Moment and maximum likelihood estimators for Weibull distributions under length- and area-biased sampling

Many of the most popular sampling schemes used in forestry are probability proportional to size methods. These methods are also referred to as size-biased because sampling is actually from a weighted form of the underlying population distribution. Length- and area-biased sampling are special cases of size-biased sampling where the probability weighting comes from a lineal or areal function of the random variable of interest, respectively. Often, interest is in estimating a parametric probability density of the data. In forestry, the Weibull function has been used extensively for such purposes. Estimating equations for method of moments and maximum likelihood for two- and three-parameter Weibull distributions are presented. Fitting is illustrated with an example from an area-biased angle-gauge sample of standing trees in a woodlot. Finally, some specific points concerning the form of the size-biased densities are reported.