Right Spread Order of the Second-Order Statistic from Heterogeneous Exponential Random Variables

Abstract

Let X 1,…, X n be independent exponential random variables with respective hazard rates λ1,…, λ n , and Y 1,…, Y n be i.i.d. exponential random variables with common hazard rate λ. It is proved that X 2: n , the second order statistic from X 1,…, X n , is larger than Y 2: n , the second order statistic from Y 1,…, Y n , with respect to the right spread order if and only if with and Λ i = Λ(1) − λ i , and X 2: n is smaller than Y 2: n with respect to the right spread order if and only if Further, the case with proportional decreasing hazard rate is also studied, and the results obtained here form nice extensions to some corresponding ones known in the literature.