## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 34, Number 1 (2020), 150-166.

### Application of weighted and unordered majorization orders in comparisons of parallel systems with exponentiated generalized gamma components

Abedin Haidari, Amir T. Payandeh Najafabadi, and Narayanaswamy Balakrishnan

#### Abstract

Consider two parallel systems, say $A$ and $B$, with respective lifetimes $T_{1}$ and $T_{2}$ wherein independent component lifetimes of each system follow exponentiated generalized gamma distribution with possibly different exponential shape and scale parameters. We show here that $T_{2}$ is smaller than $T_{1}$ with respect to the usual stochastic order (reversed hazard rate order) if the vector of logarithm (the main vector) of scale parameters of System $B$ is weakly weighted majorized by that of System $A$, and if the vector of exponential shape parameters of System $A$ is unordered mojorized by that of System $B$. By means of some examples, we show that the above results can not be extended to the hazard rate and likelihood ratio orders. However, when the scale parameters of each system divide into two homogeneous groups, we verify that the usual stochastic and reversed hazard rate orders can be extended, respectively, to the hazard rate and likelihood ratio orders. The established results complete and strengthen some of the known results in the literature.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 34, Number 1 (2020), 150-166.

**Dates**

Received: March 2018

Accepted: July 2018

First available in Project Euclid: 3 February 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1580720429

**Digital Object Identifier**

doi:10.1214/18-BJPS410

**Mathematical Reviews number (MathSciNet)**

MR4058976

**Keywords**

Exponentiated generalized gamma distribution parallel system stochastic orders unordered majorization weighted majorization

#### Citation

Haidari, Abedin; Payandeh Najafabadi, Amir T.; Balakrishnan, Narayanaswamy. Application of weighted and unordered majorization orders in comparisons of parallel systems with exponentiated generalized gamma components. Braz. J. Probab. Stat. 34 (2020), no. 1, 150--166. doi:10.1214/18-BJPS410. https://projecteuclid.org/euclid.bjps/1580720429