Author： Hu Taizhong Lu Qingshu Wen Songqiao
Publisher： Taylor & Francis Ltd
ISSN： 0361-0926
Source： Communications in Statistics: Theory and Methods, Vol.37, Iss.16, 2008-01, pp. : 2506-2515
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Abstract
Let X1, X2,…, Xn be independent exponential random variables with Xi having failure rate λi for i = 1,…, n. Denote by Di:n = Xi:n - Xi-1:n the ith spacing of the order statistics X1:n ≤ X2:n ≤ ··· ≤ Xn:n, i = 1,…, n, where X0:n ≡ 0. It is shown that if λn+1 ≤ [≥] λk for k = 1,…, n then Dn:n ≤ lr Dn+1:n+1 and D1:n ≤ lr D2:n+1 [D2:n+1 ≤ lr D2:n], and that if λi + λj ≥ λk for all distinct i,j, and k then Dn-1:n ≤ lr Dn:n and Dn:n+1 ≤ lr Dn:n, where ≤ lr denotes the likelihood ratio order. We also prove that D1:n ≤ lr D2:n for n ≥ 2 and D2:3 ≤ lr D3:3 for all λi's.
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