A new modified Weibull distribution

Abstract

We introduce a new lifetime distribution by considering a serial system with one component following a Weibull distribution and another following a modified Weibull distribution. We study its mathematical properties including moments and order statistics. The estimation of parameters by maximum likelihood is discussed. We demonstrate that the proposed distribution fits two well-known data sets better than other modified Weibull distributions including the latest beta modified Weibull distribution. The model can be simplified by fixing one of the parameters and it still provides a better fit than existing models.

Introduction

The Weibull distribution [28] has been used in many different fields with many applications, see for example [18]. The hazard function of the Weibull distribution can only be increasing, decreasing or constant. Thus it cannot be used to model lifetime data with a bathtub shaped hazard function, such as human mortality and machine life cycles. For many years, researchers have been developing various extensions and modified forms of the Weibull distribution, with number of parameters ranging from 2 to 5. The two-parameter flexible Weibull extension of Bebbington et al. [5] has a hazard function that can be increasing, decreasing or bathtub shaped. Zhang and Xie [31] studied the characteristics and application of the truncated Weibull distribution which has a bathtub shaped hazard function. A three-parameter model, called exponentiated Weibull distribution, was introduced by Mudholkar and Srivastave [17]. Another three-parameter model is by Marshall and Olkin [15] and called extended Weibull distribution. Xie et al. [30] proposed a three-parameter modified Weibull extension with a bathtub shaped hazard function. The modified Weibull (MW) distribution of Lai et al. [13] multiplies the Weibull cumulative hazard function $\alpha x^{\beta }$ by $e^{\lambda x}$, which was later generalized to exponentiated form by Carrasco et al. [6]

$$F(x)=(1-e^{-\alpha x^{\beta }{e}^{\lambda x}}{)}^{\theta },x\ge 0.$$

Recent studies of the modified Weibull include [11], [26], [27].

Among the four-parameter distributions, the additive Weibull distribution (AddW) of Xie and Lai [29] with cumulative distribution function (CDF)

$$F(x)=1-e^{-\alpha x^{\theta }-\beta x^{\gamma }},x\ge 0,$$

has a bathtub-shaped hazard function consisting of two Weibull hazards, one increasing ($\theta >1$) and one decreasing ($0<\gamma <1$). The modified Weibull distribution of Sarhan and Zaindin (SZMW) [21] can be derived from the additive Weibull distribution by setting $\theta =1$. A four-parameter beta Weibull distribution was proposed by Famoye et al. [10]. Cordeiro et al. [8] introduced another four-parameter called the Kumaraswamy Weibull distribution.

Five-parameter modified Weibull distributions include Phani's modified Weibull [20], the beta modified Weibull (BMW) introduced by Silva et al. [24] and further studied by Nadarajah et al. [19]. The latest examples include the beta generalized Weibull distribution by Singla et al. [25], exponentiated generalized linear exponential distribution by Sarhan et al. [22] and the generalized Gomprtz distribution by El-Gohary et al. [9].

We propose a new lifetime distribution based on the Weibull and the modified Weibull (MW) distributions by combining them in a serial system. The hazard function of the new distribution is the sum of a Weibull hazard function and a modified Weibull hazard function. Section 2 gives definition, motivation and usefulness of this model and lists its sub-models. Section 3 considers properties of the new distribution such as hazard, moments and order statistics. Section 4 discusses estimation of the parameters. Two real data sets are analyzed in Section 5 and the results are compared with existing distributions. Section 6 concludes the paper.