Stochastic comparisons of largest order statistics from multiple-outlier exponential models (Q2909822)

Let \(X_1,X_2,\dots,X_n\) be independent exponential random variables, and set NEWLINE\[NEWLINEX_{n:n}=\max\{X_1,X_2,\dots,X_n\}.NEWLINE\]NEWLINE Similarly, let \(Y_1,Y_2,\dots,Y_n\) be independent exponential random variables, and set \(Y_{n:n}=\max\{Y_1,Y_2,\dots,Y_n\}\). The authors find various sets of conditions, on the parameters of the above random variables, under which \(X_{n:n}\) and \(Y_{n:n}\) are ordered with respect to the ordinary stochastic order \(\leq_{\text{st}}\), the hazard rate order \(\leq_{\text{hr}}\), the reversed hazard rate order \(\leq_{\text{rh}}\), and the likelihood ratio order \(\leq_{\text{lr}}\). These conditions are stated in terms of multiple-outlier models. The results are also extended to proportional hazard rate models.