Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables (Q1006675)

Let \(X_1,X_2,\ldots,X_n\) be independent exponential random variables with possibly different hazard rates, and let \(Y_1,Y_2,\ldots,Y_n\) be independent and identically distributed exponential random variables. The authors find necessary and sufficient conditions on the hazard rates of the \(X_i\)'s and of the \(Y_i\)'s under which the second order statistic among the \(X_i\)'s is larger or smaller, than the second largest order statistic among the \(Y_i\)'s, with respect to the likelihood ratio stochastic order.