Statistics

obligatory courses

(in Polish) Oczekuje się dobrej znajomości zagadnień ujętych w sylabusach przedmiotów Analiza matematyczna II.1 oraz Rachunek prawdopodobieństwa I.

The lecture is an introduction to classical statistics and focuses on a rigorous presentation of the theoretical statistics that forms the basis of statistical techniques. The course discusses statistical models for data and their parametrisations with particular focus on exponential families. Methods for parameter estimation are discussed, confidence intervals, hypothesis testing and their theoretical properties. Gaussian linear models are treated. The theory is applied to data analysis, fitting models and using them for prediction.

Alternatively, you can choose 1000-714SAD of a more practical nature.

This course gives an introduction to classical statistics, dealing with theoretical statistics and applications to data analysis. The topics are:

1) Statistical Models, non-parametric, semi-parametric, parametric, the empirical distribution, the Kolmogorov-Smirnov test.

2) Parameters and Sufficiency: Sufficient statistics, minimal sufficient statistics, complete statistics, factorisation theorem.

3) Exponential families and their parametrisations

4) Parameter Estimation: Minimum contrast, estimating equation method, maximum likelihood, method of moments, least squares. Kullback Leibler divergence, maximum likelihood as a minimum contrast.

5) The information inequality, linear predictors.

6) Complete Sufficiency and UMVU (Uniform Minimum Variance Unbiased) estimators.

7) Asymptotic results for estimators, consistency, the Delta method.

8) Confidence Intervals: Pivot method. Hypothesis Testing: Likelihood Ratio Test, Neyman Pearson lemma, Monotone Likelihood Ratio, Rubin Karlin theorem, p-values, Confidence intervals by inverting a test statistic.

9) Gaussian Linear Models

10) Asymptotic Likelihood Ratio test, Chi squared tests, Wald statistic, Logistic regression.

There are also computer laboratories (15 hours) where the modelling techniques are applied using R.

Social Skills

The student should understand the principles of data analysis and should (using R), carry out statistical tests, be able to analyse data using Gaussian linear models and use these models for prediction.

[1] P. Bickel and K. Doksum, Mathematical Statistics: Basic ideas and selected topics, Vol. 1, 2001.

[2] J. Noble, Notatki do wykładu ze Statystyki (ang):

www.mimuw.edu.pl/~noble/courses/Statistics

1) Statistical Models, non-parametric, semi-parametric, parametric, the empirical distribution, the Kolmogorov-Smirnov test.

2) Parameters and Sufficiency: Sufficient statistics, minimal sufficient statistics, complete statistics, factorisation theorem.

3) Exponential families and their parametrisations

4) Parameter Estimation: Minimum contrast, estimating equation method, maximum likelihood, method of moments, least squares. Kullback Leibler divergence, maximum likelihood as a minimum contrast.

5) The information inequality, linear predictors.

6) Complete Sufficiency and UMVU (Uniform Minimum Variance Unbiased) estimators.

7) Asymptotic results for estimators, consistency, the Delta method.

8) Confidence Intervals: Pivot method. Hypothesis Testing: Likelihood Ratio Test, Neyman Pearson lemma, Monotone Likelihood Ratio, Rubin Karlin theorem, p-values, Confidence intervals by inverting a test statistic.

9) Gaussian Linear Models

10) Asymptotic Likelihood Ratio test, Chi squared tests, Wald statistic, Logistic regression.

Analyse data, construct statistical models, estimate parameters and use models for prediction using the R programming language, present conclusions clearly.

1) A written examination

2) Tutorial participation

3) Laboratory work.

The final grade is decided by a combination of the grades from the points above.