Preservation properties of stochastic orderings by transformation to Harris family with different tilt parameters

Abstract

Harris family of distributions models lifetime of a series system when the number of components is a positive random variable. In this paper, we reveal several stochastic comparisons in the Harris family with different tilt parameters and different baseline distributions with respect to the usual stochastic, shifted stochastic, proportional stochastic and shifted proportional stochastic orderings. Such comparisons are particularly useful in lifetime optimization of reliability systems. We shall also present two bounds for a Harris family survival function conditioned on its tilt parameter which are useful when aging properties are considered. Our results generalize several previous findings in this connection.

Preservation properties of stochastic orders by transformation to Harris family

“…t = 0 or λ = 1, or both, proofs for the other parts are immediate. By using the counterexample 3.2 of [1], the following counterexample shows that the up hazard rate order is not preserved by transformation to the Harris family, when 0 < θ < 1. COUNTEREXAMPLE 4.1.…”

“…Abbasi et al [1] compared two Harris families with different tilt parameters using stochastic orders. In this paper, we are concerned with four types of stochastic orders: simple stochastic orders, shifted stochastic orders, proportional stochastic orders and shifted proportional stochastic orders.…”