Probability density estimation is a core problem of statistics and signal processing. 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 of it, which is proved to exist and be unique subject to a simple prior that does not depend on data, and 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. Applications of the proposed density estimator in signal processing tasks are given. Simulation results validate the performance of the estimator by a comparison to several prevailing methods. 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.

翻译：概率密度估计是统计学与信号处理领域的核心问题。矩方法作为密度估计的重要途径，通常严重依赖于可行函数的选择，这极大地影响了其性能。本文提出一种基于样本矩的密度估计非经典参数化方法，该方法无需进行此类函数选择。该参数化由平方Hellinger距离诱导产生，其解在引入不依赖数据的简单先验条件下被证明存在且唯一，并可通过凸优化求解。本文给出了该密度估计量的统计特性，并基于幂矩推导了其渐近误差上界。进一步展示了该密度估计量在信号处理任务中的应用。通过与多种主流方法的比较，仿真结果验证了该估计量的性能。据我们所知，所提出的估计量是文献中首个在无需假设真实密度属于特定函数类的前提下，能精确匹配任意偶数阶样本幂矩的方法。