Tkachenko O. Polynomial methods and tools for estimating regression parameters using Non-Gaussian error models

The work was solved the scientific and technical problem of development and application of mathematical and computer modeling methods for processes of estimating regression parameters under the condition of non-Gaussian character of their errors. A new approach to the adaptive finding of parameter estimates based on the use of higher-order regression models for describing the random component is proposed, which made it possible to simply take into account deviations from Gaussian idealization in the synthesis and analysis of the effectiveness of the resulting methods. Based on the apparatus of stochastic Kunchenko polynomials and modifications of regression models obtained using instantaneous-cumulative description, the synthesis of computational methods of adaptive parameters estimation of regression models for linear, polynomial, and nonlinear type is carried out. It is shown that the general problem can be algorithmically reduced to solving a system of nonlinear stochastic equations using a numerical Newton-Rafson iterative procedure. The properties of polynomial estimates under the condition of the asymmetric and symmetric character of non-Gaussian errors are analyzed and their efficiency is compared with classical estimates of least squares and maximum likelihood. It is shown that the application of the proposed approach reduces the variance of polynomial estimates compared to the known least squares estimates, and the increase in accuracy is achieved by taking into account the non-Gaussian regression errors. The developed software package, its structure, and set of modules provide both a direct solution to the problem of finding adaptive estimates of the parameters of regression dependencies and the implementation of computer statistical modeling based on the Monte Carlo method and bootstrap analysis. The set of obtained results of statistical modeling confirms the theoretically proven efficiency of polynomial estimates. The example of the error model with exponential power distribution shows that in the absence of a priori information about the values of regression error parameters, adaptive polynomial estimates can be more accurate even compared to adaptive estimates of maximum plausibility.