Article ID Journal Published Year Pages File Type
1147110 Journal of Multivariate Analysis 2009 11 Pages PDF
Abstract

Let X1,…,XnX1,…,Xn be independent exponential random variables with respective hazard rates λ1,…,λnλ1,…,λn, and let Y1,…,YnY1,…,Yn be independent exponential random variables with common hazard rate λλ. This paper proves that X2:nX2:n, the second order statistic of X1,…,XnX1,…,Xn, is larger than Y2:nY2:n, the second order statistic of Y1,…,YnY1,…,Yn, in terms of the likelihood ratio order if and only if λ≥12n−1(2Λ1+Λ3−Λ1Λ2Λ12−Λ2) with Λk=∑i=1nλik,k=1,2,3. Also, it is shown that X2:nX2:n is smaller than Y2:nY2:n in terms of the likelihood ratio order if and only if λ≤∑i=1nλi−max1≤i≤nλin−1. These results form nice extensions of those on the hazard rate order in Paˇltaˇnea [E. Paˇltaˇnea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993–1997].

Related Topics
Physical Sciences and Engineering Mathematics Numerical Analysis
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