Abstract
We upgrade on the method used on weighted Weibull model proposed by Azzalini (1985) using the logit of Beta function by Jones (2004) to have Lehmann Type II weighted Weibull model with a view to obtaining a distribution that is more better than both weighted Weibull and Weibull distribution in terms of estimate of their characteristics and their parameter. The weighted Weibull distribution is proposed by slightly modifying the method of Azzalini (1985) on weighted distribution with additional shape and scale parameters. Some basic properties of the newly proposed distribution including moments and moment generating function, survival rate function, hazard rate function, asymptotic behaviours, and the estimation of parameters have been studied. Key words: Asymptotic, hazard rate, Lehmann Type II, moments, weighted-Weibull.
Highlights
The Weibull distribution is a well known common distribution and has been a powerful probability distribution in reliability analysis, while weighted distributions are used to adjust the probabilities of the events as observed and recorded
We upgrade on the method used on weighted Weibull model proposed by Azzalini (1985) using the logit of Beta function by Jones (2004) to have Lehmann Type II weighted Weibull model with a view to obtaining a distribution that is more better than both weighted Weibull and Weibull distribution in terms of estimate of their characteristics and their parameter
The Weibull distribution can be used as an alternative to Gamma and Log-normal distribution in reliability engineering and life testing
Summary
The Weibull distribution is a well known common distribution and has been a powerful probability distribution in reliability analysis, while weighted distributions are used to adjust the probabilities of the events as observed and recorded. Numerous authors/researchers have been worked on Weibull and exponential distribution in literature. In a nut shell, Shahbaz et al (2010) applied Azzalini’s method with the Weibull distribution that produced a new class of weighted Weibull distribution in which they obtained the probability density and cumulative density function as given below: , x > 0 (1). The graph of the probability density function of Equation (1) with is given Figure 1. Researchers/authors on Lehmann Type II are very few. This article presents the proposed distribution Lehmann Type II weighted Weibull distribution. Thereafter, moments and moment generating function is studied. This is followed by a critical discussion of the estimation of parameters showing the empirical distribution of data and the study was concluded
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