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  • Regular Issue
    Vol. 54 No. 2 (2025)

  • Special Issue by the Department of Probability, Statistics and Actuarial Mathematics at TSNU of Kyiv
    Vol. 54 No. 1 (2025)

    Editorial

    This special issue of the Austrian Journal of Statistics is the second one which is devoted to the modern achievements of Ukrainian scientists in probability and mathematical statistics. This issue, as well as the first one (52(SI), 2023), was initiated by the Editor of the Austrian Journal of Statistics Professor Matthias Templ to express solidarity with Ukraine and to support Ukrainian scientists.

    The Guest Editors who organized this issue are from the Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv: Professor, Doctor of Sciences Yuliya Mishura and Leading scientific researcher, Doctor of Sciences Lyudmyla Sakhno.

    We consider very important such support as the publication of two subsequent special issues of the Austrian Journal of Statistics dedicated to the achievements of Ukrainian scientists. This issue presents some modern trends of research of the scientific school on Probability Theory and Mathematical Statistics of Taras Shevchenko National University of Kyiv. In particular, young scientists and PhD students of the university had the opportunity to publish their results. The issue also contains the articles of scientists from the Igor Sikorsky Kyiv Polytechnic Institute, National University of “Kyiv-Mohyla Academy” and of our colleagues who worked during a long time at the Department of Probability Theory, Statistics and Actuarial Mathematics and now represent Ukrainian science in various universities of Australia, Great Britain, Sweden.

    R.Maiboroda, V.Miroshnychenko and O.Sugakova study the model of mixture with varying concentrations under the assumption that the components’ distributions are completely unknown, while the concentrations are known up to some unknown euclidean parameter. Two approaches are considered for the semiparametric estimation of this parameter: the least squares estimator and the empirical maximum likelihood estimator. The properties of these estimators are studied and compared, numerical simulations are provided.

    A.Malyarenko presents the review of the current state of the spectral theory of random functions of several variables created by Professor M. I. Yadrenko at the end of 1950s. It turns out that the spectral expansions of multi-dimensional homogeneous and isotropic random fields are governed by a pair of convex compacts and are especially simple when these compacts are simplexes. The new result of the paper gives necessary and sufficient conditions for such a situation in terms of the group representation that defines the field.

    A.Ivanov and V.Hladun consider the statistical inference problem for a time continuous statistical model of multiple chirp signal observed against the background of strongly or weakly dependent stationary Gaussian noise. Strong consistency and asymptotic normality of the least squares estimates for such a trigonometric regression model parameters are obtained.

    Yu.Mishura, K.Ralchenko and O.Dehtyar study the Vasicek model driven by a tempered fractional Brownian motion and derive the asymptotic distributions of the least-squares esti- mators (based on continuous-time observations) for the unknown drift parameters. This work continues the investigation by Mishura and Ralchenko, where these estimators were introduced and their strong consistency was proved.

    The paper by O.Prykhodko and K.Ralchenko investigates the simultaneous estimation of two drift parameters of a Cox-Ingersoll-Ross model, for which observations can be made either continuously or at discrete time instants. For continuous-time observations, the joint asymptotic normality of the strongly consistent parameter estimators is established. Additionally, the discrete counterparts of these estimators are studied and their strong consistency and joint asymptotic normality are proved.

    The paper by L.Sakhno presents conditions for the asymptotic normality of nonlinear functionals of periodograms based on tapered data. Stationary Gaussian random fields are consid- ered. Two limit theorems are stated: for the first one the certain condition of integrability of the spectral density of the field is assumed, and the second result is for spectral densities with the prescribed behavior near the points of singularities.

    The paper by A.Bilchouris and A.Olenko overviews and investigates several nonparametric methods of estimating covariograms and gives a unified approach to compare the main methods used in applied research. The main focus is on such properties of covariograms as bias, positive- definiteness and behaviour at large distances. Several theoretical properties are discussed and some surprising drawbacks of well-known estimators are demonstrated. The research is sup- ported by extensive numerical studies. The results provide an important insight and guidance for practitioners who use estimated covariograms in various applications, including kriging, monitoring network optimisation, cross-validation, and other related tasks.

    N.Leonenko, A.Liu and N.Schestyuk propose several new models in finance known as the Fractal Activity Time Geometric Brownian Motion models with Student marginals. The au- thors summarize four models that construct stochastic processes of underlying prices with short-range and long-range dependencies. Solutions of option Greeks is derived and compared with those in the Black-Scholes model. The performance of delta hedging strategy is analyzed using simulated time series data and it is verified that hedging errors are biased particularly for long-range dependence cases. The authors also apply underlying model calibration on S&P 500 index (SPX) and the U.S./Euro rate, and implement delta hedging on SPX options.

    The paper by V.Golomoziy is devoted to establishing upper bounds for a difference of n- step transition probabilities for two time-inhomogeneous Markov chains with values in a locally compact space when their one-step transition probabilities are close. This stability result is applied to the functional autoregression in Rn.

    D.Ivanenko, V.Knopova and D.Platonov extend the Asmussen-Rosinski approach for the approximation of Levy processes. To simulate the value of the process at time t, a time- dependent truncation (or dymamic cutting) of the Levy measure is introduced and followed by the simulation of the large-jump component. The authors provide the sufficient condition under which the compensated small-jump part can be replaced by a Gaussian approximation. Weak approximation rates for both approaches are derived. Numerical simulations are presented to support the study and compare the performance of the method developed in the paper with the Asmussen-Rosinski approach.

    I.Rozora and A.Melnyk consider a time-invariant continuous linear system with a real-valued impulse response function which is defined on a bounded domain. A sample input-output cross- correlogram is taken as an estimator of the response function. The input process is supposed to be a zero-mean stationary Gaussian process represented as a finite sum with uncorrelated terms. A rate of convergence of IRF estimator is obtained that gives a possibility to propose a nonparametric goodness-of-fit testing on IRF.

    We would like to express our sincere gratitude to Professor Matthias Templ for his support and for organization of this issue.

    We are thankful to all contributors for submitting their newest research results and believe that the issue will be interesting for a wide audience in view of variety of topics covered.

     

    Yuliya Mishura

    Lyudmyla Sakhno

    Taras Shevchenko National University of Kyiv

  • Regular Issue
    Vol. 53 No. 4 (2024)

  • Regular Issue
    Vol. 53 No. 3 (2024)

  • Regular Issue
    Vol. 53 No. 2 (2024)

  • Regular Issue
    Vol. 53 No. 1 (2024)

  • Recent Trends in Probability and Statistics in Ukraine
    Vol. 52 No. SI (2023)

    The special issue is devoted to Recent Trends in Probability and Statistics in Ukraine, with an emphasis on the presentation of research by young scientists. This collection of papers represents some latest results of researchers from Taras Shevchenko National University of Kyiv, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” and Kyiv School of Economics.

    The topics covered is a mixture of classical and modern ones. Some classical and new stochastic models and problems of statistical inference have received in the papers a modern treatment, that warrants addressing recent challenges from both theoretical and practical viewpoints. We believe the issue will be interesting for a wide audience.

    O. Braganets and A. Iksanov investigate a nested occupancy scheme in a random environment which can be thought of as a generalization of the classical Karlin occupancy scheme. The model is given by a nested hierarchy of boxes with hitting probabilities of boxes defined in terms of iterated fragmentation of a unit mass. Previous research resulted in a multivariate functional central limit theorem with centering for the cumulative occupancy counts as the number of balls becomes large. In the present paper a counterpart of that result is obtained, in which centering is not needed and the limit processes are no longer Gaussian.

    C. Dong, O. Marynych and V. Melnykov work out a universal approach to the analysis of random sieves and generalized leader-election procedures. A random sieve of the set of positive integers by a random set R is a nested sequence of subsets such that every set in the sequence is obtained by removing elements of the previous set with indices lying outside an independent copy of R. This model has been previously studied in two particular cases only: (a) R is a range of an increasing random walk on positive integers; (b) R is the set of record times in an infinite sample from a continuous distribution. Using a martingale approach the authors prove various limit theorems for several functionals that characterise the speed of sieving.

    I. Samoilenko, G. Verovkina and T. Samoilenko are concerned with a model of particle evolution on a complex plane, which is a generalization of the classical Goldstein-Kac model. The authors obtain a telegraph-type equation for some functionals of the evolution and construct solutions to the corresponding Cauchy problem with complex-analytic initial conditions. The method is based on reconstruction of complex-analytic functions by combination of power functions, for which the corresponding solutions are the moments of the evolution process. This approach enables avoiding analytic difficulties of the classical Riemann method for the telegraph equation. The solutions constructed in the present paper explicitly contain regular and boundary-layer components that may be useful for calculating approximate solutions.

    G. Shevchenko and A. Yaroshevskiy investigate continuous-time lattice random walks in a stationary random environment and prove a limit theorem for these walks, which is similar to that in the nonlattice case but stated under less restrictive assumptions on the distribution of jumps and under very general conditions on a random environment.

    The paper by V. Golomoziy and O. Moskanova is related to stability theory which belongs to a classical part of the theory of Markov chains. The authors are interested in recurrence properties of a time-inhomogeneous Markov chain and demonstrate that such a chain can be polynomially recurrent while exhibiting different dynamics in comparison to its homogeneous counterpart.

    A. Dzhoha and I. Rozora investigate the problem of design of clinical trials by using the multi-armed bandit problem, which is a classical example of the exploration-exploitation trade- off suited to model sequential resource allocation under uncertainty. Since the response to a procedure in clinical trials is not immediate, the authors justify the importance of adaptation of multi-armed bandit policies to delays. The Upper Confidence Bound policy is analyzed by applying such a classical tool as sub-Gaussian concentration inequalities.

    The paper by O. Hopkalo, L. Sakhno and O. Vasylyk investigates sample paths properties of random fields from the spaces of φ-sub-Gaussian random variables, which generalize Gaussian and sub-Gaussian ones. By using the entropy approach, the authors point out some bounds for the distribution tails of suprema of φ-sub-Gaussian random fields under different conditions imposed on their increments. This work is motivated and illustrated by applications to random solutions of partial differential equations.

    Tykhonenko D. and Yamnenko R. are focused on several particular classes of random processes from Orlicz spaces of exponential type and derive some estimates for the distribution of supremum of a weighted sum of such processes deviated by a continuous monotone function. Weighted sums of sub-Gaussian Wiener and fractional Brownian motion processes are considered as examples.

    A. Ivanov and V. Hladun analyze the statistical inference problem for a time continuous statistical model of multiple chirp signal observed against the background of strongly or weakly dependent stationary Gaussian noise. In this special trigonometric regression model frequencies vary with time in a non-linear fashion like quadratic functions. The main result of the paper states the strong consistency of the least squares estimates for the model parameters.

    The paper by K. Ralchenko and M. Yakovliev is devoted to the estimation of parameters of a mixed fractional Brownian motion with a linear trend. The model is driven by both a standard Brownian motion and a fractional Brownian motion. The authors consider strongly consistent estimators of unknown model parameters, which were derived in an earlier work, and prove their joint asymptotic normality. A behavior of the estimators is also analyzed numerically.

    S. Shklyar considers a classical generalized linear model and its generalizations to cover various forms of errors and incomplete data. The base model is a simple exponential regression, in which the rate parameter of the response variable linearly depends on the explanatory variable. Estimates are presented for the cases where the base model becomes more complicated by adding the censoring of the response variable and/or measurement errors in the explanatory variable. The performance of estimates is verified by simulation.

    We would like to introduce the young researchers, who contributed to the issue. Taras Shevchenko National University of Kyiv is represented by: PhD students Oksana Braganets, Andrii Dzhoha and Viacheslav Melnykov from Faculty of Computer Science and Cybernetics; PhD students Dmytro Tykhonenko and Mykyta Yakovliev and Master Degree student Olga Moskanova from Faculty of Mechanics and Mathematics; Dr. Olga Hopkalo, who received her PhD degree in 2021 and is now an Assistant Professor at Faculty of Economics. Andriy Yaroshevskiy received his Master Degree from Faculty of Mechanics and Mathematics in 2021. The Master Degree student Viktor Hladun is from Igor Sikorsky Kyiv Polytechnic Institute.

    We thank all the authors of this issue. Also, our special thanks go to the reviewers for their valuable remarks and suggestions.

    We are very grateful to the Editor of the Austrian Journal of Statistics Professor Matthias Templ, who initiated the special issue to express solidarity with Ukraine and to support Ukrainian scientists. His help with the issue preparation is highly appreciated.

    This issue is published in a very hard time for Ukraine and the Ukrainian people. Ukraine is fighting and resisting the devastating Russian invasion that began on February 24, 2022, in a war that Russia has been waging since 2014. The damage to Ukraine is impossible to describe and grasp. In all areas, including educational and research institutions and science. We greatly appreciate all the help from all over the world and all the steps taken so far to provide support to Ukraine at all levels. To respond to the severity of the consequences that have been inflicted on Ukraine and to prevent further threats, we believe that comprehensive further steps against this war, which is contrary to all the values of our civilization, are necessary and urgent.

     

    Lyudmyla Sakhno and Alexander Iksanov

    Taras Shevchenko National University of Kyiv

  • Regular Issue
    Vol. 52 No. 5 (2023)

  • Regular Issue
    Vol. 52 No. 4 (2023)

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