Abstract
We propose a method to estimate a shape parameter for a three-parameter Weibull distribution. The proposed method first derives an unbiased estimator for the shape parameter independent of the location and scale parameters and then estimates the shape parameter using a minimum-variance linear unbiased estimator. Since the proposed method is expressed using a hyperparameter, its optimal hyperparameter is searched using Monte Carlo simulations. The recommended hyperparameter used for estimating the shape parameter depends on the sample size, and this causes no problems since the sample size is known when data is obtained. The proposed method is evaluated using a bias and a root mean squared error, and the results are very promising when the population shape parameter is 2 or more in the Weibull distribution representing the wear-out failure period. A numerical dataset is analyzed to demonstrate the practical use of the proposed method.
Highlights
Weibull distribution is often used in lifetime and reliability studies
The w-maximum likelihood estimation (MLE) method used for comparison in this paper is based on Cousineau (2009a), but the weight of W3 is calculated by the following five steps based on Nagatsuka et al (2013). (i) A temporary shape parameter, m 0, is estimated by MLE for the two-parameter Weibull distribution in (X(i) − X(1)), i = 2, . . . , n. (ii) Uniform random samples, (u1, . . . , un), are generated from the standard uniform distribution. (iii) Using m 0 and (u1, . . . , un), we calculate
The location and scale parameters were estimated by the Bayesian likelihood (BL), weighted maximum likelihood estimation (w-MLE), and LSPF-MLE methods using the shape parameter estimated by the proposed method
Summary
Weibull distribution is often used in lifetime and reliability studies. The probability density function and cumulative distribution function of the three-parameter Weibull distribution are expressed as follows: g(x; m, η, γ). Maximum likelihood estimation (MLE) is often used for parameter estimation, it may not be possible to be used for a three-parameter Weibull distribution. The parameter estimation method that extends MLE have proposed for a three-parameter Weibull distribution. Nagatsuka et al (2013) proposed the location and scale parameters free maximum likelihood estimators (LSPF-MLE) method, which estimates the shape parameter using the independent statistics of the location and scale parameters. We propose a method to estimate the shape parameter, independent of the location and scale parameters. The shape parameter can be estimated first for the three-parameter Weibull distribution. 2. Unbiased Estimator of the Shape Parameter Independent of the Location and Scale Parameters.
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