Mathematics BSc
Course description
2013. Mathematical statistics
Hours
lect+pc
Credits
lect+pc
Assessment Specialization Course code
lect/pc
Semester Status
2 + 2 2 + 3 exam +
pure math. mm1c1st6m
mm1c2st6m
6 compulsory
Strong Weak Prerequisites
Practice class
Strong:
Lecture
Weak:
practice class
Course objectives
Introduction of the basic notions and methods of mathematical statistics.
Literature
Recommended:
• 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.
Syllabus
• 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.