Asymptotic Efficiency of a Nonlinear Filter Based upon the Stochastic Approximation Procedure and Its Improvement

Abstract

The performance of a state estimation algorithm of a nonlinear stochastic dynamical system based upon the stochastic approximation procedure is discussed from the viewpoint of asymptotic efficiency.
Using Fabian's theorem, we first derive the asymptotic distribution of the estimated error (x_n-x_n), and obtain the asymptotic efficiency of the state estimation algorithm based upon the stochastic approximation.
Then, we show that it is necessary to transform the deviation (y_n+1-f_n+1(g_n(x_n))) by the transformation function corresponding to the probability distribution function of the observation noise in order to improve the asymptotic efficiency.
Finally, the optimal transformation function and the improvement factor of the asymptotic efficiency are given for several distributions.