Estimation of lower tail quantiles of Weibull probability distributions for lumber strength

When the 3-parameter Weibull distribution, which contains a lower-bound parameter, is fitted to a set of lumber strength data there is an implication that the strength of a piece of lumber will never be less than the value of that parameter. This can be rather troubling and has led to the adoption of the 2-parameter Weibull model in some cases, even though it does not always yield a good fit over the entire range of the distribution. This paper presents and compares three different estimation methods for fitting the 2-parameter Weibull model to lumber strength data, but using only the smallest data values. On the basis of a simulation study, it is shown that the 2-parameter Weibull model can yield good quantile estimates in the lower tail of the distribution, even when the true population is 3-parameter Weibull with a lower-bound parameter that is quite different from zero. Recommendations are made with respect to the choice of a parameter estimation method for use in conjunction with the 2-parameter, Weibull model.

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