A nonlinear adaptive estimation recursive algorithm

A nonlinear, adaptive and recursive algorithm is derived for estimating an unknown probability density given a sequence of independent samples from the unknown density. An expansion of the unknown density in terms of a known and finite set of orthogonal functions is utilized and a Bayesian recursive learning procedure is derived for learning the coefficients of the expansion. The algorithm eliminates the need for quantization of the unknown parameter space. Adaptive estimators of population moments are also derived as known linear combinations of the recursively obtained expansion coefficients.