On the range of heterogeneous samples

  • Authors:

  • Christian Genest;Subhash C. Kochar;Maochao Xu

  • Affiliations:

  • Université Laval, Canada and Portland State University, United States;Université Laval, Canada and Portland State University, United States;Université Laval, Canada and Portland State University, United States

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2009

Quantified Score

Hi-index 0.00

Visualization

Abstract

Let R"n be the range of a random sample X"1,...,X"n of exponential random variables with hazard rate @l. Let S"n be the range of another collection Y"1,...,Y"n of mutually independent exponential random variables with hazard rates @l"1,...,@l"n whose average is @l. Finally, let r and s denote the reversed hazard rates of R"n and S"n, respectively. It is shown here that the mapping t@?s(t)/r(t) is increasing on (0,~) and that as a result, R"n=X"("n")-X"("1") is smaller than S"n=Y"("n")-Y"("1") in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X"("n") is seen to be more stochastically increasing in X"("1") than Y"("n") is in Y"("1"). In other words, the pair (X"("1"),X"("n")) is more dependent than the pair (Y"("1"),Y"("n")) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X"1,...,X"n form a random sample from a continuous distribution while Y"1,...,Y"n are mutually independent lifetimes with proportional hazard rates.