Download Анотації дисциплін вільного вибору аспіранта (Перелік №2)

Анотації дисциплін вільного вибору аспіранта (Перелік №2)
спеціальності 112 Статистика:
Lecture course:
Random processes with statistical application
Organiser: Yuriy Kozachenko
For: first year postgraduate
Prerequisites: basic course of Functional analysis, course
of Theory
Probability and Mathematical Statistics.
Short description:
This course is a introduction to theory of some space of
random variables: Orlicz spaces, spaces of Sub-Gaussian and Square-Gaussian
random variable. We discuss application of this theory for statistics of random
processes and simulation of random processes and fields. We study distribution of
same functional from random processes.
Sillabus:
1. Space of Lp(Ω) random variable and random processes from these spaces.
2. Sub-Gaussian random variables and Sub-Gaussian random processes.
3. Orlicz spaces of random variables and random processes from these spaces.
4. Conditions of continuity of sample path of random processes.
5. Boundedness and estimates for the distribution of the supremum of random
processes.
6. Stationary random processes.
7. Square Gaussion random variables.
8. Evolution of means and correlation functions of random processes.
9. Simulation of random variables.
10.Simulation of Gaussian random processes.
Required text:
1. V.V. Buldygin, Yu.V.Kozachenko (2000) Metric Characterization of
Random Variables and Random Processes, American Mathematical Society,
Providence, Rhode Island
2. Yu.V. Kozachenko, O.Pogorilyak, I.Rozora, A.Tegza (2016) Simulation of
Stochastic Processes with given Accuracy and Reliability, ISTE Press Ltd,
London and Elsevier Ltd Oxford
Recommended text:
1. I.I.Gikhman and A.V.Skorochod (1974) Theory of Random Processes, vol.1,
Springer-Verlag Berlin-Heidelberg-New York
Lecture course:
Asymptotic analysis of statistical algorithms
performance
Organiser: Rostyslav Maiboroda
For: first year postgraduate students in Statistics
Prerequisites: basic courses of Functional analysis, Probability and Statistics for
mathematical or statistical specialties of mathematical faculty or equivalent.
Short description:
This lecture course is a short introduction to asymptotic statistics, in which we
discuss how to analyze and compare accuracy of different statistical procedures for
large amount of data. Parametric, nonparametric and semiparametric procedures of
estimation, hypotheses testing, classification and prediction will be discussed from
the asymptotic perspective.
Sylabus:
1. Parametric and nonparametric inference. GEE-estimation. Statistical testing.
Approaches to the asymptotic analysis.
2. Theory of stochastic convergence. (Weak convergence in functional spaces,
weak and strong limit theorems).
3. Delta method and continuity theorems.
4. Maximum likelihood estimation. One-Step Estimators. Rate of convergence.
5. Likelihood ratios and contiguity.
6. Local asymptotic normality of estimators.
7. Efficiency of estimators. Fisher information matrix.
8. Efficiency of tests. Likelihood ratio tests. Chi-square tests.
9. Empirical processes and functional delta method. Hadamard
differentiability.
10. Nonparametric density estimation. Rate optimality.
11. Semiparametric models. Efficient score functions.
Required text:
1. van der Vaart A. (1998) Asymptotic Statistics, Cambridge U. Press.
Recommended Text:
1. Shao J. (2003) Mathematical statistics. Springer-Verlag: New York.
2. Боровков, А.А. (1997) Математическая статистика. Наука: Москва.