**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**

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

**Books:**

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

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