Mathematics and Computers in Simulation 32 (1990) 197-202 North-Holland 197 zyxwvutsrqp THE EFFECTS OF MISSPECIFICATION IN ESTIMATING THE PERCENTILES OF SOME TWO- AND THREE-PARAMETER DISTRIBUTIONS zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA J. BA I, A .J. JA KEMA N and M. McALEER * Centre zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA for Resource and Environmental Studies, Australian National University, Canberra, ACT, Australia * Department of Statistics, Australian National University and Institute of Social and Economic Research, Osaka University, Japan zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1 INTRODUCTION The gamma, Weibull and lognormal distributions have been used successfully in modelling natu- ral phenomena in areas such as reliability and life testing [l] and [2], hydrology [3], and air quality management [4] and [5]. Two (a-) and three (3-) p arameter versions of these distributions have been used because they are parsimonious in considering the shape, scale and location of the distribution, but still sufficiently flexible in fitting real data. In general, it is not known which of the 2- or S-parameter distributions is appropriate, and conventional wisdom regarding underfitting or overfitting may not be a good guide to selecting one of t,hese distributions, especially if upper percentiles of the distribution are required. It is therefore important to take account of the sorts of errors that might be made in fitt’ing the distribution. Specifically, the consequences of misspecifying the distribution should be evaluated. Such misspecifications arise when a 2- (3-) p arameter distribution is estimated when the 3- (2-) parameter version is correct. For a given distribution, the statistical decision might be to simply estimate the 2- or 3-parameter variant of the distribution, or to use discrimination and/ or testing criteria to choose one of the two distributions. The three discrimination methods considered in this paper are the likelihood ratio (LR) test, Akaike’s information criterion (AIC), and Schwarz’s information criterion (SIC) based on Bayesian methods. The primary aim of the experiments is to observe the magnitudes of the errors in the upper percentiles obtained by fitting the incorrect distribution (by overfitting or underfitting), by fitting the correct distribution, and by fitting a distribution that is selected by the LR method, AIC or SIC. The experiments are conducted for three different distributions ( the gamma, Weibull and lognormal ) and different parameter sets, especially for different values of the shape parameter. 2 DISTRIBUTION FUNCTIONS AND STATISTICAL CRITERIA For a sample of n independently and identically distributed random observations, the log- likelihood functions for the 3-parameter gamma, Weibull and lognormal distributions are well known, see e.g. [4]. The three parameters of interest are the scale p, shape cy and location y. The 2-parameter versions of these functions are obtained by setting the location parameter y to zero in each case. The density ‘The third author wishes to acknowledge the research support of the Australian Research Council. 037%4754/90/$3.50 0 1990, IMACS/EI sevier Science Publishers B.V. (North-Holland)