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Syllabus designed by:
Introduction of the basic notions and methods of mathematical statistics.
- E. L. Lehmann, G. Casella: Theory of Point Estimation. 2nd Edition, Springer Texts in Statistics, 2003.
- E. L. Lehmann, J. P. Romano: Testing Statistical Hypotheses. Springer Texts in Statistics, 2010.
- Statistical field, random sample, statistics, empirical distribution, the Glivenko-Cantelli theorem.
- The multivariate normal distribution and its properties. Student’s distribution, chi-square distribution, F-distribution.
- Dominated families of distributions. Sufficient statistics, the Neyman-Fisher factorization theorem. Complete statistics. The Fisher information and its properties.
- Point estimations, loss functions, risk. Unbiasedness, consistency, admissibility, efficiency. Minimax estimators. The Blackwell-Rao theorem. The Cramér-Rao information. inequality. Asymptotic version, Bahadur’s theorem.
- Method of moments. Maximum likelihood estimation. Bayesian estimation.
- Elements of hypothesis testing. The Neyman-Pearson theorem on the optimality of the likelihood ratio test.
- Samples from normal distribution. Independence of the sample mean and the sample variance. Testing hypotheses on the parameters of the normal distribution.
- Chi-square tests for testing goodness of fit, homogeneity, and independence in the discrete case. Kolmogorov-Smirnov tests for goodness of fit and homogeneity in the continuous case.
- Linear models. Least squares estimators, the Gauss-Markov theorem. Testing linear hypotheses in linear models with Gaussian errors.