arXiv subjects

Yury A. Kutoyants

Publications and source records attributed to Yury A. Kutoyants.

At least 19 recordsLinked to original sources

arXiv2023

Hidden AR Process and Adaptive Kalman Filter

The model of partially observed linear system depending on some unknown parameters is considered. An approximation of the unobserved component is proposed. This approximation is realized in three steps. First an estimator of the method of moments of unknown parameter is constructed. Then this estimator is used for defining the One-step MLE-process and finally the last estimator is substituted to the equations of Kalman filter. The solution of obtained equations provide us the approximation (adaptive Kalman filter). The asymptotic properties of all mentioned estimators and MLE and Bayesian estimators of the unknown parameters are described. The asymptotic efficiency of adaptive filtering is discussed.

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arXiv2023

Hidden Ergodic Ornstein-Uhlenbeck Process and Adaptive Filter

The model of partially observed linear stochastic differential equations depending on some unknown parameters is considered. An proximation of the unobserved component is proposed. This approximation is realized in three steps. First an estimator of the thod of moments of unknown parameter is constructed. Then this estimator is used for defining the One-step MLE-process and nally the last estimator is substituted to the equations of Kalman-Bucy (K-B) filter. The solution of obtained K-B equations ovide us the approximation (adaptive K-B filter). The asymptotic properties of all mentioned estimators and MLE and Bayesian timators of the unknown parameters are described. The asymptotic efficiency of the proposed adaptive filter is shown.

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arXiv2022

Volatility Estimation of Hidden Markov Processes and Adaptive Filtration

The partially observed linear Gaussian system of stochastic differential equations with low noise in observations is considered. A kernel-type estimators are used for estimation of the quadratic variation of the derivative of the limit of the observed process. Then this estimator is used for nonparametric estimation of the integral of the square of volatility of unobservable component. This estimator is also used for construction of substitution estimators in the case where the drift in observable component and the volatility of the state component depend on some unknown parameter. Then this substitution estimator and Fisher-score device allows us to introduce the One-step MLE-process and adaptive Kalman-Bucy filter.

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arXiv2020

Hidden Markov Model Where Higher Noise Makes Smaller Errors

We consider the problem of parameter estimation in a partially observed linear Gaussian system with small noises in the state and observation equations. We describe asymptotic properties of the MLE and Bayes estimators in the setting with state and observation noises of possibly unequal intensities. It is shown that both estimators are consistent, asymptotically normal with convergent moments and asymptotically efficient. This model has an unusual feature: larger noise in the state equation yields smaller estimation error. The proofs are based on asymptotic analysis of the Kalman-Bucy filter and the associated Riccati equation in particular.

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arXiv2020

Approximation of BSDE with Hidden Forward Equation and Unknown Volatility

In the present paper the problem of approximating the solution of BSDE is considered in the case where the solution of forward equation is observed in the presence of small Gaussian noise. We suppose that the volatility of the forward equation depends on an unknown parameter. This approximation is made in several steps. First we obtain a preliminary estimator of the unknown parameter, then using Kalman-Bucy filtration equations and Fisher-score device we construct an one-step MLE-process of this parameter. The solution of BSDE is approximated by means of the solution of PDE and the One-step MLE-process. The error of approximation is described in different metrics.

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arXiv2020

On Multi-step Estimation of Delay for SDE

We consider the problem of delay estimation by the observations of the solutions of several SDEs. It is known that the MLE for these models are consistent and asymptotically normal, but the likelihood ratio functions are not differentiable w.r.t. parameter and therefore numerical calculation of the MLEs has certain difficulties. We propose One-step and Two-step MLE, whose calculation has no such problems and provide estimator asymptotically equivalent to the MLE. These constructions are realized in two or three steps. First we construct preliminary estimators which are consistent and asymptotically normal, but not asymptotically efficient. Then we use these estimators and modified Fisher-score device to obtain One-step and Two-step MLEs. We suppose that its numerical realization is much more simple. Stochastic Pantograph equation is introduced and related statistical problems are discussed.

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arXiv2019

On Parameter Estimation of the Hidden Gaussian Process in perturbed SDE

We present results on parameter estimation and non-parameter estimation of the linear partially observed Gaussian system of stochastic differential equations. We propose new one-step estimators which have the same asymptotic properties as the MLE, but much more simple to calculate, the estimators are so-called "estimator-processes". The construction of the estimators is based on the equations of Kalman-Bucy filtration and the asymptotic corresponds to the small noises in the observations and state (hidden process) equations. We propose conditions which provide the consistency and asymptotic normality and asymptotic efficiency of the estimators.

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arXiv2019

On Parameter Estimation of Hidden Ergodic Ornstein-Uhlenbeck Process

We consider the problem of parameter estimation for the partially observed linear stochastic differential equation. We assume that the unobserved Ornstein-Uhlenbeck process depends on some unknown parameter and estimate the unobserved process and the unknown parameter simultaneously. We construct the two-step MLE-process for the estimator of the parameter and describe its large sample asymptotic properties, including consistency and asymptotic normality. Using the Kalman-Bucy filtering equations we construct recurrent estimators of the state and the parameter.

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arXiv2018

Method of Moments Estimators and Multu-step MLE for Poisson Processes

We introduce two types of estimators of the finite-dimensional parameters in the case of observations of inhomogeneous Poisson processes. These are the estimators of the method of moments and multi-step MLE. It is shown that the estimators of the method of moments are consistent and asymptotically normal and the multi-step MLE are consistent and asymptotically efficient. The construction of multi-step MLE-process is done in two steps. First we construct a consistent estimator by the observations on some learning interval and then this estimator is used for construction of one-step and two-step MLEs. The main advantage of the proposed approach is its computational simplicity.

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arXiv2018

Poisson Source Localization on the Plane. Change-Point Case

We present a detection problem where several spatially distributed sensors observe Poisson signals emitted from a single source of unknown position. The measurements at each sensor are modeled by independent inhomogeneous Poisson processes. A method based on Bayesian change-point estimation is proposed to identify the location of the source's coordinates. The asymptotic behavior of the Bayesian estimator is studied. In particular the consistency and the asymptotic efficiency of the estimator are shown. The limit distribution and the convergence of the moments are also described. The similar statistical model could be used in GPS localization problems.

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arXiv2018

Poisson Source Localization on the Plane. Smooth Case

We consider the problem of localization of Poisson source by the observations of inhomogeneous Poisson processes. We suppose that there are $k$ detectors on the plane and each detector provides the observations of Poisson processes whose intensity functions depend on the position of the emitter. We describe the properties of the maximum likelihood and Bayesian estimators. We show that under regularity conditions these estimators are consistent, asymptotically normal and asymptotically efficient. Then we propose some simple consistent estimators and this estimators are further used to construct asymptotically efficient One-step MLE-process.

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arXiv2018

On Cusp Location Estimation for Perturbed Dynamical Systems

We consider the problem of parameter estimation in the case of observation of the trajectory of diffusion process. We suppose that the drift coefficient has a singularity of cusp-type and the unknown parameter corresponds to the position of the point of the cusp. The asymptotic properties of the maximum likelihood estimator and Bayesian estimators are described in the asymptotics of {\it small noise}, i.e., as the diffusion coefficient tends to zero. The consistency, limit distributions and the convergence of moments of these estimators are established.

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arXiv2016

On Multi-step MLE-process for Markov Sequences

We consider the problem of the construction of the estimator-process of the unknown finite-dimensional parameter in the case of the observations of nonlinear autoregressive process. The estimation is done in two or three steps. First we estimate the unknown parameter by a learning relatively short part of observations and then we use the one-step MLE idea to construct an-estimator process which is asymptotically equivalent to the MLE. To have the learning interval shorter we introduce the two-step procedure which leads to the asymptotically efficient estimator-process too. The presented results are illustrated with the help of two numerical examples.

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arXiv2015

On ADF Goodness-of-Fit Tests for Perturbed Dynamical Systems

We consider the problem of the construction of the goodness-of-fit tests for diffusion processes with small noise. The basic hypothesis is composite parametric and our goal is to obtain asymptotically distribution free tests. We propose two solutions. The first one is based on the change of time and the second test we obtain with the help of some linear transformation of the "natural" statistics.

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arXiv2015

On Multi-Step MLE-Process for Ergodic Diffusion

We propose a new method of the construction of the asymptotically efficient estimator-processes asymptotically equivalent to the MLE and the same time much more easy to calculate. We suppose that the observed process is ergodic diffusion and that there is a learning time interval of the length negligeable with respect to the whole time of observations. The preliminary estimator obtained after the learning time is then used in the construction of one-step and two-step MLE processes. We discuss the possibility of the applications of the proposed estimation procedure to several other observations models.

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arXiv2014

On Score-Functions and Goodness-of-Fit Tests for Stochastic Processes

The problems of the construction of the asymptotically distribution free goodness-of-fit tests for three models of stochastic processes are considered. The null hypothesis for all models is composite parametric. All tests are based on the score-function processes, where the unknown parameter is replaced by the MLE. We show that a special change of time transforms the limit score-function processes into the Brownian bridge. This property allows us to construct the asymptotically distribution free tests for the following three models of stochastic processes : dynamical systems with small noise, ergodic diffusion processes, inhomogeneous Poisson processes and nonlinear AR time series.

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arXiv2013

On Approximation of the Backward Stochastic Differential Equation

We consider the problem of approximation of the solution of the backward stochastic differential equation in the Markovian case. We suppose that the trend coefficient of the diffusion process depends on some unknown parameter and the diffusion coefficient of this equation is small. We propose an approximation of this solution based on the one-step MLE of the unknown parameter and we show that this approximation is asymptotically efficient in the asymptotics of "small noise".

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arXiv2013

Goodness-of-Fit Tests for Ornstein-Uhlenbeck Process

We consider the goodness of fit testing problem for linear stochastic differential equation (Ornstein-Uhlenbeck process). The basic hypothesis is supposed to be composite with two-dimensional unknown parameter. We study two goodness of fit tests of Cramer-von Mises type based on empirical distribution function and on local time estimator of the invariant density. It is shown that the limit distributions of the underlying statistics under hypothesis do not depend on the unknown parameter.

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