A stationary two-component markovian process
(Xn; Sn)n>1 is considered with the first component observable
and the second one non-observable. The problem of filtering a
stochastic signal (Sn)n>1 from the mixture with a noise by observations
Xn 1 = X1; ¢ ¢ ¢ ;Xn is solved in non-parametric uncertainty
regarding the distribution of the desired signal. This means that
the probabilistic parametric model of the useful signal (Sn) is assumed
to be completely unknown. Under these assumptions, in
general, it is impossible to build an optimal Bayesian estimator.
However, for a more restricted class of observation models, in
which the conditional density f(xnjsn; xn¡1
1 ) belongs to conditionally-
exponential family of distribution densities, the Bayesian
estimator is a solution of some nonrecurrent equation which depends
only on probabilistic characteristics of the observable process
(Xn). These unknown characteristics can be restored from
observations Xn
1 by using stable non-parametric estimation procedures
adapted to dependent data. In detail the nonlinear multiplicative
observation model with non-Gaussian noise is considered,
and the non-parametric estimator of an unknown gain coefficient
is constructed. The results of the model experiment show
that the quality of the non-parametric estimator, built for nonlinear
observation model, is slightly worse than the quality of
the Bayesian estimator (which is naturall), but better than the
quality of optimal linear estimator. When building a stable nonparametric
procedures the choice of smoothing and regularization
parameters plays the crucial role. We propose the optimal choice
of these parameters, which leads to an automatic algorithm of
non-parametric filtering.
