SUFFICIENT CONDITION FOR COINCIDENCE OF THE LS AND AITKEN ESTIMATIONS OF PARAMETER OF QUADRATIC REGRESSION IN CASE HETEROSCEDASTIC DEVIATIONS

Abstract

In the paper in case heteroscedastic independent deviations a regression model whose function has the form $f(x) = ax^2+bx+c$, where $a$, $b$ and $c$ are unknown parameters, is studied. Approximate values (observations) of functions $f(x)$ are registered at equidistant points of a line segment. The theorem which is proved at the paper gives a sufficient condition on the variance of the deviations at which the Aitken estimation of parameter $a$ coincides with its estimation of the LS in the case of odd number of observation points and bisymmetric covariance matrix. Under this condition, the Aitken and LS estimations of $b$ and $c$ will not coincide. The proof of the theorem consists of the following steps. First, the original system of polynomials is simplified: we get the system polynomials of the second degree. The variables of both systems are unknown variances of deviations, each of the solutions of the original system gives a set variances of deviations at which the estimations of Aitken and LS parameter a coincide. In the next step the solving of the original system polynomials is reduced to solving an equation with three unknowns, and all other unknowns are expressed in some way through these three. At last it is proved that there are positive unequal values of these three unknowns, which will be the solution of the obtained equation. And all other unknowns when substituting in their expression these values will be positive.