Mathematical Statistics                                                                                       Fall AY 2016-2017

Lecturer: Marianna Bolla

No. of Credits: 3 and no. of  ECTS credits: 6

Prerequisites: Undergraduate Calculus and Basic Probability

Course Level:  intermediate

Brief introduction to the course:

While probability theory describes random phenomena, mathematical statistics teaches us how to behave in the face of uncertainties, according to the famous mathematician Abraham Wald.  We will learn strategies of treating chances in everyday life.

The goals of the course:

The course gives an introduction to the theory of estimation and hypothesis testing.

The main concept is that our inference is based on a randomly selected sample from a large population, and hence, our observations are treated as random variables. Therefore,

we  use methods and laws of  probability. On this basis, applications are also discussed, mainly on a theoretical basis, but we make the students capable of solving numerical exercises.

The learning outcomes of the course:

Students will be able to find the best possible estimator for a given parameter by investigating the bias, efficiency, sufficiency, and consistency of an estimator on the basis of theorems and theoretical facts. Students will gain familiarity with basic methods of estimation and will be able to construct statistical tests for simple and composite hypotheses. They will become familiar with applications to real-world data and will be able to choose the most convenient method for given real-life problems.

More detailed display of contents

1. Statistical space, statistical sample. Basic statistics, empirical distribution function, Glivenko-Cantelli theorem.
2. Descriptive study of data, histograms. Ordered sample, Kolmogorov-Smirnov Theorems.
3. Sufficiency, Neyman-Fisher factorization. Completeness, exponential family.
4. Theory of point estimation: unbiased estimators, efficiency, consistency.
5. Fisher information. Cramer-Rao inequality, Rao-Blackwellization.
6. Methods of point estimation: maximum likelihood estimation, method of moments, Bayes estimation. Interval estimation: confidence intervals.
7. Theory of hypothesis testing, Neyman-Pearson lemma for simple alternative and its extension to composite hypotheses.
8. Parametric inference: z, t, F, chi-square, Welch, Bartlett tests.
9. Nonparametric inference: chi-square, Kolmogorov-Smirnov, Wilcoxon tests.
10. Sequential analysis, Wald-test, Wald-Wolfowitz theorem.
11. Theory of least squares, regression analysis, correlation, Gauss-Markov theorem.
12. One-way analysis of variance and analyzing categorized data.

Books:

C.R. Rao, Linear statistical inference and its applications. Wiley, New York, 1973.

G. K. Bhattacharyya, R. A. Johnson, Statistical concepts and methods. Wiley, New York, 1992.

C. R. Rao, Statistics and truth. World Scientific, 1997.

Handouts: tables of notable distributions (parameters and quantile values of the distributions).

E-material: brief summary of the lectures and exercises are available on the lecturer’s homepage.

Teaching format: lectures combined with problem solving and  SPSS practice.

Assessment:

Attendance is mandatory.

Homework: will be assigned every second week (altogether worth 25%)

Tests and grading: There will be two midterm tests (worth 25%) and a final exam (worth 50%) which together with the homework give the base of the final grade. However, the final grade can be improved by participation in class discussions.

Final exam: within two weeks after the final lecture

Office hours: by appointment

Contact details:

Marianna Bolla

Institute of Mathematics

Budapest University of Technology and Economics

1111. Budapest, Egry Jozsef u. 1. Bldg. H5/2.

marib@math.bme.hu

http://www.math.bme.hu/~marib/ceu2