Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X 1 , X 2 , … , X n be independent exponential random variables such that X i has failure rate λ for i = 1 , … , p and X j has failure rate λ * for j = p + 1 , … , n , where p ≥ 1 and q = n - p ≥ 1 . Denote by D i : n ( p , q ) = X i : n - X i - 1 : n the ith spacing of the order statistics X 1 : n ⩽ X 2 : n ⩽ ⋯ ⩽ X n : n , i = 1 , … , n , where X 0 : n ≡ 0 . It is shown that D i : n ( p , q ) ⩽ lr D i + 1 : n ( p , q ) for i = 1 , … , n - 1 , and that if λ ⩽ λ * then D i : n ( p , q ) ⩽ lr D i + 1 : n + 1 ( p + 1 , q ) , D i : n + 1 ( p , q + 1 ) ⩽ lr D i : n ( p , q ) and D i : n ( p , q ) ⩽ lr D i : n ( p + 1 , q - 1 ) for i = 1 , … , n , where ⩽ lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.