Conditional Confidence Interval for the Scale Parameter of a Weibull Distribution

Abstract

A two-sided conditional confidence interval for the scale parameter θ of a Weibull distribution is constructed. The construction follows the rejection of a preliminary test for the null hypothesis: θ = θ0 where θ0 is a given value. The confidence bounds are derived according to the method set forth by Meeks and D’Agostino (1983) and subsequently used by Arabatzis et al. (1989) in Gaussian models and more recently by Chiou and Han (1994, 1995) in exponential models. The derived conditional confidence interval also suits non large samples since it is based on the modified pivot statistic advocated in Bain and Engelhardt (1981, 1991). The average length and the coverage probability of this conditional interval are compared with whose of the corresponding optimal unconditional interval through simulations. The study has shown that both intervals are similar when the population scale parameter is far enough from θ0. However, when θ is in the vicinity of θ0, the conditional interval outperforms the unconditional one in terms of length and also maintains a reasonably high coverage probability. Our results agree with the findings of Chiou and Han and Arabatzis et al. which contrast with whose of Meeks and D’Agostino stating that the unconditional interval is always shorter than the conditional one. Furthermore, we derived the likelihood ratio confidence interval for θ and compared numerically its performance with the two other interval estimators.