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Estimation of the shape and scale parameters of the Weibull distribution. (English) Zbl 0111.15803
Summary: Estimates \(\hat c\) and \(\hat b\) are proposed for the shape parameter \(c\) and the scale parameter \(b\) of the Weibull distribution on the assumption that the location parameter is known: \(\hat c\) obtained by first finding an estimate \(\hat d\) of \(1/c\), and then setting \(\hat c = 1/\hat d\). When \(b\) is unknown, \(\hat d\) is a consistent and non-negative estimate of \(d\), with a bias which tends to vanish as the sample size increases and with an asymptotic efficiency of about 55%. When \(b\) is known, \(\hat d\) is an unbiased, non-negative and consistent estimate of \(d\), and its efficiency is approximately 84%. An estimate \(\widehat{\log b}\) of \(\log b\) is found. Its asymptotic efficiency is 95%. It is proposed that \(\exp(\widehat{\log b})\) be used to estimate \(b\).

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