THE NECESSARY CONDITION FOR COINCIDENCE OF LS AND AITKEN ESTIMATIONS OF THE HIGHER COEFFICIENT OF THE LINEAR REGRESSION MODEL IN THE CASE OF CORRELATED DEVIATIONS

Abstract

At the paper a linear regression model whose function has the form $f (x)=ax + b$, $a$ and $b$ — unknown parameters, is studied. Approximate values (observations) of functions $f(x)$ are registered at equidistant points $x_0,x_1,...,x_n$ of a line segment. It is also assumed that the covariance matrix of deviations is the symmetric Toeplitz matrix. Among all Toeplitz matrices, a family of matrices is selected for which all diagonals parallel to the main, starting from the $(k+1)$th, are zero, $k=n/2$, $n$ — even. Elements of the main diagonal are denoted by $\lambda$, elements of the $k$th diagonal are denoted by $c$, elements of the $j$th diagonal are denoted by $c_{k-j}$, $j=1,2,...,k-1$. The theorem proved in the article states that the following condition on the elements of such covariance matrix $c_j=\bigl(k/(k+1)\bigr)^j c$, $j=1,2,...,k-1$, is necessary for the coincidence of the LS and Aitken's estimations of the parameter $a$ of this model. Values $\lambda$ and $c$ are any that ensure the positive definiteness of such matrix.