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Let X1,…,Xn be independent exponential random variables with respective hazard rates λ1,…,λn, and let Y1,…,Yn be independent exponential random variables with common hazard rate λ. This paper proves that X2:n, the second order statistic of X1,…,Xn, is larger than Y2:n, the second order statistic of Y1,…,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...
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