×

Maximum likelihood and the Weibull distribution. (English) Zbl 1009.62016

Summary: The Weibull distribution has three parameters, location \(a\), scale \(b\) and shape \(c\). Maximum lilelihood estimators are \(\widehat a, \widehat b,\widehat c\), and solutions may not always exist; for example the location estimate \(\widehat a\) must be less than the smallest member of the sample. We consider three estimation problems:
(1) Estimation of one parameter when the other two are assumed to be known. (2) Estimating the scale and shape parameters when the location parameter is known. (3) Estimating the three parameters simultaneously.
Results being based on the covariance matrix and its cofactors, we give explicit expressions for the asymptotic bias, 2nd order variances, skewness to order \(1/ \sqrt N\), and asymptotic kurtosis to order \(1/N\), \(N\) being the sample size. Except for the simultaneous estimation of \(a,b,c\), the expressions for these asymptotic moments and moment ratios are simple in form involving gamma and Riemann zeta functions. They provide a new basic supplement to our knowledge of maximum likelihood estimator moments.
A surprising discovery is the part played by the location parameter whenever it has to be estimated. For the three parameter estimation case it is already known that asymptotic covariance only exist if \(c>2\). It turns out that the asymptotic skewness only exists if \(c>3\), and the asymptotic kurtosis only exists if \(c>4\). This applies to the asymptotic distribution of \(\widehat a\), \(\widehat b\), and \(\widehat c\). The source of this characteristic is the singularity appearing in the expectation of logarithmic derivatives. When less than 3 parameters are to be estimated the problem arises whenever \(\widehat a\) intrudes.
For the 3 parameter case, a new expression is developed for the asymptotic variance of \(\widehat c\). Lastly, whereever possible simulation studies are invoked for verification purposes.

MSC:

62F10 Point estimation
62E20 Asymptotic distribution theory in statistics
62N05 Reliability and life testing
PDFBibTeX XMLCite