A New Flexible Bathtub-Shaped Modification of the Weibull Model: Properties and Applications

Qinghu Liao,G G Hamedani,Zubair Ahmad,Eisa Mahmoudi

doi:10.1155/2020/3206257

Abstract

Many studies have suggested the modifications and generalizations of the Weibull distribution to model the nonmonotone hazards. In this paper, we combine the logarithms of two cumulative hazard rate functions and propose a new modified form of the Weibull distribution. The newly proposed distribution may be called a new flexible extended Weibull distribution. Corresponding hazard rate function of the proposed distribution shows flexible (monotone and nonmonotone) shapes. Three different characterizations along with some mathematical properties are provided. We also consider the maximum likelihood estimation procedure to estimate the model parameters. For the illustrative purposes, two real applications from reliability engineering with bathtub-shaped hazard functions are analyzed. The practical applications show that the proposed model provides better fits than the other nonnested models.

Highlights

Lai et al [8] proposed a new modification of the Weibull distribution called modified Weibull (MW) by multiplying eλx with the cumulative hazard rate function (CHRF) of the Weibull model given by Mathematical Problems in Engineering
We present certain characterizations of new flexible extended Weibull (NFEW) distribution. e first characterization is based on the hazard function, the second one is based on the ratio of two truncated moments, and the third is based on conditional expectation of certain function of the random variable
A new flexible extended Weibull distribution with a nonmonotone hazard rate function is proposed and investigated by taking into account a convex combination of the logarithms of two cumulative hazard functions. e resulting hazard rate function of the proposed model is capable of accommodating different shapes of the failure rates including bathtub shape to describe the failure behavior of a variety of real lifetime data sets

Summary

Basic Properties

Some basic properties of the proposed model are derived. 3.1. E expression for the qth quantile, say xq, of the NFEW model is given by βxαq −. (3) e proposed model is capable of modeling the last phase of the modified unimodal-shaped failure rate function closely (see Figure 2). (5) It is capable of modeling the last phase of the bathtub-shaped failure rate closely (see Figure 3). E expression for generating random numbers from NFEW distribution is given by βxα −. If X has the NFEW distribution with parameters vector (α, β, σ, λ), the rth moment of X is derived as. E hth order negative moment of the NFEW random variable X is derived as μ/− h 􏽚 x− hf(x)dx, μ/− h xr􏼠αβxα−. Xk be a random sample from NFEW distribution with parameters (α, β, σ, λ) and let X(1:k) ≤ .

Maximum Likelihood Estimation

Characterizations

Applications

Concluding Remarks