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Last update: T_KPMS (06.05.2014)
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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. |
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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.
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Last update: T_KPMS (16.05.2013)
Lecture + exercises. |
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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. |