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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.