A class of regression models for parallel and series systems with a random number of components

TL;DR

This paper introduces the extended Weibull power series (EWPS) class of distributions, generalizing the Weibull power series models to include series and parallel systems with a random number of components, offering greater flexibility for modeling positive data.

Contribution

The paper develops the EWPS distribution class, extending WPS models to include Weibull as a special case and applicable to a broader range of systems and data.

Findings

EWPS distributions are more flexible than WPS distributions.

The Weibull distribution is a special case of EWPS.

Maximum likelihood estimators are consistent and asymptotically normal.

Abstract

In this paper we extend the Weibull power series (WPS) class of distributions and named this new class as extended Weibull power series (EWPS) class of distributions. The EWPS distributions are related to series and parallel systems with a random num- ber of components, whereas the WPS distributions (Morais and Barreto-Souza, 2011) are related to series systems only. Unlike the WPS distributions, for which the Weibull is a limiting special case, the Weibull law is a particular case of the EWPS distributions. We prove that the distributions in this class are identifiable under a simple assumption. We also prove stochastic and hazard rate order results and highlight that the shapes of the EWPS distributions are markedly more flexible than the shapes of the WPS distributions. We define a regression model for the EWPS response random variable to model a scale parameter and its quantiles. We present the maximum likelihood estimator and prove its consistency and normal asymptotic distribution. Although the construction of this class was motivated by series and parallel systems, the EWPS distributions are suitable for modeling a wide range of positive data sets. To illustrate potential uses of this model, we apply it to a real data set on the tensile strength of coconut fibers and present a simple device for diagnostic purposes.