Abstract
The aims of this paper is to propose a new approach for fitting a three-parameter weibull distribution to data from an independent and identically distributed scheme of sampling. This approach use a likelihood function based on the n - 1 largest order statistics. Information loss by dropping the first order statistic is then retrieved via an MM-algorithm which will be used to estimate the model’s parameters. To examine the properties of the proposed estimators, the associated bias and mean squared error were calculated through Monte Carlo simulations. Subsequently, the performance of these estimators were compared with those of two concurrent methods.
Highlights
The Weibull distribution is a continuous probability distribution orginaly used as a model for material breaking strength
Griffiths (1980) point out that the log-likelihood is unbounded when the shape parameter is smaller than unity and the location one closed to the smallest order statistic
An marginalized log-likelihood function based on the n − 1 largest order statistics is given as follows: n n l (θ | x2:n, · · ·,xn:n) = ln (n!) + (n − 1) ln λ − (n − 1) λ ln β − ln + λ ln r=2 xr:n − ν λ
Summary
The Weibull distribution is a continuous probability distribution orginaly used as a model for material breaking strength. Two versions of the Weibull probability density function (pdf) are in common use: the two parameter pdf and the three parameter pdf. A cumulative distribution function (cdf) is indexed by a vector of three real parameters θ = (λ, β, ν) and is defined as follows:. The corresponding probability density function (pdf) is f (x|θ) =. Where β > 0, λ > 0 and ν < x are the scale, shape and threshold parameters respectively. If the threshold parameter νis to estimate, the three-parameter Weibull distribution is non-regular and several undesirable situations can arise when the maximum likelihood method is used to fit the model to data. In the recent past some several works including those of Cheng and Iles (1990); Cheng (1987); Smith (1985) & al.Nagatsuka and Balakrishnan (2012) have developed methods that aim to provide with reliable estimators as those based on complete and censored samples
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