Improved Maximum Likelihood Estimation for the Weibull Distribution Under Length-Biased Sampling

We consider the estimation of the parameters of the Weibull distribution when the data arise from "length-biased" sampling. Specifically, the appropriate weighted density is formulated and we analyze the finite-sample properties of the maximum likelihood estimators for its parameters. The analytic Cox-Snell "corrective" approach is used to reduce the biases of these estimators, and we find that this can be done effectively and without detrimental consequences for the mean squared errors. Bootstrap bias-correction is also found to be effective. Simulation results also illustrate the severe consequences of failing to allow for "length-biased" sampling, even in very large samples.