Suhakova O. Adaptive methods of statistical analysis.

The thesis is devoted to the development of adaptive statistical analysis algorithms for inhomogeneous data. The methods are developed for Euclidean parameters and nonparametric part estimation in the models of symmetric distribution observed with admixture, the mixture of two symmetric distributions, semiparametric and nonparametric models of mixture with varying mixing probabilities. Asymptotic behavior of obtained estimators is investigated. Advantages of adaptive estimation are shown in comparison with simple non-adaptive approaches. Adaptive methods of empirical Bayesian classification are developed in the case, when the teaching sample is taken form a mixture with different components. Convergence rates of the obtained classifiers to the theoretically optimal ones are investigated. Asymptotic behavior of DP-estimators for change-points detection is investigated in the case of many change-points in a sequence of independent observations. It is shown that DP-estimates possess optimal properties, which were previously established only for the case of one change-point. New estimates are obtained for the distributions of sums of a random number of differently distributed random variables. These results can be applied to the statistical analysis of medical, biological and sociological data.