# Department of Mathematical Statistics

**General information**

The Department of Mathematical Statistics was established at the faculty of Computational Mathematics and Cybernetics at Kazan State University on April 20,1988 as experts in mathematical statistics were in keen demand. The first head of the department was Professor I.N.Volodin. Up to date he is the academic advisor of the department. Presently there are 20 members of staff, 15 of them have academic degrees of Candidate and Doctor of science.

**History**

Theory of probability as the science has its roots in the Middle Ages and the first attempts of mathematical analysis of gambling (a coin toss, dice throw, roulette). The earliest works by scientists in the field of probability theory are dated to the XVII century. Studying forecasting of a gambling gain Blaise Pascal and Pierre Fermat introduced the first probabilistic patterns when throwing dice. Jacob Bernoulli made the important contribution to theory of probability: he proved the law of large numbers in the simplest case of independent trials. In the first half of the 19th century probability theory started being applied to the analysis of response errors. Laplace and Poisson proved the first limit theorems. In the latter half of the 19th century the Russian scientists P. L. Chebyshev, A.A. Markov and A.M. Lyapunov made the main contribution to the development in this area of knowledge. At that time law of large numbers, central limit theorem were demonstrated and also theory of Markov's chains was developed. The modern look of probability theory is due to the axiomatization offered by Andrey Kolmogorov.

Probability theory originates the base to develop models of real phenomena the basis for which are relations between frequencies of certain events occurrence. Having probabilistic model we can calculate probabilities (relative frequencies) of these events and optimize thus the behavior under uncertainty. The mathematical statistics develops models of inductive behavior in these conditions on the basis of the available probabilistic models. The main challenge is in according to supervision of elementary outcomes (it is usually a value of observed random variables) giving a selection method of actions when the frequency of mistakes would be the smallest. Clearly, this problem mates with the solution of complicated problems on an extremum, however even when they cannot be solved the probability theory gives a method to calculate average losses which we will have using the concrete inductive behavior rule we have chosen. Thus, mathematical statistics is theory of making optimal decision when consequences from actions taken on the basis of these decisions have a casual character. Mathematical statistics uses probability theory methods to calculate frequency of the "wrong" decisions or, in other words, for the amount of average losses inevitably arising in conditions of incidental. The majority of problems of mathematical statistics lead to an assessment problem of parameters or a problem of hypotheses checking that is a problem of selection of one of the several alternative statements about the object being studied. Presently the solution of these tasks is even more often is carried out using information technologies.

**About Head of Department**

The head of Department is Candidate of Sciences, Associate Professor Ekaterina Turilova. Her scientific interests are connected with theory of operator algebras, quantum structures of the functional analysis and their applications in various areas. E. Turilova is the member of the International Association of Quantum Structures and regularly participates in international conferences (including activity in organizing committees). She is the leading expert of Unified State Examination in Mathematics.

**Educational Activities**

The lecturers of the department conduct classes in mathematical analysis, probability theory and mathematical statistics for all bachelor's degree at the Institute of Computational Mathematics and Information Technologies and also teach various disciplines in mathematical statistics for other institutes of Kazan Federal University. The Department trains bachelors in 01.03.04 "Applied mathematics" as well. The department provides the unique courses for students specializing on the mathematical statistics: General Theory of Statistical Inference, Multidimensional Statistical Analysis, Theory of Stochastic Processes, Actuarial Mathematics, Stochastic Models in Economics and Finances.

Within the Master's degree program 01.04.02 "Applied Mathematics and Computer Science (Informatics)" the Master's program "Methods of Applied Mathematical Statistics" started in 2015. The subjects being taught can be compared to programs of the leading world universities. Courses of study developed and supported by the department are highly applied in economics and science (finance, medicine, industry, scientific research) and are widely demanded by business. The students training here have an excellent opportunity to use the gained theoretical knowledge and skills in the field of mathematical statistics and programming when solving practical tasks arising during medical research and financial organizations activity.

The Department also provides postgraduate students with training in 01.01.05 Probability theory and Mathematical Statistics.

**Scientific Activities**

The department performs high-quality research activity of its unique fundamental mathematical statistics area of "d-posterior approach to the problem of statistical inference". This innovative statistical methodology develops within Bayesian paradigm of statistical inference and is of high interest not only in scientific problems, but also in plenty applications and practical situations connected with quality control, medical diagnostics, genetic researches. Practical usefulness of this approach was due to qualitatively new definition of assurance of statistical conclusion in the specified appendices. Unlike the classical procedures of quality control which impose restrictions on probability of rejecting a good product (i.e. control the risk of a producer), the d-posterior procedures restrict the share of bad products among the accepted ones (i.e. controls the risk of a consumer). The wide spread of statistical methods in life sciences and genetic researches results in appearance of many natural applications for d-posterior approach methods (for example, the problem of identifying the genes responsible for pathologies of a human body, including oncological).

Other scientific directions successfully and actively develop at the Department as well. Within one of them the research in the field of numerical analysis is being conducted and is directed to construction, theoretical and numerical research of algorithms of the task solutions with free boundary, variational inequalities and problems of optimal control. It is also oriented to application of obtained methods and algorithms to applied problems of mechanics and physics. Besides of that, researches in the field of the general theory of operator algebras, theories of measure and non-commutative integration, quantum structures of functional analysis and their application in quantum informatics and quantum probability are being carried out.

**Alumni**

Our alumni work for government agencies of statistics, large industrial enterprises of the Republic of Tatarstan, health care providing institutions, insurance companies and banks. It should be noted that experts in the field of mathematical statistics are most demanded in the bank sphere that supports their serious competitive advantages at recruitment. The large number of graduates (including postgraduate study) works in the sphere of higher education in Russia and overseas (Canada, Lithuania, Mexico, etc.).

**Interesting Facts**

Staff members have close international relations both in educational activity and in the sphere of academic interests. Contacts with University of Oulu (Finland), Czech Technical University in Prague (Czech Republic), University of Veracruz (Mexico), University Regina (Canada) are made.

Besides, we are coming into contacts with Professor of University of Stanford B. Efron in connection with the applications of the d-posterior statistical methods in genetics. Using the empirical way along with his colleagues he anew received a number of the results which were published by the department staff in the 1980s.

**References **

The Department site: https://kms.kpfu.ru/en