SubjectsSubjects(version: 867)
Course, academic year 2019/2020
Selected Probability Topics for Statistics - NMTP563
Title: Vybrané partie pravděpodobnosti pro statistiku
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2016
Semester: summer
E-Credits: 5
Hours per week, examination: summer s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: not taught
Language: English, Czech
Teaching methods: full-time
Guarantor: prof. RNDr. Jana Jurečková, DrSc.
Class: M Mgr. PMSE
M Mgr. PMSE > Volitelné
Classification: Mathematics > Probability and Statistics
Annotation -
Last update: T_KPMS (06.05.2014)
The course supplements the basic course Probability 1 for expert and practical knowledge which a statistician, but also a probabilitist can need in their own research. As such, the course can be considered as an alternative to Probability 2. It is focused on the conditional probability and conditional expectation in the Kolmogorov sense, on dominated systems of probability distributions, on important probability inequalities and lower/upper bounds, on the contiguity of probability measures, on relations between probability measures, and on the empirical processes.
Aim of the course -
Last update: T_KPMS (06.05.2014)

To extend the basic knowledge of probability for expert and practical knowledge which a statistician, but also a probabilitist can need in their own research.

Literature - Czech
Last update: prof. RNDr. Jana Jurečková, DrSc. (06.09.2013)

[1] Shorack, G. R. Probability for Statisticians. Springer 2000

[2] Pollard, D. A User’s Guide to Measure Theoretic Probability. Cambridge University Press 2002.

[3] Lehmann, E.L. Testing Statistical Hypotheses. Springer 1986.

[4] Csörgö, M. and Révész, P. Strong Approximations in Probability and Statistics.

Akadémiai Kiadó, Budapest 1981.

[5] Grenander, U. Abstract Inference. J.Wiley 1981.

[6] Jurečková, J., Sen, P. K. and Picek. J. Methodology in Robust and Nonparametric Statistics. Chapman & Hall/CRC Press 2013.

Teaching methods -
Last update: T_KPMS (16.05.2013)

Lecture + exercises.

Syllabus -
Last update: T_KPMS (05.05.2015)

1. Conditioning: Conditional probability and conditional expectation in the Kolmogorov conception. Conditions under which there exists a genuine conditional probability distribution and its density. Sufficiency, sufficient statistics, factorization. Existence of a nontrivial sufficient statistic. Completness. Basu Theorem. Completness of the vector of order statistics for the system of all absolutely continuous distributions. Examples.

2. Dominated systems of probability measures. Existence of a countable equivalent subsystem. Dominated systems of probability measures and sufficient statistics. The least favorable probability measures. Examples.

3. Some important inequalities, lemmas and upper bounds: Bernstein, Billingsley, Birnbaum-Marshall, Borel-Cantelli, Chebyshev, convexity lemma, C_r- inequality, Doob, entropy inequality, Hájek-Rényi, Hoeffding, Jensen, Hájek-Hoeffding projection, Kolmogorov maximal inequality, and others. Aplications and examples.

4. Contiguity of probability measures, Hájek-LeCam Theorem, local asymptotic normality, Convolution Theorem. Aplications.

5. Inter-relations of probability measures (coupling): Relations of Poisson and binomial probability distributions, Theorem of Komlós-Májor-Tusnády, Strassen Theorem.

6. Empirical processes and their applications in the statistical inference.

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