Some Ordering Results for the Marshall and Olkin’s Family of Distributions

Usual stochastic ordering of the sample maxima from dependent distribution‐free random variables

“…where F is the baseline distribution function, 𝜶 = (𝛼 1 , … , 𝛼 n ) is the frailty vector, and 𝝀 = (𝜆 1 , … , 𝜆 n ) is the scale vector. We then denote X = (X 1 , … , X n ) ∽ SPRH(F, 𝜶, 𝝀, 𝜙) as the variables having the Archimedean copula with generator 𝜙 and following a SPRH model with X i ∽ F 𝛼 i (𝜆 i x), i = 1, … , n. It is readily evident that the PRH model (see Gupta et al (2007) and Di Crescenzo (2000)) is a special case of the SPRH models, and SPRH is a particular case of the modified proportional reversed hazard rate scale model (see Das and Kayal (2021)). Li et al (2020) investigated stochastic comparisons of parallel systems (corresponding to the largest order statistics) with respect to the reversed hazard rate and likelihood ratio orders for the proportional reversed hazard rate model.…”

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.…”

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 were considered by Das and Kayal (2019a), Das and Kayal (2019b) and Das et al (2019) to study ordering properties of extreme order statistics in the context of reliability studies. 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.…”