Sergiy Shklyar

Taras Shevchenko National University of Kyiv · Faculty of mechanics and mathematics

A simple exponential regression model is considered where the rate parameter of the response variable linearly depends on the explanatory variable. We consider complications of the model: censoring of the response variable (either upper censoring or interval observations), the additive classical error or multiplicative Berkson error in the explanat...

This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analyti...

The paper is devoted to three-parametric self-similar Gaussian Volterra processes that generalize fractional Brownian motion. We study the asymptotic growth of such processes and the properties of long- and short-range dependence. Then we consider the problem of the drift parameter estimation for Ornstein–Uhlenbeck process driven by Gaussian Volter...

This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analyti...

We study Volterra processes Xt=∫0tK(t,s)dWs, where W is a standard Wiener process, and the kernel has the form K(t,s)=a(s)∫stb(u)c(u-s)du. This form generalizes the Volterra kernel for fractional Brownian motion (fBm) with Hurst index H>1/2. We establish smoothness properties of X, including continuity and Hölder property. It happens that its Hölde...

In this paper the study of a three-parametric class of Gaussian Volterra processes is continued. This study was started in Part I of the present paper. The class under consideration is a generalization of a fractional Brownian motion that is in fact a one-parametric process depending on Hurst index H. On the one hand, the presence of three paramete...

The stochastic process of the form \[ {X_{t}}={\int _{0}^{t}}{s^{\alpha }}\left({\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\right)\hspace{0.1667em}d{W_{s}}\] is considered, where W is a standard Wiener process, $\alpha >-\frac{1}{2}$, $\gamma >-1$, and $\alpha +\beta +\gamma >-\frac{3}{2}$. It is proved that the process X is...

The increased risk of thyroid cancer among individuals exposed during childhood and adolescence to Iodine-131 (¹³¹I) is the main statistically significant long-term effect of the Chornobyl accident. Several radiation epidemiological studies have been carried out or are currently in progress in Ukraine, to assess the risk of radiation-related health...

The special-purpose software implementation for estimating the subpixel shift between satellite images using advanced computer technology is described in this paper. The automatic calculation of the mutual subpixel shift between a pair of digital satellite images by correlation algorithm is performed. The proposed implementation was tested on a sta...

We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space D [ 0 , T ] . Then we investigate the convergence of the related multiplicative scheme to a process that can be interpreted as an asset price with memory. As an example, we study an additive scheme that converge...

We study Volterra processes $X_t = \int_0^t K(t,s) dW_s$, where $W$ is a standard Wiener process, and the kernel has the form $K(t,s) = a(s) \int_s^t b(u) c(u-s) du$. This form generalizes the Volterra kernel for fractional Brownian motion (fBm) with Hurst index $H>1/2$. We establish smoothness properties of $X$, including continuity and Holder pro...

A novel physically justified method for spatial resolution enhancement of satellite dual-polarization synthetic aperture radar data is proposed. The method starts from the conversion of the specific land surface radar backscattering into the land surface dielectric permittivity in each polarization band separately. Said conversion is founded on a w...

A new physically conditioned method for spatial resolution enhancement of satellite dual-polarization synthetic aperture radar data is considered. The method is based on the conversion of the specific Earth's surface backscattering into the dielectric permittivity in each polarization band separately and resulting permittivity field superresolution...

We consider the distance between the fractional Brownian motion defined on the interval [0,1] and the space of Gaussian martingales adapted to the same filtration. As the distance between stochastic processes, we take the maximum over [0,1] of mean-square deviances between the values of the processes. The aim is to calculate the function a in the G...

This chapter describes the main properties of fractional Brownian motion (fBm), including its integral representations. It formulates the minimizing problem simplifying it at the same time. The chapter proposes a positive lower bound for the distance between fBm and the space of Gaussian martingales. The main problem of minimization procedure is th...

This chapter first considers representation of fractional Brownian motion (fBm) via the uniformly convergent series of special Lebesgue integrals. It then presents the approximation of a fBm by semimartingales. The chapter shows that pathwise stochastic integral with respect to fBm can be approximated by the stochastic integrals with respect to sem...

This chapter describes the procedure of evaluation of the minimizing function and the distance between fractional Brownian motion (fBm) and the respective class of Gaussian martingales in some cases. The cases include: integrand is a constant function; it is a power function with a fixed exponent; it is a power function with arbitrary non‐negative...

The purpose of the article is twofold. Firstly, we review some recent results on the maximum likelihood estimation in the regression model of the form \(X_t = \theta G(t) + B_t\), where B is a Gaussian process, G(t) is a known function, and \(\theta \) is an unknown drift parameter. The estimation techniques for the cases of discrete-time and conti...

The purpose of the article is twofold. Firstly, we review some recent results on the maximum likelihood estimation in the regression model of the form $X_t = \theta G(t) + B_t$, where $B$ is a Gaussian process, $G(t)$ is a known function, and $\theta$ is an unknown drift parameter. The estimation techniques for the cases of discrete-time and contin...

Both mathematical model and software module for automatic estimating subpixel shift of aerial image acquired from quadcopter are described. The said shift henceforth will be required for super-resolution of fused aerial image.

This paper deals with a homoskedastic errors-in-variables linear regression model and properties of the total least squares (TLS) estimator. We partly revise the consistency results for the TLS estimator previously obtained by the author [18]. We present complete and comprehensive proofs of consistency theorems. A theoretical foundation for constru...

The data used in estimation of radiation exposure dose are not known exactly. We discuss different types of errors which are present in radio-epidemiological models we work with. We illustrate these types or errors in simplified models and share our experience on how the errors of different types affect the estimates of radiation risk parameters. K...

We investigate the regression model Xt = θG(t) + Bt, where θ is an unknown parameter, G is a known nonrandom function, and B is a centered Gaussian process. We construct the maximum likelihood estimators of the drift parameter θ based on discrete and continuous observations of the process X and prove their strong consistency. The results obtained g...

We consider a structural linear regression model with measurement errors. A new parametrization is proposed in which the expectation of the response variable plays the role of a new parameter instead of the intercept. This enables us to form three groups of asymptotically independent estimators in the case where the ratio of variances measurement e...

This monograph discusses statistics and risk estimates applied to radiation damage under the presence of measurement errors. The first part covers nonlinear measurement error models, with a particular emphasis on efficiency of regression parameter estimators. In the second part, risk estimation in models with measurement errors is considered. Effic...

This is the second part of the author paper published in Theor. Probability and Math. Statist. 92 (2016), 147–161. The first part considers the functional version of the conic section fitting problem and states the asymptotic normality of the ALS2 estimator for the coefficients of the conic section. In the present paper, a similar theorem on the as...

The paper deals with the regression model X_t = \theta t + B_t , t\in[0, T ], where B=\{B_t, t\geq 0\} is a centered Gaussian process with stationary increments. We study the estimation of the unknown parameter $\theta$ and establish the formula for the likelihood function in terms of a solution to an integral equation. Then we find the maximum lik...

The conic section fitting problem is considered. True points are assumed to lie on a conic section. The points are observed with additive errors, which are independent and have bivariate normal distribution N(0, σ²I) with unknown σ². We study asymptotic properties of the estimator of conic section parameters introduced by Kukush, Markovsky, and Van...

Spatial resolution is the main specification of satellite imagery quality, including the thermal infrared imagery. Many technical restrictions opposed to realize the high spatial resolution imagers of thermal infrared band. The software-based method for imagery spatial resolution enhancement is presented. The method used two sub-pixel shifted image...

We consider the two-line fitting problem. True points lie on two straight lines and are observed with Gaussian perturbations. For each observed point, it is not known on which line the corresponding true point lies. The parameters of the lines are estimated. This model is a restriction of the conic section fitting model because a couple of two line...

In this paper, the influence of measurement errors in exposure doses in a regression model with binary response is studied. Recently, it has been recognized that uncertainty in exposure dose is characterized by errors of two types: classical additive errors and Berkson multiplicative errors. The combination of classical additive and Berkson multipl...

The mathematical and physical models of the new frame infrared spectroradiometer based on microbolometer array sensor with subpixel image registration are presented. It is planned to include the radiometer into onboard instrumentation of the future “Sich” satellite system for the land surface physical characterization by enhanced spatial resolution...

The best uniform approximation of a Wiener process by integrals of the Form (formula presented)is established in the space L∞([0, T];L2(Ω)), where {BtH, t ∈ [0, T]} is the fractional Brownian motion with the Hurst index H and f(s)= k·sα, s ∈ [0, T], for k > 0 and α = H − 1/2.

We consider the Berkson model of logistic regression with Gaussian and homoscedastic error in regressor. The measurement error variance can be either known or unknown. We deal with both functional and structural cases. Sufficient conditions for identifiability of regression coefficients are presented. Conditions for identifiability of the model are...

Ignoring errors in exposure doses leads to reducing of the radiation risk estimates, and therefore, is a reason for underestimation of unhealthy action of the exposure. The first part of the monograph is devoted to nonlinear measurement error models and parameter estimation in the models. The second part deals with the problem of risk estimation in...

The autoregressive model with errors in variables and with a normally distributed control sequence is considered. For the main sequence, two cases are considered: (a) the main sequence has a stationary distribution, and (b) the initial distribution is arbitrary, independent of the control sequence, and has a finite fourth moment. Here the elements...

In this chapter we deal with a regression model in which there is Gaussian error in the regressor and the response variable has an exponential distribution. We consider three methods of estimation: Sufficiency estimator, Conditional Score estimators developed by Stefanski and Carroll (Biometrika 74, 703–716 1987), and Corrected Score estimator deve...

The Poisson regression Berkson type model with a Gaussian error in the regressor is studied. Simple score and quasi-likelihood estimators of the regression parameters are considered. Sufficient conditions for the strong consistency of the estimators and sufficient conditions for the uniqueness of a solution of estimating equations are found. The pr...

The methods, model, algorithm and special software for subpixel image processing from the frame infrared camera for remote sensing satellite system are considered. Subpixel processing of image sequence provides a significant improvement in spatial resolution and minimum resolvable temperature difference (MRTD) of the infrared camera. Described resu...

The 1986 accident at the Chernobyl nuclear power plant remains the most serious nuclear accident in history, and excess thyroid cancers, particularly among those exposed to releases of iodine-131 remain the best-documented sequelae. Failure to take dose-measurement error into account can lead to bias in assessments of dose-response slope. Although...

Objective: To estimate the influence of Berkson errors in exposure doses on the results of risk analysis within example of radiation epidemiological studies of the thyroid cancer prevalence. Materials and methods: The impact of Berkson errors of the thyroid doses in a dose-response analysis is studied by the method of stochastic simulation. Res...

Development of new satellite imagery processing algorithms for resolution enhancement and subpixel analysis allows to increase efficiency of remote sensing applications. Insufficiency of the spatial resolution leads to uncertainty in satellite imagery classification and analysis. In this paper we present the technique for satellite imagery spatial...

We consider a Berkson model of logistic regression with a single regressor and normally distributed homoscedastic errors in the regressor (the so-called Berkson model). The variance of the errors is assumed to be known. Sufficient conditions for the uniqueness of a solution of the limit estimating equation in the structural model, and sufficient co...

We present a method for the sensor spectral response calibration of the “Sich-2” multispectral satellite system on the basis of satellite imaging for ground calibration test sites. A special parameterization of spectral response functions of the multispectral sensor is proposed. The parameterization allows one to solve analytically a system of equa...

We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exist a unique martingale closest to fractional Brownian motion in a specific sense. It shown that this martingale has a specific form. Numerical results concerning the approximation problem are given.

A new approach to subpixel analysis of hyperspectral imagery is offered. The traditional endmember spectral unmixing, the spatial reallocation of endmember subpixel fractions is performed taking their mutual position on the reference higher spatial resolution multispectral image. The mixing of reallocated endmember fractions in pixels is executed;...

With a binary response Y, the dose-response model under consideration is logistic in flavor with pr(Y=1 | D) = R (1+R)(-1), R = λ(0) + EAR D, where λ(0) is the baseline incidence rate and EAR is the excess absolute risk per gray. The calculated thyroid dose of a person i is expressed as Dimes=fiQi(mes)/Mi(mes). Here, Qi(mes) is the measured content...

A homoscedastic errors-in-variables linear regression model is considered. The total least squares estimator is studied. New conditions for the consistency and strong consistency of the total least squares estimator are proposed. These conditions are weaker than those proposed by Kukush and Van Huffel (Metrika 59 (2004), 75- 97).

We consider an error-in-variables model for a polynomial regression with Gaussian errors. We assume that the covariance matrix of the measurement errors of the regressor and the echo is known up to a scalar factor. We consider the moment estimator of regression coefficients proposed by Cheng and Schneeweiss. Sufficient conditions for the strong con...

Adjusted least squares (ALS) estimators for the conic section problem are considered. Consistency of the translation invariant version of ALS estimators is proved. The similarity invariance of the ALS estimator with the estimated noise variance is shown. The conditions for consistency of the ALS estimator are relaxed compared with the ones of the p...

We consider a polynomial regression model, where the covariate is measured with Gaussian errors. The measurement error variance is supposed to be known. The covariate is normally distributed with known mean and variance. Quasi score (QS) and corrected score (CS) are two consistent estimation methods, where the first makes use of the distribution of...

The rigorous algorithm of separation of mixtures of spectral components in pixels of hyperspectral aerospace imagery on the basis of least-squares method with restrictions, is given. The use of separation of mixtures with the known spectrums of components while land cover classification. Subpixel classification is useful for solving different subje...

We consider a Poisson model, where the mean depends on certain covariates in a log-linear way with unknown regression parameters. Some or all of the covariates are measured with errors. The covariates as well as the measurement errors are both jointly normally distributed, and the error covariance matrix is supposed to be known. Three consistent es...

A structural errors-in-variables model is investigated, where the response variable follows a Poisson distribution. Assuming the error variance to be known, we consider three consistent estimators and compare their relative efficiencies by means of their asymptotic covariance matrices. The comparison is made for arbitrary error variances. The struc...

We compare two consistent estimators of the parameter vector beta of a general exponential family measurement error model with respect to their relative efficiency. The quasi score (QS) estimator uses the distribution of the regressor, the corrected score (CS) estimator does not make use of this distribution and is therefore more robust. However, i...