Statistics

Completed courses Analysis 1, Analysis 2a and Analysis 2b.

Central limit theorem.

Sampling: introductory examples, simple random sampling, sampling distribution, standard error and confidence intervals, stratified sampling.

Parameter estimation: statistical model, estimators, the properties of estimators, maximum-likelihood estimation, asymptotic properties of estimators.

Regression models: linear regression model, estimators, the Gauss-Markov theorem, logistic regression.

Hypothesis testing: basic definitions and examples, the power of the test, variance analysis, Wilks' theorem, nonparametric tests.

G. Grimmett, D. Welsh: Probability : An Introduction, Oxford Univ. Press, Oxford, 1986.
J. Pitman: Probability, Springer, New York, 1999.
D. Stirzaker: Probability and Random Variables : A Beginner’s Guide, Cambridge Univ. Press, Cambridge, 1999.
R. Lupton: Statistics in Theory and Practice, Princeton Univ. Press, Princeton, 1993.
J. A. Rice: Mathematical Statistics and Data Analysis, 2nd edition, Duxbury Press, Belmont, 1995.

We introduce the basic concepts of statistics such as an estimator, sampling distribution, standard error and confidence interval. We continue with the concept of statistical model and parameter estimation. Regression models of various types are among the most common statistical models in applications. Finally, a student is acquainted with the basic concepts and examples of hypothesis testing.

Knowledge and understanding: Statistics is a standard part of mathematical education, and, on the other hand, the starting point for applications in a wide range of disciplines from biology, economics, financial and actuarial mathematics. It has become a tool in almost every science, where one has to analyze quantitative data. Knowledge of basic concepts of statistics is thus a necessary part of education of any mathematician.
Application: The use of concepts of statistics extends to many areas of science and social science. Statistics is the language of economics, and it is also indispensable in medical research.
Reflection: Understanding of theoretical concepts in many applications.
Transferable skills: The ability to identify the probability and statistical concepts in other sciences (physics, economics, finance, actuarial science, medicine, biology, industry statistics).

Lectures, exercises, midterm exams seminar work, consultations

2 midterm exams instead of written exam, written exam
Seminar work and oral exam

PERMAN, Mihael, WERNER, Wendelin. Perturbed Brownian motions. Probability theory and related fields, ISSN 0178-8051, 1997, let. 108, št. 3, str. 357-383. [COBISS-SI-ID 7848537]
HUZAK, Miljenko, PERMAN, Mihael, ŠIKIĆ, Hrvoje, VONDRAČEK, Zoran. Ruin probabilities and decompositions for general perturbed risk processes. Annals of applied probability, ISSN 1050-5164, 2004, vol. 14, no. 3, str. 1378-1397. [COBISS-SI-ID 13168985]
HUZAK, Miljenko, PERMAN, Mihael, ŠIKIĆ, Hrvoje, VONDRAČEK, Zoran. Ruin probabilities for competing claim processes. Journal of Applied Probability, ISSN 0021-9002, 2004, vol. 41, no. 3, str. 679-690. [COBISS-SI-ID 13207641]
SMREKAR, Jaka, YAMASHITA, Atsushi. Function spaces of CW homotopy type are Hilbert manifolds. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2009, vol. 137, no. 2, str. 751-759. [COBISS-SI-ID 14965849]
SMREKAR, Jaka. Periodic homotopy and conjugacy idempotents. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2007, vol. 135, no. 12, str. 4045-4055. [COBISS-SI-ID 14382681]
CENCELJ, Matija, DYDAK, Jerzy, SMREKAR, Jaka, VAVPETIČ, Aleš, VIRK, Žiga. Algebraic properties of quasi-finite complexes. Fundamenta mathematicae, ISSN 0016-2736, 2007, vol. 197, str. 67-80. [COBISS-SI-ID 14502233]
CHEN, Louis Hsiao Yun, RAIČ, Martin, THÀNH, Lê Vǎn. On the error bound in the normal approximation for Jack measures. Bernoulli. Feb. 2021, vol. 27, no. 1, str. 442-468. ISSN 1350-7265. [COBISS-SI-ID 38707203]
RAIČ, Martin. A multivariate Berry-Esseen theorem with explicit constants. Bernoulli. 2019, vol. 25, no. 4a, str. 2824-2853. ISSN 1350-7265. [COBISS-SI-ID 18772825]
EICHELSBACHER, Peter, RAIČ, Martin, SCHREIBER, Tomasz. Moderate deviations for stabilizing functionals in geometric probability. Annales de l'I.H.P. Probabilités et statistiques. 2015, vol. 51, no. 1, str. 89-128. ISSN 0246-0203. [COBISS-SI-ID 17294681]