On stochastic orderings between distributions and their sample spacings

Let X1:n[less-than-or-equals, slant]X2:n[less-than-or-equals, slant]...[less-than-or-equals, slant]Xn:n denote the order statistics of a random sample X1,X2,...,Xn from a probability distribution with distribution function F. Similarly, let Y1:n[less-than-or-equals, slant]Y2:n[less-than-or-equals, slant]...[less-than-or-equals, slant]Yn:n denote the order statistics of an independent random sample Y1,Y2,...,Yn from G. The corresponding spacings are defined by Ui:n[reverse not equivalent]Xi:n-Xi-1:n and Vi:n[reverse not equivalent]Yi:n-Yi-1:n, for i=1,2,...,n, where X0:n=Y0:n[reverse not equivalent]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 Ui:n's is stochastically smaller than the vector of Vi: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 Ui:n is smaller than Vi:n in the hazard rate sense for 1[less-than-or-equals, slant]i[less-than-or-equals, slant]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 Ui:n's is smaller than the corresponding random vector of Vi:n's in the sense of multivariate likelihood ratio ordering.