arXiv subjects

Yury A. Kutoyants

Publications and source records attributed to Yury A. Kutoyants.

28 records · Page 2Linked to original sources

arXiv2013

On Asymptotic Distribution of Parameter Free tests for Ergodic Diffusion Processes

We consider two problems of constructing of goodness of fit tests for ergodic diffusion processes. The first one is concerned with a composite basic hypothesis for a parametric class of diffusion processes, which includes the Ornstein-Uhlenbeck and simple switching processes. In this case we propose asymptotically parameter free tests of Cramér-von Mises type. The basic hypothesis in the second problem is simple and we propose asymptotically distribution free tests for a wider class of trend coefficients.

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arXiv2010

On Identification of the Threshold Diffusion Processes

We consider the problems of parameter estimation for several models of threshold ergodic diffusion processes in the asymptotics of large samples. These models are the direct continuous time analogues of the well-known in time series analysis threshold autoregressive (TAR) models. In such models the trend is switching when the observed process atteints some (unknown) values and the problem is to estimate it or to test some hypotheses concerning these values. The related statistical problems correspond to the singular estimation or testing, for example, the rate of convergence of estimators is $T$ and not $\sqrt{T}$ as in regular estimation problems. We study the asymptotic behavior of the maximum likelihood and bayesian estimators and discuss the possibility of the construction of the goodness of fit test for such models of observation.

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arXiv2010

On Parameter Estimation of Threshold Autoregressive Models

This paper studies the threshold estimation of a TAR model when the underlying threshold parameter is a random variable. It is shown that the Bayesian estimator is consistent and its limit distribution is expressed in terms of a limit likelihood ratio. Furthermore, convergence of moments of the estimators is also established. The limit distribution can be computed via explicit simulations from which testing and inference for the threshold parameter can be conducted. The obtained results are illustrated with numerical simulations.

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arXiv2009

On the Goodness-of-Fit Testing for Ergodic Diffusion Processes

We consider the goodness of fit testing problem for ergodic diffusion processes. The basic hypothesis is supposed to be simple. The diffusion coefficient is known and the alternatives are described by the different trend coefficients. We study the asymptotic distribution of the Cramer-von Mises type tests based on the empirical distribution function and local time estimator of the invariant density. At particularly, we propose a transformation which makes these tests asymptotically distribution free. We discuss the modifications of this test in the case of composite basic hypothesis.

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arXiv2009

Goodness-of-Fit Tests for Perturbed Dynamical Systems

We consider the goodness of fit testing problem for stochastic differential equation with small diffiusion coefficient. The basic hypothesis is always simple and it is described by the known trend coefficient. We propose several tests of the type of Cramer-von Mises, Kolmogorov-Smirnov and Chi-Square. The power functions of these tests we study for a special classes of close alternatives. We discuss the construction of the goodness of fit test based on the local time and the possibility of the construction of asymptotically distribution free tests in the case of composite basic hypothesis.

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arXiv2009

On Properties of Estimators in non Regular Situations for Poisson Processes

We consider the problem of parameter estimation by observations of inhomogeneous Poisson process. It is well-known that if the regularity conditions are fulfilled then the maximum likelihood and Bayesian estimators are consistent, asymptotically normal and asymptotically efficient. These regularity conditions can be roughly presented as follows: a) the intensity function of observed process belongs to known parametric family of functions, b) the model is identifiable, c) the Fisher information is positive continuous function, d) the intensity function is sufficiently smooth with respect to the unknown parameter, e) this parameter is an interior point of the interval. We are interested in the properties of estimators when these regularity conditions are not fulfilled. More precisely, we preset a review of the results which correspond to the rejection of these conditions one by one and we show how the properties of the MLE and Bayesian estimators change. The proofs of these results are essentially based on some general results by Ibragimov and Khasminskii.

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arXiv2009

Hypotheses Testing: Poisson Versus Self-exciting

We consider the problem of hypotheses testing with the basic simple hypothesis: observed sequence of points corresponds to stationary Poisson process with known intensity. The alternatives are stationary self-exciting point processes. We consider one-sided parametric and one-sided nonparametric composite alternatives and construct locally asymptotically uniformly most powerful tests. The results of numerical simulations of the tests are presented.

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arXiv2009

On the Goodness-of-Fit Tests for Some Continuous Time Processes

We present a review of several results concerning the construction of the Cramer-von Mises and Kolmogorov-Smirnov type goodness-of-fit tests for continuous time processes. As the models we take a stochastic differential equation with small noise, ergodic diffusion process, Poisson process and self-exciting point processes. For every model we propose the tests which provide the asymptotic size $α$ and discuss the behaviour of the power function under local alternatives. The results of numerical simulations of the tests are presented.

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arXiv2007

Moment estimation for ergodic diffusion processes

We investigate the moment estimation for an ergodic diffusion process with unknown trend coefficient. We consider nonparametric and parametric estimation. In each case, we present a lower bound for the risk and then construct an asymptotically efficient estimator of the moment type functional or of a parameter which has a one-to-one correspondence to such a functional. Next, we clarify a higher order property of the moment type estimator by the Edgeworth expansion of the distribution function.

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arXiv2006

Hypotheses Testing: Poisson Versus Self-correcting

We consider the problem of hypotheses testing with the basic simple hypothesis: observed sequence of points corresponds to stationary Poisson process with known intensity against a composite one-sided parametric alternative that this is a self-correcting point process. The underlying family of measures is locally asymptotically quadratic and we describe the behavior of score function, likelihood ratio and Wald tests in the asymptotics of large samples. The results of numerical simulations are presented.

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