Likelihood ratio and mean residual life orders for order statistics of heterogeneous random variables (Q2731309)

Let \(X_{1:n}\leq X_{2:n} \leq\dots\leq X_{n:n}\) denote the order statistics of a set of independent, but not necessarily identically distributed random variables \(X_1,X_2,\dots,X_n\). Under some regularity conditions it is shown that if \(X_1\leq_{\text{lr}} X_2\leq_{\text{lr}}\cdots \leq_{\text{lr}} X_n\) then \(X_{k-1:n-1} \leq_{\text{lr}} X_{k:n}\) for \(k=2,3, \dots, n\), and if \(X_1\geq_{\text{lr}}X_2\geq_{\text{lr}} \cdots \geq_{\text{lr} }X_n\) then \(X_{k:n} \leq_{\text{lr}} X_{k:n-1}\) for \(k=1,2, \dots,n-1\), where \(\leq_{\text{lr}}\) denotes the likelihood ratio stochastic order. Regarding the mean residual life stochastic order \(\leq_{\text{mrl}}\), it is shown that if \(X_i\leq_{\text{mrl}}X_n\) for \(i=1,2, \dots,n-1\), then \(X_{n-1:n-1}\leq_{\text{mrl}} X_{n:n}\). Counterexamples show, regarding the order \(\leq_{\text{mrl}}\), that under the above assumptions it may not be true that \(X_{k-1:n-1}\leq_{\text{mrl}}X_{k:n}\) for \(k<n\).