Optimal fourier-hermite expansion for estimation

The purpose of the paper is to present a systematic method for developing an approximate recursive estimator which is optimal for the given structure and approaches the best estimate, when the order of approximation increases.The minimal variance estimate is projected onto the Hilbert subspace of all Fourier-Hermite (FH) series, driven by the observations, with the same given index set. The projection results in a system of linear algebraic equations for the FH coefficients, the parameters of the desired approximate estimator.The estimator consists of finitely many Wiener integrals of the observations and a memoryless nonlinear postprocessor. The postprocessor is an arithmetic combination of the Hermite polynomials evaluated at the Wiener integrals. A couple of recursive methods for calculating the Wiener integrals are included.