Regularization of Positive Signal Nonparametric Filtering in Multiplicative Observation Model

A solution to the problem of useful random signal extraction from a mixture with a noise in the multiplicative observation model is proposed. Unlike conventional filtering tasks, in the problem under consideration it is supposed that the distribution (and the model) of the useful signal is unknown. Therefore, in this case one cannot apply such well-known techniques like Kalman filter or posterior Stratonovich-Kushner evolution equation. The new paper is a continuation and development of the author's article, reported at the First ISNPS Conference (Halkidiki' 2012), where the filtering problem of positive signal with the unknown distribution had been solved using the generalized filtering equation and nonparametric kernel techniques. In the present study, new findings are added concerning the construction of stable procedures for filtering, the search for optimal smoothing parameter in the multidimensional case and some of the convergence results of the proposed techniques. The main feature of the problem is the positive distribution support. In this case, the classical methods of nonparametric estimation with symmetric kernels are not applicable because of large estimator bias at the support boundary. To overcome this drawback, we use asymmetric gamma kernel functions. To have stable estimators, we propose a regularization procedure with a data-driven optimal regularization parameter. Similar filtering algorithms can be used, for instance, in the problems of volatility estimation in statistical models of financial and actuarial mathematics.