Stochastic Comparisons and Dependence of Spacings from Two Samples of Exponential Random Variables |
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| Authors: | Taizhong Hu Feng Wang Zegang Zhu |
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| Affiliation: | 1. Department of Statistics and Finance , University of Science and Technology of China , Hefei , Anhui , P.R. China thu@ustc.edu.cn;3. Department of Statistics and Finance , University of Science and Technology of China , Hefei , Anhui , P.R. China;4. Department of Industrial Engineering and Operations Research , University of California , Berkeley , California , USA |
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| Abstract: | ![]()
Let X 1,X 2,…,X n be independent exponential random variables such that X i has hazard rate λ for i = 1,…,p and X j has hazard rate λ* for j = p + 1,…,n, where 1 ≤ p < n. Denote by D i:n (λ, λ*) = 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 the spacings (D 1,n ,D 2,n ,…,D n:n ) are MTP2, strengthening one result of Khaledi and Kochar (2000), and that (D 1:n (λ2, λ*),…,D n:n (λ2, λ*)) ≤ lr (D 1:n (λ1, λ*),…,D n:n (λ1, λ*)) for λ1 ≤ λ* ≤ λ2, where ≤ lr denotes the multivariate likelihood ratio order. A counterexample is also given to show that this comparison result is in general not true for λ* < λ1 < λ2. |
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| Keywords: | Dependent Likelihood ratio order MTP2 Multivariate likelihood ratio order Permanent TP2 |
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