vj:m≡(m-j+1)(Yj:m-Yj-1:m), for i=1,...,n and j=1,...,m, where X0:n=Y0: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 decreasing failure rate (DFR) distribution, then Ui:n is stochastically smaller than Vj:m for i≤j and n-i≥m-j. If instead, we assume that X is smaller than Y in the likelihood ratio order and if either F or G is DFR, then this result can be strengthened from stochastic ordering to hazard rate ordering. Finally, under a stronger assumption on the shapes of the distributions that either F or G has log-convex density, it is proved that X being smaller than Y in the likelihood ratio order implies that Ui:n is smaller than Vj:m in the sense of likelihood ratio ordering for i≤j and n-i=m-j. © 1999 Elsevier Science B.V.

publication date

  • August 15, 1999

published in

Digital Object Identifier (DOI)

start page

  • 161

end page

  • 166

volume

  • 44

issue

  • 2