Nonparametric Probability Density Estimation by Discrete Maximum Penalized- Likelihood Criteria

A nonparametric probability density estimator is proposed that is optimal with respect to a discretized form of a continuous penalized-likelihood criterion functional. Approximation results relating the discrete estimator to the estimate obtained by solving the corresponding infinite-dimensional problem are presented. The discrete estimator is shown to be consistent. The numerical implementation of this discrete estimator is outlined and examples displayed. A simulation study compares the integrated mean square error of the discrete estimator with that of the well-known kernel estimators. Asymptotic rates of convergence of the discrete estimator are also investigated.