Неклассические методы вероятностного и статистического анализа моделей смеси распределений Панов Владимир Александрович

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Панов, Владимир Александрович.
Неклассические методы вероятностного и статистического анализа моделей смеси распределений = Non-classical methods of probabilistic and statistical analysis of the mixture models : Non-classical methods of probabilistic and statistical analysis of the mixture models : диссертация ... доктора физико-математических наук : 01.01.05 / Vladimir Panov; [Место защиты: ФГАОУ ВО «Национальный исследовательский университет «Высшая школа экономики»]. - Москва, 2022. - 221 с. : ил.
Оглавление диссертациидоктор наук Панов Владимир Александрович
Contents
Introduction
Acknowledgments
1 Semiparametric estimation in mixture models
1.1 Inference in normal variance-mean Gaussian mixtures
1.1.1 Historical overview
1.1.2 Mellin transform
1.1.3 Estimation of ^
1.1.4 Estimation of G with known ^
1.1.5 Estimation of G with unknown ^
1.1.6 Numerical example
1.1.7 Real data example: diamond sizes
1.1.8 Discussion
1.2 Inference in continuous-time moving average Levy processes
1.2.1 Set-up and historical remarks
1.2.2 Setup
1.2.3 Mellin transform approach for moving average processes
1.2.4 Convergence rates
1.2.5 Example
1.2.6 Mixing properties of the Levy-based MA processes
1.2.7 Numerical example
2 Stochastic time-changed models
2.1 Estimation of the Blumenthal-Getoor indices
2.1.1 Formulation of the problem
2.1.2 Assumptions on the model
2.1.3 Several examples
2.1.4 The characteristic function of xa
2.1.5 Main idea of the estimation procedure
2.1.6 The case of known a
2.1.7 The case of unknown a
2.1.8 Numerical examples
2.2 Multivariate subordinated models
2.2.1 Subordination of stable processes: general idea
2.2.2 Subordination of stable processes for financial modelling
2.2.3 Main properties of the class of stable processes
2.2.4 Dependence structures for stable processes and related models
2.2.5 Multivariate subordination of stable processes
2.2.6 Main theorem
2.2.7 Empirical analysis
3 Construction of honest confidence sets
3.1 Density estimation
3.1.1 Statement of the problem
3.1.2 Asymptotic behaviour of the maximum of Gaussian processes
3.1.3 Projection estimates for densities
3.1.4 Relation to the extreme value theory for Gaussian processes
3.1.5 SBR-type theorem for projection density estimates
3.1.6 Sequence of accompanying laws
3.1.7 Numerical example
3.2 Confidence bands for the Levy density estimators
3.2.1 Motivation of this research
3.2.2 Projection estimates for Levy densities
3.2.3 Some assumptions on the basis
3.2.4 Relation to the theory of Gaussian processes
3.2.5 Asymptotic behaviour of the corresponding Gaussian process
3.2.6 Sequence of accompanying laws
3.2.7 Asymptotic confidence bands
3.2.8 Discussion
4 Limit laws and phase transitions in the mixture models
4.1 Preliminaries: classical REM
4.1.1 Asymptotic analysis of classical REM
4.1.2 Anderson parabolic problem
4.2 Alloy-type REM
4.2.1 Definition
4.2.2 Limit laws for the alloy-type REM
4.2.3 Free energy for the alloy-type REM
5 Analysis of the "purity" of a distribution
5.1 Statement of the problem
5.2 Discrete Dickman-Goncharov distribution
5.2.1 Definition of the Dickman-Goncharov distribution
5.2.2 Discrete Dickman-Goncharov distribution. Erdos problem
Conclusion
Notational Conventions
Abbreviations
References