Some New Results on Likelihood Ratio Orderings for Spacings of Heterogeneous Exponential Random Variables |
| |
| Authors: | Taizhong Hu Qingshu Lu Songqiao Wen |
| |
| Institution: | 1. Department of Statistics and Finance , University of Science and Technology of China , Anhui , P.R. China thu@ustc.edu.cn;3. Department of Statistics and Finance , University of Science and Technology of China , Anhui , P.R. China |
| |
| Abstract: | Let X 1, X 2,…, X n be independent exponential random variables with X i having failure rate λ i for i = 1,…, n. Denote by D i:n = 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 if λ n+1 ≤ ≥] λ k for k = 1,…, n then D n:n ≤ lr D n+1:n+1 and D 1:n ≤ lr D 2:n+1 D 2:n+1 ≤ lr D 2:n ], and that if λ i + λ j ≥ λ k for all distinct i,j, and k then D n?1:n ≤ lr D n:n and D n:n+1 ≤ lr D n:n , where ≤ lr denotes the likelihood ratio order. We also prove that D 1:n ≤ lr D 2:n for n ≥ 2 and D 2:3 ≤ lr D 3:3 for all λ i 's. |
| |
| Keywords: | Exponential distributions Likelihood ratio order Order statistics Permanent Spacings |
|
|
