Stochastic comparisons of series systems with heterogeneous Weibull components

Let X1,…,Xn be independent random variables with Xi∼W(α,λi), where W(α,λi) denotes a Weibull distribution with shape parameter α and scale parameter λi, i=1,…,n. Let Y1,…,Yn be a random sample of size n from a Weibull distribution with shape parameter α and a common scale parameter λ. Firstly, we prove that the smallest order statistic X1:n is greater than the smallest order statistic Y1:n according to the convex transform order. Secondly, we prove that λ≥(1n∑i=1nλiα)1α implies Y1:n≤dispX1:n; and λ=(∏i=1nλi)1n implies X1:n≤rhY1:n. Let X1∗,…,Xn∗ be independent random variables with Xi∗∼W(α,λi∗),i=1,…,n. Then (λ1∗,…,λn∗)≤m(λ1,…,λn) implies that X1:n≤rhX1:n∗ for α>1 and X1:n∗≤rhX1:n for 0<α≤1.