Robustness and precision of parametric and distribution-free tolerance limits for two lifetime distributions

Exact parametric tolerance limits or confidence limits on reliability are not available for the gamma distribution, and it is difficult to determine the approximate methods which are accurate for all parameter values. The precision lost by using the distribution-free tolerance-limit method based on the first order statistic, compared to using an approximate gamma tolerance limit method, is studied. The robustness of the approximate gamma tolerance limit when the true model is Weibull and the robustness of a Weibull tolerance limit when the true model is gamma are also studied. The efficiency of the distribution-free method ranges from about 0.60 to 0.90 in the range of values studied. Neither the Weibull nor the gamma method is very robust for the alternate model, but the Weibull tolerance limit is conservative for the gamma model for parameter values commonly encountered in life-testing problems and may be preferable to the distribution-free method in this case