On the Nonconsistency of Maximum Likelihood Nonparametric Density Estimators

One criterion proposed in the literature for selecting the smoothing parameter(s) in RosenblattParzen nonparametric constant kernel estimators of a probability density function is a leave-out-one-at-a-time nonparametric maximum likelihood method. Empirical work with this estimator in the univariate case showed that it worked quite well for short tailed distributions. However, it drastically oversmoothed for long tailed distributions. In this paper it is shown that this nonparametric maximum likelihood method will not select consistent estimates of the density for long tailed distributions such as the double exponential and Cauchy distributions. A remedy which was found for estimating long tailed distributions was to apply the nonparametric maximum likelihood procedure to a variable kernel class of estimators. This paper considers one data set, which is a pseudo-random sample of size 100 from a Cauchy distribution, to illustrate the problem with the leave-out-one-at-a-time nonparametric maximum likelihood method and to illustrate a remedy to this problem via a variable kernel class of estimators.