Nonhomogeneous Exponential Distribution Random Variable Interval Likelihood Ratio Order

Let X₁,X₂, ... ,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 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. Similarly, if λ ≥ λ*, then D_i:n(p,q) ≤lr D_i+1:n+1(p,q+1), D_i:n+1(p+1,q)≤lr D_i:n(p,q) and D_i:n(p,q) ≤lr D_i:n(p - 1,q+ 1) for i = 1, ... ,n. Furthermore, we give applications of the main results and investigate likelihood ratio orderings of spacings of general heterogeneous exponential random variables.