AIPENOVA AZIZA SRAILOVNA NONPARAMETRIC ESTIMATION OF DERIVATIVES OF DENSITIES USING BESOV NORMS ABSTRACT of the dissertation of Aipenova A.S. for the scientific degree of Doctor of Philosophy (PhD) in the specialty 6D060100 - Mathematics Relevance of the theme of dissertation. Among the tools of mathematical statistics, nonparametric estimation occupies a central place. It replaced the parametric estimates, which are much inferior to it in accuracy and flexibility. The first results in this area have appeared about 50 years ago in the works of Rosenblatt and Parsen. Currently, the basic theory is well developed, but at the same there are a number of areas where significant improvement is possible. Often the improvement can be achieved through the application of the theory of functions, which are not well known among statisticians. Considered dissertation belongs to the class of works in this direction. At the same time can improve even acknowledged classical results. The asymptotic properties of ̂ have been studied by, among others, Bhattacharya (1967), Schuster (1969) and Silverman (1978). Schuster (1969) showed that the uniform continuity of was necessary for uniform consistency, under the condition ∑ . This condition is substantially weakened in Silverman (1978). Singh (1977, 1979, 1987) show that it is possible to reduce bias and improve the mean integrated squared error (MISE) of ̂ by considering restrictions on the class of kernels used in its construction. Similar efforts have been undertaken by Muller (1984), Coppejans and Sieg (2005), Bearse and Rilstone (2009) and Henderson and Parmeter (2012). In the theory of nonparametric estimation, there are many different methods. Recently, Mynbaev and Martins-Filho (2010) proposed a class of nonparametric kernel density estimators that attains bias reduction relative to ̂ by imposing global higher order Lipschitz conditions on . In this dissertation we extend the nonparametric class of kernel density estimators proposed by Mynbaev and Martins-Filho (2010) to the estimation of order density derivatives. Contrary to some existing derivative estimators, the estimators in our proposed class have a full asymptotic characterization, including uniform consistency, asymptotic normality and exact convergence rates. In addition, we discuss the criterion of the bandwidth choice that minimizes an asymptotic approximation for the estimators' integrated squared error. Further, we give an integral representation for the bias and exact orders of the bias and variance of the proposed derivative estimator. Also, we consider in the dissertation bias reduction using higher order kernels which is popular in statistics, and provide construction of higher order kernels. In addition, we extend the nonparametric estimation method to estimation of a differential expression. It determines the relevance of the theme of the dissertation. Research objective. Research objective is to extend the nonparametric class of kernel density estimators proposed by Mynbaev and Martins-Filho (2010) to the estimation of order density derivatives and to construct new higher-order kernels that reduce bias. We show the effectiveness of the proposed methods by the Monte-Carlo method. Subject of research. Subject of research is the nonparametric class of kernel density estimators proposed by Mynbaev and Martins-Filho (2010) and construction of higher order kernels by way of multiplication of densities by polynomials. Purpose of research: Purpose of research is to obtain exact asymptotic characterization for density derivative estimators and show the effectiveness of the proposed method by the Monte-Carlo method. Scientific novelty of research: The dissertation presents the following results: The first result: the nonparametric class of density estimators proposed by Mynbaev and Martins-Filho (2010) is extended to the estimation of -order density derivatives. The second result: the asymptotic properties of the proposed derivative estimator are found, including the uniform consistency, asymptotic normality and exact convergence rates. The third result: the integral representation for the bias of the proposed derivative estimator is found. The fourth result: the criterion of the optimal bandwidth choice is proved. The fifth result: the exact orders of the bias and variance of the proposed derivative estimator are obtained. The sixth result: the effectiveness of the proposed method is shown by the Monte-Carlo method. The seventh result: the method for improving bias in kernel density estimation using higher order kernels is obtained and the effectiveness of the method is shown by the Monte-Carlo method. The eighth result: the nonparametric estimation method is extended to the estimation of differential expressions. The nineth result: the method to construct higher order kernels is obtained. In addition, the expressions and the moments for the higher order kernels are obtained. Theoretical significance of research: Results of the dissertation make a significant contribution to the theory nonparametric kernel density estimation. Practical significance of the results: The work can be used in mathematical statistics and in numerous applications in engineering, finance, medicine and biology. Structure and volume of the dissertation: The dissertation consists of introduction, three chapters, conclusion and list of references. The work is presented in 76 pages, including 6 figures, 10 tables and a list of references of the 47 items.