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

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