The Workshop is organized by the Department of Probability Theory, Statistics and Actuarial Mathematics of Taras Shevchenko National University of Kyiv
All sessions are organized as Zoom meetings
Time is everywhere local time in Kyiv (GMT+3)
October 11: official program, scientific talks, 14:15-18:40
October 12: scientific talks, 14:00-18:00
Program committee: Kestutis Kubilius, Yuliya Mishura
Organizing committee: Iryna Bodnarchuk, Kostiantyn Ralchenko
Thomas Augustin, Ludwig-Maximilians-Universität München, Germany
Title of the talk: Survival Analysis under Generalized Measurement Error Models
Survival models provide a very simple but practically highly relevant example of counting processes with intensities that depend on certain characteristics (covariates) of the units investigated. This talk considers Cox's proportional hazards model and some accelerated failure time models under the frequently occurring complication that covariates may be subject to measurement error. For the classical additive measurement error model, powerful correction methods for unbiased parameter estimation have been developed in the literature. We examine in detail one of these correction methods, namely the principle of corrected score functions, aiming at extending it to more complex and "realistic" error models. In this context, we discuss first ideas to handle internal dependencies, error structures of Berkson type, and eventually set-valued error models based on the framework of imprecise probabilities.
Sandor Baran, University of Debrecen, Hungary
Title of the talk: K-optimal designs for regression models driven by Ornstein-Uhlenbeck processes and fields
Properties of the K-optimal design minimizing the condition number of the Fisher information matrix of the unknown parameters of regression models driven by Ornstein-Uhlenbeck processes and sheets are investigated and the differences compared with the classical D-optimal sampling are highlighted. The problems of existence of K-optimal designs are studied and the dependence of the two designs on the covariance parameters of the driving processes are clarified. Finally, a simulation study displaying the superiority of the K-optimal design for large parameter values of the driving random process is presented.
Taras Bodnar, Stockholm University, Sweden
Title of the talk: Singular Conditional Autoregressive Wishart Model for Realized Covariance Matrices
Realized covariance matrices are often constructed under the assumption that richness of intra-day return data is greater than the portfolio size, resulting in non-singular matrix measures. However, when for example the portfolio size is large, assets suffer from illiquidity issues, or market microstructure noise deters sampling on very high frequencies, this relation is not guaranteed. Under these common conditions, realized covariance matrices may obtain as singular by construction. Motivated by this situation, we introduce the Singular Conditional Autoregressive Wishart (SCAW) model to capture the temporal dynamics of time series of singular realized covariance matrices, extending the rich literature on econometric Wishart time series models to the singular case. This model is furthermore developed by covariance targeting adapted to matrices and a sectorwise BEKK-specification, allowing excellent scalability to large and extremely large portfolio sizes. Finally, the model is estimated to a 20 year long time series containing 50 stocks and to a 10 year long time series containing 300 stocks, and evaluated using out-of-sample forecast accuracy. It outperforms the benchmark models with high statistical significance and the parsimonious specifications perform better than the baseline SCAW model, while using considerably less parameters.
Khalifa Es-Sebaiy, Kuwait University, Kuwait
Title of the talk: Berry-Esseen bounds of second moment estimators for Gaussian processes observed at high frequency
Let Z be a centered stationary Gaussian process. We study two second moment estimators of the variance of Z0 based on continuous-time and discrete-time observations of Z, using tools from analysis on Wiener space. We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem. We apply these results to asymptotically stationary Gaussian processes and estimate the drift parameter for Gaussian Ornstein-Uhlenbeck processes.
Alexander Ivanov, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Ukraine
Title of the talk: LSE Consistency of the Symmetric Textured Surface Parameters
A multivariate trigonometric regression model is considered. Several discrete modifications of the similar bivariate trigonometric model were studied in detail in the literature on signal and image processing thanks to multiple applications in the analysis of symmetric textured surfaces. In the lecture strong consistency of the least squares estimator for amplitudes and angular frequencies is obtained in such a multivariate model on the assumption that random noise is a homogeneous and Isotropic Gaussian, specifically, strongly dependent random field on M-dimensional Euclidean space.
Kestutis Kubilius, Vilnius university, Lithuania
Title of the talk: Stochastic differential equation with a soft wall
Recently, much attention has been paid to SDEs driven by fractional Brownian motion (fBm). In our research we are interested in fractional SDE with a soft wall. What do we mean by such type of an equation? It has been established that SDE with reflection can be imagined as equations having a hard wall. Now by introducing repulsion instead of reflection one gets an SDE with a soft wall. In contrast to the SDE with reflection, where the process cannot pass the hard wall, the soft wall is repulsive but not impenetrable. As the process crosses the soft wall boundary, it experiences the force of a chosen magnitude in the opposite direction. When the process is far from the wall, the force acts weakly.
We are trying to find conditions under which SDE with a soft wall has a unique solution and to construct an implicit Euler approximation with a rate of convergence for this equation. Using the example of the fractional Vasicek process with soft walls, we illustrate the dependence of the behavior of the solution on the repulsion force.
Rostyslav Maiboroda, Taras Shevchenko National University of Kyiv, Ukraine
Title of the talk: Estimation of concentrations parameters in models of mixtures with varying concentrations
Finite Mixture Models are useful in statistical analysis of heterogeneous data, in cluster analysis and pattern recognition. We will consider the model at which the concentrations of mixture components (mixing probabilities) are different for different observations. The components distributions are completely unknown, but there is some parametrical model of the concentrations. Least squares and smoothed empirical likelihood estimators will be discussed for the estimation of these parameters. Results of simulations will be presented.
Lutz Mattner, Universität Trier, Germany
Title of the talk: A convolution inequality, yielding a sharper Berry-Esseen theorem for summands Zolotarev-close to normal
The classical Berry-Esseen error bound, for the normal approximation to the law of a sum of independent and identically distributed random variables, is improved by replacing the standardised third absolute moment by a weak norm distance to normality. We thus sharpen and simplify two results of Ulyanov (1976) and of Senatov (1998), each of them previously optimal, in the line of research initiated by Zolotarev (1965) and Paulauskas (1969).
Our proof is based on a seemingly incomparable normal approximation theorem of Zolotarev(1986), combined with our main technical result:
The Kolmogorov (supremum norm of distribution function) distance between a convolution of two laws and a convolution of two Lipschitz laws is bounded homogeneously of degree 1 in terms of the Wasserstein distances (L1 norms of distribution functions) of the corresponding factors.
Yuliya Mishura, Taras Shevchenko National University of Kyiv, Ukraine
Title of the talk: Statistical estimation in the models with memory
We investigate the mixed fractional Brownian motion with trend, driven by a standard
Brownian motion and a fractional Brownian motion. We develop and compare two approaches to estimation of
all unknown parameters by discrete observations. Also, we consider the model with trend and two
fractional Brownian motions with different Hurst indices.
Ostap Okhrin, Technical University of Dresden, Germany
Title of the talk: Infinitely stochastic micro reserving (Jointly with Matúš Maciak and Michal Pešta)
Stochastic forecasting and risk valuation are now front burners in a list of applied and theoretical sciences. In this work, we propose an unconventional tool for stochastic prediction of future expenses based on the individual (micro) developments of recorded events. Considering a firm, enterprise, institution, or any entity, which possesses knowledge about particular historical events, there might be a whole series of several related subevents: payments or losses spread over time. This all leads to an infinitely stochastic process at the end. The aim, therefore, lies in predicting future subevent flows coming from already reported, occurred but not reported, and yet not occurred events. The emerging forecasting methodology involves marked time-varying Hawkes process with marks being other time-varying Hawkes processes. The estimated parameters of the model are proved to be consistent and asymptotically normal under simple and easily verifiable assumptions. The empirical properties are investigated through a simulation study. In the practical part of our exploration, we elaborate a specific actuarial application for micro claims reserving.
Kostiantyn Ralchenko, Taras Shevchenko National University of Kyiv, Ukraine
Title of the talk: Drift parameters estimation in the Cox–Ingersoll–Ross model
The talk is devoted to the drift parameters estimation in the Cox–Ingersoll–Ross model.
We prove the strong consistency of the maximum likelihood estimators based on the continuous-time
observations and obtain their rate of convergence in probability. Then we introduce the discrete versions of
these estimators and investigate their asymptotic behavior. In particular we establish the conditions for weak
and strong consistency, asymptotic normality and get the rate of convergence in probability. The quality of
the estimators is illustrated by simulation results.
Shalabh, Indian Institute of Technology Kanpur, India
Title of the talk: Goodness of Fit Statistic in Non-parametric Measurement Error Model
The coefficient of determination (R2) is used for judging the goodness of fit in a
multiple linear regression model only when the observations are correctly observed without any
measurement error. In the non-parametric regression model with the presence of measurement errors in the data,
the conventional R2 provides invalid results. To avoid this issue, a goodness of fit statistic
for non-parametric multiple measurement error models has been proposed in this talk.
Sergiy Shklyar, Taras Shevchenko National University of Kyiv, Ukraine
Title of the talk: Sufficiency estimator in the inverse exponential regression
Consider an inverse exponential regression with measurement errors. As the exponential
distribution makes a one-parameter exponential family, and the error-free inverse exponential regression
model is a classical generalized linear model, conditional score methods are applied in the model with
Gaussian errors. Thus, a conditional maximum likelihood ("sufficiency") estimator and conditional-score
estimators are constructed. Conditions for consistency of the sufficiency estimator are provided. This model
and its generalizations can be used in cohort studies.
Nakahiro Yoshida, University of Tokyo, Japan
Title of the talk: Adaptive and non-adaptive estimation for degenerate diffusion processes
We consider a degenerate system of stochastic differential equations. The first component of
the system has a parameter θ1 in a non-degenerate diffusion coefficient and a parameter
θ2 in the drift term. The second component has a drift term parameterized by
θ3 and no diffusion term. Parametric estimation of the degenerate diffusion system from
time-discrete observations is discussed under ergodicity. Asymptotic normality is proved in three different
situations for an adaptive estimator for θ3 with some initial estimators for
(θ1,θ2), an adaptive one-step estimator for
(θ1,θ2,θ3) with some initial estimators for them, and
a joint quasi-maximum likelihood estimator for (θ1,θ2,θ3)
without any initial estimator. The estimators incorporate information of the increments of both components.
By this construction, we show the asymptotic variance of the estimator for θ1 is smaller
than the standard one based on the first component only. The convergence of the estimator for
θ3 is much faster than the other parameters. The resulting asymptotic variance is smaller
than that of a seemingly natural estimator only using the increments of the second component.
This is a joint work with Arnaud Gloter.