Empirical models for evaluating errors in fitting extremes of a probability distribution

The paper examines how empirical models can be constructed to describe the dependence of the error in fitting data to parametric models of probability distributions on the type of distribution, sample size, parent parameter values and percentiles of interest. Such models are important in evaluating the goodness-of-fit of some distributional forms to air pollution data and, once calibrated, require only simple calculations. The procedure and results are described for the three-parameter gamma distribution, although they can also be readily applied to other distributions such as the Weibull and lognormal. Monte Carlo simulations are used to infer the true errors used as dependent variables to calibrate the parameters of the empirical model, and a variety of model selection criteria are used to examine the performance of the model. The use of such models can dramatically improve the efficiency of assessment procedures in air quality management.