EQUALITY 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 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 $λ$, 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 at the paper states that if $c_j = (k/(k + 1))^j c$, $j = 1, 2,..., k−1$, that the LS estimation and the Aitken estimation of the $a$ parameter of this model coincide for any values $λ$ and $c$, which provide the positive definiteness of the resulting matrix.