# 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

• Marta Savkina Institute of Mathematics of NASU, Kyiv, Ukraine
Keywords: least square method, regression model, Aitken estimation

### 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.

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Savkina M. Yu. Equality of least squares method and Aitken senior coefficient estimates of the linear regression model in the case of correlated deviations. Journal of Numerical and Applied Mathematics. 2021. No. 2 (136). P. 64–72.

Published
2023-02-01
Section
Articles