Ordering Results on Extremes of Exponentiated Location-Scale Models

Abstract

In this paper, we consider exponentiated location-scale model and obtain several ordering results between extreme order statistics in various senses. Under majorization type partial order-based conditions, the comparisons are established according to the usual stochastic order, hazard rate order and reversed hazard rate order. Multiple-outlier models are considered. When the number of components are equal, the results are obtained based on the ageing faster order in terms of the hazard rate and likelihood ratio orders. For unequal number of components, we develop comparisons according to the usual stochastic order, hazard rate order, and likelihood ratio order. Numerical examples are considered to illustrate the results.

Ordering results for smallest claim amounts from two portfolios of risks with dependent heterogeneous exponentiated location-scale claims

“…The following example provides a demonstration of Theorem 3.5. , 𝑥 > 0, be the generator of the Gumbel-Barnett survival copula. Set the three-dimensional vectors 𝝀 = 𝝁 = (0.6, 1.1, 2), 𝜽 = (𝑒 5 , 𝑒 7 , 𝑒 9 ), 𝜹 = (𝑒 6 , 𝑒 8 , 𝑒 10 ), 𝒑 = (0.11, 0.21, 0.31) and 𝒑 * = (0.62, 0.70, 0.82). Furthermore, we can check that 𝜓 satisfies Conditions (ii) and (iii) in Theorem 3.5.…”

“…The general exponentiated location-scale model serves this purpose very well due to its great flexibility. Das et al [9], Das and Kayal [7] and Das et al [8] considered (1.1) to obtain different stochastic ordering results.…”

“…The general exponentiated location-scale model serves this purpose very well due to its great flexibility. Das et al [9], Das and Kayal [7] and Das et al [8] considered (1.1) 1:𝑛 with respect to the usual stochastic and hazard rate orders are discussed under some conditions involving the concept of vector majorization and related orders. The study first carries out the required stochastic ordering when the dependence structure is modeled by an Archimedean survival copula and then by a general survival copula.…”

Ordering results between the largest claims arising from two general heterogeneous portfolios

“…To the best of our knowledge, stochastic comparisons of the largest claim amounts when random claims have ELS models have not been addressed in the literature so far. However, some generalized models to study ordering properties of extreme order statistics in the context of reliability studies can be found in [7][8][9][10][11]16]. In this paper, we address this problem and derive sufficient conditions for the stochastic comparison of the largest claim amounts in the sense of various stochastic orderings.…”