Weibull distribution: Some stochastic comparisons results

Let X1,…,Xn be independent random variables such that Xi has Weibull distribution with shape parameter α and scale parameter λi, i=1,…,n. Let X1*,…,Xn* be another set of independent Weibull random variables with the same common shape parameter α, but with scale parameters as λ*=(λ1*,…,λn*). Suppose that λ≽mλ*. We prove that when 0<α<1, (X(1),…,X(n))≽st(X(1)*,…,X(n)*). For α⩾1, we prove that X(1)⩽hrX(1)*, whereas the inequality is reversed when α⩽1. Let Y1,…,Yn be a random sample of size n from a Weibull distribution with shape parameter α and scale parameter λ˜=(∏i=1nλi)1/n, the geometric mean of the λi's. It is shown that X(n)⩾hrY(n) for all values of α and in case α⩽1, we also have that X(n) is greater than Y(n) according to dispersive ordering. In the process, we also prove some new results on stochastic comparisons of order statistics for the proportional hazards family.