Probability and Mathematical Statistics 2

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The subject is devoted to the statistical techniques for estimation and testing within parametric and nonparametric models such as Maximum likelihood principle, Uniformly most powerful tests, Goodness of fitness tests of models, confidence regions, etc. We focus on real practical applications of these statistical techniques in frame of the specific examples.

Requirements:

Basic course of Calculus and Probability (in the extent of the courses 01MAA3-4 or 01MAB3-4, 01PRA1 nebo 01PRST held at the FNSPE CTU in Prague).

Syllabus of lectures:

Unbiased minimum variance estimates, Fisher information matrix, Rao-Cramér inequality, Bhattacharrya inequality. Moment estimators, Maximum likelihood principle, consistency, asymptotic normality and efficiency of MLE. Testing of simple and composite hypotheses. The Neyman-Pearson lemma. Uniformly most powerful tests. Randomized testing, generalized Neyman-Pearson lemma The likelihood ratio test, t-test, F-test. Nonparametric models, empirical distribution and density function, their properties, histogram and kernel density estimate. Pearson goodness of fit test, Kolmogorov-Smirnov test. Confidence sets and intervals, pivotal quantities, acceptance regions, Pratt theorem.

Syllabus of tutorials:

1.Parameter Estimation for specific distributions. 2. Testing hypotheses in normal model, t-test, F-test applied to data sets from steel industry. 3. Randomized testing - task from epidemiology. 4. Variance analysis - task from agriculture. 5. Nonparametric models - goodness of fit test for data from chemical industry. 6. Confidence intervals in normal models - application to temperature data set.

Study Objective:

Knowledge: In frame of the course, to provide students with the knowledge necessary for the following future subjects using stochastic models. To give a deeper insight into the field in the area of point statistical parameter estimation and testing statistical hypothesis in parametric and nonparametric probabilistic models.

Abilities: Orientation in majority of standard notions of the statistics and capabilities of practical applications in actual stochastic computations.

Study materials:

Key:

[1] Anděl J., Základy matematické statistiky, MatFyzPress, Praha, 2005.

[2] Schervish M.J., Theory of Statistics, Springer, 1995.

Recommended:

[3] Shao J., Mathematical Statistics, Springer, 1999.

[4] Lehmann E.L., Point Estimation, Wiley, N.Y., 1984.

[5] Lehmann E.L., Testing Statistical Hypotheses, Springer, N.Y., 1986.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: