An F approximation for two‐parameter weibull and log‐normal tolerance bounds based on possibly censored data

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Abstract
An approximation suggested in Mann, Schafer and Singpurwalla [18] for obtaining small‐sample tolerance bounds based on possibly censored two‐parameter Weibull and lognormal samples is investigated. The tolerance bounds obtained are those that effectively make most efficient use of sample data. Values based on the approximation are compared with some available exact values and shown to be in surprisingly good agreement, even in certain cases in which sample sizes are very small or censoring is extensive. Ranges over which error in the approximation is less than about 1 or 2 percent are determined. The investigation of the precision of the approximation extends results of Lawless [8], who considered large‐sample maximum‐likelihood estimates of parameters as the basis for approximate 95 percent Weibull tolerance bounds obtained by the general approach described in [18]. For Weibull (or extreme‐value) data the approximation is particularly useful when sample sizes are moderately large (more than 25), but not large enough (well over 100 for severely censored data) for asymptotic normality of estimators to apply. For such cases simplified efficient linear estimates or maximum‐likelihood estimates may be used to obtain the approximate tolerance bounds. For lognormal censored data, best linear unbiased estimates may be used, or any efficient unbiased estimators for which variances and covariances are known as functions of the square of the distribution variance.