Stochastic comparisons of series systems with heterogeneous Weibull components

Let X1,…,XnX1,…,Xn be independent random variables with Xi∼W(α,λi)Xi∼W(α,λi), where W(α,λi)W(α,λi) denotes a Weibull distribution with shape parameter αα and scale parameter λiλi, i=1,…,ni=1,…,n. Let Y1,…,YnY1,…,Yn be a random sample of size nn from a Weibull distribution with shape parameter αα and a common scale parameter λλ. Firstly, we prove that the smallest order statistic X1:nX1:n is greater than the smallest order statistic Y1:nY1: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:nX1: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>1α>1 and X1:n∗≤rhX1:n for 0<α≤10<α≤1.