In this paper we prove that a parallel system consisting of Weibull components with different scale parameters ages faster than a parallel system comprising Weibull components with equal scale parameters in the convex transform order when the lifetimes of components of both systems have different shape parameters satisfying some restriction. Moreover, while comparing these two systems, we show that the dispersive and the usual stochastic orders, and the right-spread order and the increasing convex order are equivalent. Further, some of the known results in the literature concerning comparisons of k-out-of-n systems in the exponential model are extended to the Weibull model. We also provide solutions to two open problems mentioned by Balakrishnan and Zhao (2013) and Zhao et al. (2016).

A parallel system with heterogeneous exponential component lifetimes is shown to be more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. As a consequence, equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and homogeneous exponential samples in the sense of the dispersive order and the right-spread order are established. A sufficient condition is also given for the proportional hazard rate model.

For nonnegative random variables X and Y we write X ≤TTTY if ∫0F-1(p)(1-F(x))dx ≤ ∫0G-1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.