Method of least squares using exhaustive search

Abstract

The estimation of an unknown finite-dimensional parameter of a nonlinear nonstationary regression using exhaustive search through the sum of the squares of the residuals in a finite lattice of points that approximates a known bounded set of values of this parameter is considered. A technique for the a priori analysis of the accuracy of such an estimation is proposed for the case of partial statistical uncertainty in which some standard regression function possessing a connected structure that approximates the true regression function with a known error and a numerical sequence that majorizes the spectrum of the covariance matrices of the random vectors formed by the first observation errors are known. The technique is based on an analysis of the semimetric generated by the standard regression function and, without having to use a lengthy Monte-Carlo simulation experiment, makes it possible to obtain robust upper limits (which may be strengthened if it is further assumed that the observation noise is normal) for the estimation error quantiles and for risks relative to losses of an extremely general form. The use of the technique is illustrated by an example in which a Burgers [slip] vector is estimated in one version of an extremal correlation navigation search system based on the approximation of a topographical field by means of two-dimensional Bernstein polynomials.