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 λ for i=1,…,p and Xj has failure rate λ* 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≡0. It is shown that Di:n(p,q)⩽lrDi+1:n(p,q) for i=1,…,n-1, and that if λ⩽λ* then , and for i=1,…,n, where ⩽lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.