Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X1,X2,...,Xn be independent exponential random variables such that Xi has failure rate [lambda] for i=1,...,p and Xj has failure rate [lambda]* for j=p+1,...,n, where p>=1 and q=n-p>=1. Denote by Di:n(p,q)=Xi:n-Xi-1:n the ith spacing of the order statistics , where X0:n[reverse not equivalent]0. It is shown that Di:n(p,q)[less-than-or-equals, slant]lrDi+1:n(p,q) for i=1,...,n-1, and that if [lambda][less-than-or-equals, slant][lambda]* then , and for i=1,...,n, where [less-than-or-equals, slant]lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.