# Advanced Statistics I - Prof. Dr. Matei Demetrescu # Syllabus

### I. Course description:

The course provides a rigorous foundation in the principles of probability and mathematical statistics underlying statistical inference in the field of economics and business. Special emphasis is given to the study of parametric families of distributions, univariate as well as multivariate, and to basic asymptotics for sample averages.

This course is a prerequisite for the lecture Advanced Statistics II, which focuses on the methods of statistical inference including parameter estimation and hypothesis testing.
Furthermore, it provides the foundation for the specialization courses in statistics and econometrics (Time Series Analysis, Statistics for Financial Markets, Microeconometrics, Multivariate Statistics, etc.).

### II. Prerequisities:

Calculus (no excuses!), some algebra.

Written exam.

### IV. Outline:

1. Elements of Probability Theory:

1.1  Sample Space and Events

1.2  Probability

1.3  Properties of the Probability Function

1.4  Conditional Probability

1.5  Independence

1.6  Total Probability Rule and Bayes' Rule

2. Random Variables and their Probability Distributions:

2.1  Univariate Random Variables

2.2  Univariate Cumulative Distribution Functions

2.3  Multivariate Random Variables

2.4  Marginal and Conditional Distributions

2.5  Independence of Random Variables

3. Moments of Random Variables:

3.1  Expectation of a Random Variable

3.2  Expectation of a Function of Random Variables

3.3  Conditional Expectation

3.4  Moments of a Random Variable

3.5  Moment-Generating Functions

3.6  Joint Moments and Moments of Linear Combinations

3.7  Means and Variances of Linear Combinations of Random Variables

4. Parametric Families of Density Functions:

4.1 Discrete Density Functions

4.2  Continuous Density Functions

4.3  Normal Family of Densities

4.4  Exponential Class of Distributions

5. Basic Asymptotics:

5.1  Convergence of Number and Function Sequences

5.2  Convergence Concepts for Sequences of Random Variables

5.3  Weak Laws of Large Numbers

5.4  Central Limit Theorems

5.5  Asymptotic Distributions of Functions for Asymptotically Normally Distributed Random Variables

6. Sample Moments and their Distributions

6.1  Random Sampling

6.2  Empirical Distribution Function

6.3  Sample Moments

6.4  Sample Mean and Variance from Normal Random Samples

6.5  Probability Density Functions of Functions of Random Variables

6.6 Order Statistics

V. Materials:

• Lecture notes (OLAT)
• Slides will be made available in due time (OLAT again)
• Main textbooks

Mood, A. M., F.A. Graybill and D.C. Boes (1974, 3rd ed.). Introduction to the Theory of Statistics. McGraw-Hill.

Mittelhammber, R.C. (1996). Mathematical Statistics for Economics and Business. Springer.

Other useful ones (by far not an exhaustive list)

• Casella, G. and R. Berger (2002, 2nd ed.). Statistical Inference. Duxbury.
• Dudewicz, E.J. and S.N. Mishra (1988). Modern Mathematical Statistics. John Wiley & Sons.
• Hogg, R.V. and R. Craig (1995, 5th ed.). Introduction to Mathematical Statistics. Prentice Hall.
• Rohatgi, V.K. und A.K. Saleh (2001, 2nd ed.). An Introduction to Probability Theory and Mathematical Statistics. John Wiley & Sons.

VI. Schedule:

• course, 2 hrs. per week
• tutorial (Anna Titova/Felix Kießner), 2 hrs. per week; you have the choice between two different time slots.
• non-compulsory computer class: every second week (tutored by Felix Kießner)
(for this you'll have to choose a slot and will need an account, details in class)
• see univis for exact times and location.