Regularized nonparametric filtering of signal with unknown distribution in nonlinear observation model

Resumo

This paper solves the problem of usefull random signal filtering in a discrete-time nonlinear observation model. As such model, we consider the multiplicative model with nonnegative signals and noises. In contrast to standard filtering problems, our statement assumes that the distribution and equation of the useful signal are unknown. To solve the nonlinear filtering problem, the idea is to employ a generalized optimal filtering equation with a feature that the optimal estimator is expressed only through characteristics of the observed process. The role of such characteristics in the equation belongs to the logarithmic derivative of the conditional multidimensional density of observations. We find the solution of the equation using nonparametric kernel estimation methods with nonsymmetrical gamma kernel functions defined on the positive semiaxis. Moreover, we establish convergence conditions for the kernel estimator of the logarithmic derivative of the multidimensional density by dependent observations, as well as derive explicit formulas for optimal bandwidths and construct a stable (regularized) filtering estimator.