The condition for the coincidence of LS and Aitken estimations of both parameters of the linear regression model in the case of tridiagonal bisymmetric covariance matrix

Abstract

At the paper a linear regression model whose function has the form f(x) = ax + b, where a and b are unknown parameters, is studied. Approximate values (observations) of functions f(x) are registered at equidistant points of a line segment. It is also assumed that the covariance matrix of deviations is a tridiagonal bisymmetric matrix. In the theorem proved in the paper, necessary and sufficient condition for the elements of such matrix is found, which ensures the equality of LS and Aitken estimations both parameters of this model simultaneously. All elements are expressed in terms of two, which correspond to the variances of random deviations at the first two observation points. A sufficient condition for the connection between these two elements of the matrix to be positive definite was also found.