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Comparative Analysis of M and MM-Estimates in Regression Models with Stochastic Regressors

Student: Sergey Demchenko

Supervisor: Elena R. Goryainova

Faculty: Faculty of Computer Science

Educational Programme: Applied Mathematics and Information Science (Bachelor)

Final Grade: 10

Year of Graduation: 2017

In regression analysis the use of classical methods like the ordinary least squares or the least absolute deviation method would not be acceptable in solving problems containing extreme observations, so-called outliers. Thus, robust analysis was invented in the 1960-1970’s to solve this kind of problems. The main idea of robust analysis is the development of estimates that are resistant to outliers. And from another hand, this robust estimations must show a result that is close to classical methods on datasets which are not containing atypical observations. In the course of the diploma work, the several main robust estimations such as M-, MM-, LTS-, LMS- and S-estimations are described. Moreover, the robust machine learning method RANSAC is described in this paper. Then, using the Monte Carlo method, a numerical comparative analysis is carried out of two robust M- and MM-estimates, in order to identify advantages and disadvantages each of them. Thus, to solve this problem, two models of one-dimensional and multidimensional regressions are considered. For each case, the error distributions with “heavy” tails, such as Tukey and Cauchy distributions, are modeled. Also, a comparative analysis is carried out on the simulated data containing a cluster of outliers. In each case, recommendations of using robust estimates are given. In the final chapter, all considered estimates are applied to three real datasets.

Full text (added May 28, 2017)

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