Identification of Possible Estimates Areas for Parameters of Fully connected Linear Regression Models

This article is devoted to the study of fully connected linear regression models, in which the observed variables contain errors, and the pairs of true variables are interconnected by linear functional dependencies. When estimating fully connected regressions, the main problem is the correct choice of the error variances ratios of the variables. If the choice is made incorrectly, then the fully connected regression estimates will be biased. The purpose of this article is to find the dependence of main parameters possible estimates areas on the possible error variances ratios of the variables in fully connected regressions. For the first time, with the help of matrix algebra elements, the inverse problem is solved - analytical dependences of the error variances ratios of variables on the main parameters are obtained. These dependences make it possible to identify the parameters possible estimates areas in which the necessary condition for the extremum of the objective function is satisfied. It is proved that, under certain conditions, for any error variances ratios of the variables, the parameters estimates always lie inside an open convex polygon located only in one of the orthants of the multidimensional space. In this case, the signs of the estimates always agree with the signs of the corresponding correlation coefficients. A numerical experiment was carried out, confirming the correctness of the results obtained.