A METHOD FOR ESTIMATING PARAMETERS AND QUANTILES OF DISTRIBUTIONS OF CONTINUOUS RANDOM-VARIABLES

Classical estimation methods (e.g., maximum likelihood and method of moments) work well, for example, in cases where the distribution belongs to the exponential family. In many other cases, they may not exist, they may be computationally difficult, and/or they may produce unsatisfactory results. Examples of cases where the classical methods encounter difficulties include cases where: (a) the range of the random variable depends on the parameters, (b) the moments either do not exist or they are complicated functions of the parameters, (c) the likelihood function either may have no local maximum or may be unbounded, and (d) the estimates require multidimensional numerical search. In this paper we propose a method for estimating the parameters and quantiles of continuous distributions. The estimators are obtained in two steps. First, some elemental estimates are obtained by solving equations relating the cumulative distribution function or the survival function to their percentile values for some elemental subsets of the observations. These elemental estimates are then combined in a suitable way to obtain a statistically more efficient estimates of the parameters and quantiles. The method has several advantages including: (a) the estimates are easy to derive and to compute, (b) the estimates are unique and well-defined for all parameter and sample values, and (c) the estimates exist in cases where other classical estimators do not exist. The method is illustrated by several examples of continuous univariate and multivariate random variables. In some cases (e.g., when X is uniform (0, theta)), the method gives rise to a uniformly minimum variance unbiased estimate. In other cases, where analytical comparisons cannot be made, we perform simulation experiments to compare the proposed estimates with the classical ones. The results seem to indicate that the method performs well for estimating both the parameters and the quantiles of several continuous univariate and multivariate distributions.