Estimation of Weibull Parameters from Two-Order Statistics

Summary The use of a computationally simple statistic for estimating, from small samples, the important reciprocal-shape parameter, γ, of the Weibull distribution is described. The estimator uses two suitably selected order statistics from a sample of N failure times. The choice of the two has been made to minimize the variance of the estimator; however, concise formulae are given to permit the use of any two order statistics in the case of censored samples. The statistic is unbiased, its efficiency relative to the Cramér–Rao bound appears to approach 70 per cent and its exact distribution function is given. The statistic is applied to hypothesis tests of the reciprocal-shape parameter when the location parameter is unknown. The relationship between this test and other tests for Weibull parameters are discussed in the Introduction.