Sample range: Some stochastic comparisons results (Q2771470)

Let \(X_1,X_2,\dots,X_n\) be independent nonnegative random variables following the proportional hazards model; that is, for some survival function \(\overline F(\cdot)\), the random variable \(X_i\) has the survival function \(\overline F^{\lambda_i} (\cdot)\) where \(\lambda_i>0\), \(i=1,2, \dots,n\). Let \(Y_1,Y_2, \dots,Y_n\) be a random sample from a distribution with survival function \(\overline F^{\widetilde \lambda}(\cdot)\), where \(\widetilde \lambda= (\prod^n_{i=1} \lambda_i)^{1/n}\). The authors show that if \(\overline F\) is new worse than used (NWU), then the sample range of the \(X_i\)'s is stochastically larger than the sample range of the \(Y_i\)'s.