MINIMAX ROOT–MEAN–SQUARE ESTIMATES OF MATRIX PARAMETERS IN LINEAR REGRESSION PROBLEMS UNDER UNCERTAINTY

Abstract

The issues of parameter estimation in linear regression problems with random matrix coefficients were researched. Given that random linear functions are observed from unknown matrices with random errors that have unknown correlation matrices, the problems of guaranteed mean square estimation of linear functions of matrices were investigated. The estimates of the upper and lower guaranteed standard errors of linear estimates of observations of linear functions of matrices were obtained in the case when the sets are found, for which the unknown matrices and correlation matrices of observation errors are known. It was proved that for some partial cases such estimates are accurate. Assuming that the sets are bounded, convex and closed, more accurate two-sided estimates have been gained for guaranteed errors. The conditions when the guaranteed mean squared errors approach zero as the number of observations increases were found. The necessary and sufficient conditions for the unbiasedness of linear estimates of linear functions of matrices were provided. The notion of quasi-optimal estimates for linear functions of matrices was introduced, and it was proved that in the class of unbiased estimates, quasi-optimal estimates exist and are unique. For such estimates, the conditions of convergence to zero of the guaranteed mean-square errors were obtained. Also, for linear estimates of unknown matrices, the concept of quasi-minimax estimates was introduced and it was confirmed that they are unbiased. For special sets, which include an unknown matrix and correlation matrices of observation errors, such estimates were expressed through the solution of linear operator equations in a finite-dimensional space. For quasi-minimax estimates under certain assumptions, the form of the guaranteed mean squared error of the unknown matrix was found. It was shown that such errors are limited by the sum of traces of the known matrices. An example of finding a minimax unbiased linear estimation was given for a special type of random matrices that are included in the observation equation.