A non-classical parameterization for density estimation using sample moments

Abstract

Probability density estimation is a core problem in statistics and data science. Moment methods are an important means of density estimation, but they are generally strongly dependent on the choice of feasible functions, which severely affects the performance. In this paper, we propose a non-classical parametrization for density estimation using sample moments, which does not require the choice of such functions. The parametrization is induced by the squared Hellinger distance, and the solution minimizing it, which is proved to exist and be unique subject to a simple prior that does not depend on data, and which can be obtained by convex optimization. Statistical properties of the density estimator, together with an asymptotic error upper bound, are proposed for the estimator by power moments. Simulation results validate the performance of the estimator by a comparison to several prevailing methods. The convergence rate of the proposed estimator is proved to be \(m^{-1/2}\) (m being the number of data samples), which is the optimal convergence rate for parametric estimators and exceeds that of the nonparametric estimators. To the best of our knowledge, the proposed estimator is the first one in the literature for which the power moments up to an arbitrary even order exactly match the sample moments, while the true density is not assumed to fall within specific function classes.