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# Department of Mathematical Statistics

Acting Head of the Department: Korolev Victor, Professor, Dr.Sc., Vice Dean for public relations

An important role in organizing the Department was also played by L.N. Bolshev (1922 - 1978). The Department is focused on the study of mathematical models that consider statistical regularities in random factor effects. The current research is carried out in different probability theory and mathematical statistics areas of fundamental theoretical and practical importance, including limit theorems in the probability theory, asymptotic methods in mathematical statistics, theory and statistical analysis of random processes, and the queuing theory.

In recent years, along with the research in classical fields of knowledge, the Department has been conducting extensive studies into a number of topical areas including financial and actuarial mathematics, computer security and software reliability.

Fundamental applied problems involving modeling of plasma turbulence and the analysis of information flows in complex information and telecommunication systems are also considered. Research results find application in physics, medicine, biology, psychology, financial and insurance activities, in designing communication systems and novel computers (with their hardware and software).

The Department includes the Laboratory of statistical analysis which is aimed to address a wide range of applied problems related to mathematical modeling of the behavior of complex financial and economic systems.

The high level of academic background of our graduate and postgraduate students is supported by the close collaboration with leading Russian companies and research centers.

### Staff members:

- Bening Vladimir, Professor, Dr.Sc., Scientific Secretary of the Department
- Kruglov Victor, Professor, Dr.Sc.
- Ulyanov Vladimir, Professor, Dr.Sc.
- Ushakov Vladimir, Professor, Dr.Sc., Head of the Laboratory of the statistical analysis
- Khokhlov Yury, Associate Professor, Dr.Sc.
- Kudryavtsev Alexey, Associate Professor, PhD
- Matveev Victor, Associate Professor, PhD
- Ponomarenko Lubov, Associate Professor, PhD
- Shestakov Oleg, Associate Professor, Dr.Sc.
- Zakharova Tatyana, Senior Lecturer, PhD
- Pagurova Vera, Senior Research Fellow, PhD
- Shevtsova Irina, Assistant Professor, PhD
- Naumov Aleksey, Researcher, PhD
- Nefedova Yulia, Researcher, PhD
- Stolyarova Marina, Engineer

### Regular courses:

- Probability theory and mathematical statistics by Prof. Korolev, Prof. Ushakov, Prof. Ulyanov, 64 lecture hours and 64 seminar hours, 3rd and 4th semesters.
- Additional chapters of mathematical statistics by Prof. Bening, 104 lecture hours, 6th and 7th semesters.
- Mathematical foundations of probability theory by Prof. Ulyanov, 68 lecture hours, 5th and 6th semesters.
- Applied problems of probability theory by Dr. Kudryavtsev, 32 lecture hours, 8th semester.
- Stochastic processes by Prof. Khokhlov and Prof. Kruglov, 100 lecture hours, 6th – 8th semesters.
- Statistics of stochastic processes by Prof. Khokhlov, 36 lecture hours, 9th semester.
- Probability models by Assoc. Prof. Shestakov, 32 lecture hours, 5th semester.
- Probability models by Dr. Nefedova, 14 lecture hours, master program.

### Special courses:

- Paradoxes in probability theory by Prof. Ulyanov, 32 lecture hours, 3rd – 10th semesters.
- Univariate and multivariate analysis methods by Dr. Ufimtsev, 64 lecture hours, 3rd – 10th semesters.
- Selected issues of statistical and stochastic tomography theory by Assoc. Prof. Shestakov, 32 lecture hours, 3rd – 10th semesters.
- Mathematical models for the protection against malicious computer attacks by Prof. Grusho, 64 lecture hours, 3rd – 10th semesters.
- Econometrics by Prof. Khokhlov, 32 lecture hours, 3rd – 10th semesters.
- Normal approximation by Dr. Shevtsova, 64 lecture hours, 3rd – 10th semesters.
- Mathematical models of the evolution of financial indexes by Prof. Korolev, 32 lecture hours, 5th – 10th semesters.
- Statistical methods in recommendation systems by Dr. Nefedova, 32 lecture hours, 5th – 10th semesters.

### Special scientific seminars:

- Risk theory and related topics by Prof. Bening and Prof. Korolev.
- Probabilistic and statistical methods in information and computer security by Prof. Grusho.
- Financial engineering by Dr. Doynikov and Dr. Nazarov.
- Probabilistic and statistical methods of signal and image processing by Assoc. Prof. Shestakov and Dr. Zakharova.

### Master programs (MSc)

Probability theory, mathematical statistics and their applications by Prof. Bening.

## Main Scientific Directions

Along with the classical problems of probability theory and mathematical statistics such as theory of limit theorems, asymptotic statistics, theory of stochastic processes, queuing theory, some prospective problems are considered by the Department’s team. Some of them are described below in detail.

### Contemporary methods of risk theory and reliability theory

(Academician of RAS Yu.V. Prohorov, Professors V.Yu. Korolev and V.E. Bening)

Novel randomized models and methods to study risk and reliability in situations when the analyzed system operates in a non-stationary environment are proposed. Their implication is that for statistical analysis of risk situations and complex system reliability, not only the statistical data itself, but also the sample size and the distribution parameters of the characteristics under study should be considered arbitrary due to the non-closedeness of the analyzed objects. These studies make extensive use of the asymptotic approach based on limit theorems of the probability theory. Within the framework of this approach, we succeeded in obtaining new and more flexible mathematical models of risk situations. These results are of considerable importance to solving risk analysis problems in conditions of stochastic uncertainty. Analytical mechanisms of the origins of these models are described. It has been shown that these mechanisms are the limiting processes for the statistics constructed from samples of a random volume and sums of a random number of random variables. They successfully explain deviations of the actually observed (empirical) distributions from classical models. Specifically, they provide a consistent explanation for the origins of so-called “heavy tails” which consists in the fact that large values of the observed characteristics are far more common in practice than the classical models prescribe.

The Department has developed adequate mathematical models to evaluate successfully the risks related to non-uniform flows of extreme events. In particular, the asymptotic theory of generalized risk processes describing reserve dynamics of insurance companies has been worked out. Specific procedures that calculate probabilistic characteristics of disasters, including forecasts for the “expected time” of a disaster and the duration of a period with the negligible disaster probability, have been established. In particular, the developed methods have been successfully applied to probability forecasting for a catastrophic Earth collision with potentially dangerous asteroids, comets, and other celestial bodies, which made it possible to analytically and statistically prove the statement, known in paleobiology as a “Shiva hypothesis”. According to this statement, once every 25 million years on average the range of biological species inhabiting the Earth is radically renewed.

We have also developed the mathematical theory of reliability for modifiable information systems which encompasses probabilistic and statistical methods of the software reliability analysis. New and unique methods that are based on novel non-conventional approaches of mathematical statistics, have been devised to estimate and predict software reliability. The uniqueness of these methods lies in the fact that the estimated parameters initially depend on the sample volume which prohibits the use of classical methods of mathematical statistics.

A series of papers on analytical methods in the risk theory was honoured by the Lomonosov Prize (MSU). Research in this area is carried out in the close collaboration with the Institute of Informatics Problems of the Russian Academy of Sciences.

Refs:

- V.Bening, V. Korolev, Generalized Poisson Models and their Application in Insurance and Finance. - Utrecht: VSP, 2002, xix + 434 pp.
- V.Korolev and I.Shevtsova, An improvement of the Berry-Esseen inequality with applications to Poisson and mixed Poisson random sums // Scandinavian Actuarial J., 2010.

### Methods for probabilistic and statistical analysis of chaotic processes

(Prof. V.Yu. Korolev)

Exponential productivity growth of high-performance computing systems highlights the fundamental problem of developing novel mathematical and algorithmic techniques that solve mathematical modeling and optimization problems applicable to objects of high complexity, the techniques that could not have been developed earlier due to the lack of appropriate computer technology. A promising direction in this area is the development of novel methods and technologies in the study of random processes which describe information flows in complex systems.

Fundamentally new and efficient statistical algorithms have been developed to solve multivariate problems, including so-called “grid methods” of separation of mixtures of probability distributions, which make it possible to solve the problem of chaotic process volatility decomposition in a real-time mode.

The developed methods have been successfully applied to stochastic structure analysis of some specific complex systems, to plasma turbulence and information flows in complex telecommunication and computing systems in particular.

Research in this area is carried out in the close collaboration with the General Physics Institute and the Institute of Informatics Problems of the Russian Academy of Sciences.

Refs:

- V. Korolev, N. Skvortsova (Eds.). Stochastic Models of Structural Plasma Turbulence. - Utrecht: VSP, 2006, ix + 400 pp.

### Probabilistic and statistical methods in computer security and computer system protection

(Corresponding Member of the Academy of Cryptography of RF, Prof. А.А. Grusho)

As a computer or a computerized system penetrates into every area of human activity, the issues of computer security and computer system protection are becoming increasingly important. It is essential that we understand what computer system security is and what we should take into consideration to be sure that the required security is in place. Historical experience shows that in order to formalize both security definition and security justification we should use mathematical methods within the framework of mathematical models.

The Department has developed mathematical methods of security analysis for distributed computer systems. For the first time, examples of provably secure computer systems were built in Russia. Significant results have been obtained in the development of methods for information flow control in computer systems. It was the first time when the properties of hidden information flows in distributed computer systems had been investigated. The secure communication has been proved to be possible even through such strong security solutions as firewalls and cryptography. These results were verified in practice. Having proved the statements of non-existence for consistent sequences of statistical tests for feature detection in target channels, we were able to prove “invisibility” of some information flows. The applied methods are novel to mathematical statistics.

### Multivariate statistical analysis for high-dimensional observations and its applications

(Prof. V.V. Ulyanov)

Whenever we make a phone call, use a bank card to pay for a purchase, show a bonus or discount card, we leave “footprint information”: the time, duration and recipient of the call, the purchase sum, and the list of products we buy etc. This information is not only stored by mobile companies, banks, and shops, but it can also be used and is used to make decisions based on statistical analysis of the collected data.

Conventional methods of multivariate statistics are based on approximations involving a considerable amount of input data whose accuracy decreases dramatically with the growth of the data dimensionality. However, complex stochastic systems with a large or theoretically infinite number of parameters occur in an increasing number of applications, specifically in the analysis of financial markets and institutions, in sociology, genetics, biology, and mathematical physics. Thus, there is an urgent need to revise the classical approaches in multivariate statistics with a view of creating a new set of tools to solve applied problems with high-dimensional data. The issue of approximation accuracy of the distributions of the emerging statistics is utterly important for these problems. It is common practice to manage solely with qualitative approximation characteristics when distributions of statistics can be said to approximately belong to a certain distribution class under certain conditions. In the best case one is provided with the order of accuracy of this approximation in terms of a sample volume.

However, from the perspective of practical application this information is not sufficient since dealing with applied problems we are concerned with quite specific finite values of observation dimensionality and sample volume. Therefore it is necessary to have computable error estimates for the proposed approximations.

In multivariate statistics of particular importance is the problem of dimensionality reduction, the selection of the most informative variables (the main components) for better visualizing and interpreting of statistical conclusions. One approach is to use decision trees which can help to analyze data of different nature: numerical, ordinal, nominal, and also data with missing values, which is important to applications.

For high-dimensional observations it is preferable to make use of random forests (a fairly recent technique) rather than separate decision trees. Most applied research which is based on the use of random forests, is related to data processing in bioinformatics, genetics, chemistry, astronomy, and medicine, for example, in early disease detection by biomarkers.

The latest results in this area are presented in the following monograph and papers:

- Y. Fujikoshi, V.V. Ulyanov, R. Shimizu R. Multivariate Statistics: High-Dimensional and Large-Sample Approximations. - Hoboken, NJ, John Wiley & Sons, Inc., 2010.
- V.V.Ulyanov, Cornish-Fisher expansions // International Encyclopedia of Statistical Science, Berlin, Germany: Springer Verlag, pp. 613-618, 2011.
- V.V.Ulyanov, Short asymptotic expansions in the CLT in Euclidian spaces: a sharp estimate for its accuracy // Proc. 2011 World Congress on Engineering and Technology. Shanghai, China: IEEE Press, pp. 260-262, 2011.

### Probabilistic and statistical methods of signal and image processing

(Dr. О.V. Shestakov and Dr. T.V. Zakharova)

The Department is also engaged in the research of a new scientific area of image processing and analysis by a wavelet-based statistical method. This is a relatively new area of applied mathematics which has been recognized by the scientific community. The research is carried out in the close collaboration with the Institute of Mechanics (MSU), the Federal State Institution/FSI “Russian center of Forensic Medical Expertise”, and the Space Research Institute of the Russian Academy of Sciences.

Wavelets have found a wide range of applications in the modern world. The wavelet analysis is essential in many fields of physics, technology, and industry, in every field where signal and image processing is used. The classical applications of wavelets, data compression and noise removal, are complemented with applications in such subject areas as medicine, meteorology, communication, forecasting and analysis of financial markets.

A new way to process and study signals is given by wavelets and wavelet transforms whose development has only recently been made possible by the emergence of high-speed computers (since this technique is computer-intensive). By convention, a wavelet transform can be considered as a three-dimensional spectrum, where X-axis represents time, Y-axis represents frequency, and Z-axis represents harmonic amplitude with a given frequency at a given time.

Methods of wavelet analysis proved very useful in solving some problems of computerized tomography. In particular, these methods can help to partially localize the problem of tomographic image reconstruction, thereby reducing the radiation dose necessary to collect projections. In addition, the threshold processing of wavelet coefficients makes it possible to implement non-linear regularization of the reconstruction method and remove noise from the image under reconstruction. Methods of threshold risk analysis allow one to quantify errors in tomographic images.

Multiresolution wavelet analysis is applied to study the phenomenon of atmospheric turbulence characterized by localized events. Wavelet processing of experimental data allows ones to estimate quantitatively the behavior of turbulent flows at different scales. In addition, we carried out the processing of aerodynamic images of streamlined bodies, having performed both qualitative and quantitative image analysis. The problem of chromaticity restoration in colored shadow patterns of the experiment has been solved, thus resulting in higher accuracy calculation of the aerodynamic flow density.

The Department carries out research that concerns another interesting application of wavelet processing – that is the analysis of satellite images of the Earth’s water surface. A technique to identify subtle patterns in the field of the Earth’s water vapor has been developed. This makes it possible to localize homogeneous zones and substantially specify the boundaries of the field. The Department pioneered in constructing the so-called frequency map of the water surface which can serve as a new tool in forecasting certain atmospheric phenomena, tropical cyclones in particular. The Department «Earth remote sensing» at the Space Research Institute of the Russian Academy of Sciences has observed that the proposed processing method is unique in geophysical problems and has no analogues in the world.

### Recent publications:

• 2013

- Bening V.E. On the assymptotic deficiency of some Bayesian criteria // J. Math. Sciences. 2013. 189. N 6. P. 967-975.
- Bening V.E., Korolev V.Yu. On estimation of Student distribution center with a small number of degrees of freedom // J. Math. Sciences. 2013. 189. N 6. P. 889-898.
- Bening V.E., Korolev V.Yu. Statistical estimation of parameters of fractionally stable distributions // J. Math. Sciences. 2013. 189. N 6. P. 899-902.
- Korolev V.Yu., Zaks L.M., Zeifman A.I. On convergence of random walks generated by compound Cox processes to Levy processes // Statistics and Probability Letters. 2013. 83. N 10. P. 2432-2438.
- Kruglov V.M. A characterisation of the Gaussian distribution // Stochastic Analysis and Applications. 2013. 31. N 5. P. 872-875.
- Nefedova Yu.S., Shevtsova I.G. On nonuniform convergence rate estimates in the central limit theorem // Theory Probab. and its Appl. 2013. 57. N 1. P. 28-59.
- Shevtsova I.G. A square bias transformation of probability distributions: some properties and applications // Doklady Mathematics. 2013. 88. N 1. P. 388-390.
- Shevtsova I.G. Moment-type estimates with an improved structure for the accuracy of the normal approximation to distributions of sums of independent symmetric random variables // Theory Probab. and its Appl. 2013. 57. N 3. P. 468-496.
- Shevtsova I.G. On the accuracy of approximation of the complex exponential by the first terms of its Taylor expansion and applications to the Fourier-Stieltjes transform // Doklady Mathematics. 2013. 88. N 1. P. 409-412.
- Ushakov A.V., Ushakov V.G. Limiting expectation time distribution for a critical load in a system with relative priority // Moscow University Computational Mathematics and Cybernetics. 2013. 37. N 1. P. 42-48.
- Christoph G., Ulyanov V., Fujikoshi Y. Accurate approximation of correlation coefficients by short Edgeworth-Chebyshev expansion and its statistical applications // Prokhorov and Contemporary Probability Theory. Springer Proceedings in Mathematics and Statistics. N 33. N.Y., USA: Springer, 2013. P. 239-260.
- Prokhorov Yu.V., Ulyanov V.V. Some approximation problems in statistics and probability // Limit Theorems in Probability, Statistics and Number Theory. Springer Proceedings in Mathematics and Statistics. N 42. Heidelberg, Germany: Springer, 2013. P. 236-249.

• 2012

- Gerasimov M., Kruglov V., Volodin A. On negatively associated random variables // Lobachevskii J. Mathematics. 2012. 33. N 1. P. 47-55.
- Korolev V.Yu. On the relationship between the generalized student t-distribution and the variance gamma distribution in statistical analysis of random-size samples // Doklady Mathematics. 2012. 86. N 1. P. 566-570.
- Korolev V.Yu., Popov S.V. Improvement of convergence rate estimates in the central limit theorem under weakened moment conditions // Doklady Mathematics. 2012. 86. N 1. P. 506-511.
- Korolev V.Yu., Shevtsova I.G. An improvement of the Berry-Esseen inequality with applications to Poisson and mixed Poisson random sums // Scandinavian Actuarial J. 2012. 2012. N 2. C. 81-105.
- Kruglov V.M. A characterization of the convolution of Gaussian and Poisson Distributions // Sankhya: the Indian J. Statistics. 2012. 74-A. N 1. P. 1-9.
- Shestakov O.V. Asymptotic normality of adaptive wavelet thresholding risk estimation // Doklady Mathematics. 2012. 86. N 1. P. 556-558.
- Shestakov O.V. Regularizing the method of reconstructing a function on the basis of its spherical Radon transform // Moscow University Comput. Mathematics and Cybern. 2012. 36. N 1. P. 23-27.
- Shevtzova I. Moment-type estimates with asymptotically optimal structure for the accuracy of the normal approximation // Annales Mathematicae et Informaticae. 2012. 39. P. 241-307.
- Ushakov A.V., Ushakov V.G. Limiting expectation time distribution for a critical load in a sys-tem with relative priority // Moscow University Comput. Mathematics and Cybern. 2012. 36. N 4. P. 169-175.
- Ushakov A.V., Ushakov V.G. Queue length in an absolute priority system with hyperexponential input flow // Moscow University Comput. Mathematics and Cybern. 2012. 36. N 1. P. 28-35.
- Ushakov V.G., Ushakov N.G. On bandwidth selection in kernel density estimation // J. Nonparam. Statistics. 2012. 24. N 2. P. 419-428.
- Zakharova T., Khaziakhmetov M. The using of wavelet analysis in climatic challenges // Annales Mathematicae et Informaticae. 2012. 39. C. 159-172.
- Ulyanov V.V. Accurate approximation of correlation coefficients by short edgeworth-chebyshev expansion and its statistical applications // Prokhorov and Contemporary Probability Theory. Springer Proc. in Math. & Statistics. N 33. N.Y.: Springer, 2012. P. 239-260.