Least squares method of relative errors for parameter estimation of systems under non-classical assumptions
DOI:
https://doi.org/10.17721/1812-5409.2025/2.31Keywords:
regression model, parameter estimation, non-classical assumption, Moore – Penrose pseudo-inversion operator, least squares method, weighted least squares method, least squares method of relative errorsAbstract
The problem of optimal estimation of regression model parameters based on available observations of the dependent variable and all regressors is considered. It is suggested to use the least squares method of relative residuals to solve this problem. The estimate of this method is the point at which the sum of squares of relative residuals of the regression model reaches its minimum, not the sum of squares of absolute residuals of the model, as in the ordinary least squares method, which can be called the least squares method of absolute residuals. The value of the quality functional of the considered method no longer depends on the units of measurement of the available observations.
In the paper, the explicit form of the representation of the optimal estimate of the least squares method of relative errors in a situation where the classical assumption is valid, which guarantees its uniqueness, is extended to the case when this assumption is violated and the estimate is no longer unique. In the latter situation, there is a need to use the Moore-Penrose matrix pseudo-inversion operator. Corresponding explicit expressions for the estimation error by this method are also presented for both cases.
In addition, as a consequence, for the weighted least squares method, in the case of using an arbitrary positive definite matrix as the weight matrix, for both of the above-mentioned situations, explicit forms of representation of the corresponding optimal estimate of this method and the expression for the estimation error are also given by analogy.
Pages of the article in the issue: 197 - 201
Language of the article: Ukrainian
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