Characterization Ordering Results for Largest Order Statistics From Heterogeneous and Homogeneous Exponentiated Generalized Gamma Variables

Abstract

The main aim of this paper is to present two new results concerning the characterization of likelihood ratio and reversed hazard rate orders between largest order statistics from two sets of independent heterogeneous and homogeneous exponentiated generalized gamma distributed random variables. These characterization results complete and strengthen some previous ones in the literature.

Stochastic comparisons of largest-order statistics for proportional reversed hazard rate model and applications

“…We give the necessary and sufficient conditions for the stochastic comparisons of the largest-order statistics for the EGG distribution and the exponentiated Pareto distribution. Our results on the EGG distribution recover the recent result established by Haidari and Najafabadi [18] and strengthen the result for the Weibull distribution given by Zhao, Zhang, and Qiao [37]. To the best of our knowledge, the result on the (exponentiated) Pareto distribution is novel.…”

“…However, it is not the case for Weibull distributions (see Section 3.1). Recently, Haidari and Najafabadi [18] generalized the above result to exponentiated generalized gamma distributions, which will be stated in Proposition 3 in Section 3.1.…”

“…From Theorem 1, we can immediately get the following corollary for the EGG distribution. The result has been given by Haidari and Najafabadi [18], whose proof is technical and requires several lemmas. Recall that a random variable X is said to have the EGG distribution with shape parameters θ > 0, λ > 0, r > 0, and scale parameter μ > 0, denoted by X ∼ EGG(θ, λ, r, μ), if it has the following cumulative distribution function:…”

“…Stochastic comparisons 833 gamma, Weibull, etc. ; see [4], [8], [18], [21], [23], [26], [35], [36], and [37], and the references therein.…”

Ordering results of the smallest and largest order statistics from independent heterogeneous exponentiated gamma random variables

“…Stochastic comparisons of parallel and series systems with heterogeneous components have been studied by many authors. For example, see the works of Balakrishnan and Zhao (); Balakrishnan, Haidari, and Masoumifard (); Fang, Ling, and Balakrishnan (); Gupta, Patra, and Kumar (); Haidari and Payandeh Najafabadi (); Kundu and Chowdhury (); Li and Li (); Torrado and Kochar (); and Zhao and Balakrishnan ().…”