Yury A. Kutoyants

Statistical Inference for Ergodic Diffusion Processes

1 Diffusion Processes and Statistical Problems.- 2 Parameter Estimation.- 3 Special Models.- 4 Nonparametric Estimation.- 5 Hypotheses Testing.- Historical Remarks.- References.

Identification of dynamical systems with small noise

Introduction. 1. Auxiliary Results. 2. Asymptotic Properties of Estimator in Standard and Nonstandard Situations. 3. Expansions. 4. Nonparametric Estimation. 5. The Disorder Problem. 6. Partially Observed Systems. 7. Minimum Distance Estimation. Remarks. References. Index.

Delay estimation for some stationary diffusion-type processes

In this paper the asymptotic behaviour of the maximum likelihood and Bayesian estimators of a delay parameter is studied. The observed process is supposed to be the solution of a linear stochastic differential equation with one time delay term. It is shown that these estimators are consistent and their limit distributions are described. The behaviour of the estimators is similar to the behaviour of corresponding estimators in change-point problems. The question of asymptotical efficiency is also discussed.

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.

Efficiency of the empirical distribution for ergodic diffusion

We consider the problem of stationary distribution function estimation at a given point by the observations of an ergodic diffusion process on the interval [0, T] as T 0 o. First we introduce a lower (minimax) bound on the risk of all estimators and then we prove that the empirical distribution function attains this bound. Hence this estimator is asymptotically efficient in the sense of the given bound.

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 Cramér–von Mises type tests based on the empirical distribution function and local time estimator of the invariant density. Particularly, we propose a transformation which makes these tests asymptotically distribution-free.

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.

Asymptotically Efficient Estimation of the Derivative of the Invariant Density

The problem of estimation of the derivative of the invariant density is considered for a one-dimensional ergodic diffusion process. The lower minimax bound on the L2-type risk of all estimators is proposed and an asymptotically efficient (up to the constant) in the sense of this bound kernel-type estimator is constructed.

On parameter estimation for switching ergodic diffusion processes

Abstract We consider the problem of trend parameter estimation by the observations of ergodic diffusion process in the situation when the trend coefficient of the process is switching, i.e., is discontinuous function of the unknown parameter. In the asymptotics of large samples we prove the consistency of the MLE and Bayes estimators, describe their limiting distributions and show the convergence of moments of these estimators. Then we discuss several generalizations of these results (many jumps, simultaneous estimation of the smooth and discontinuous parameters, multidimentional case and the asymptotics of MLE in misspecified swithching system).

Small noise asymptotics of the GLR test for off-line change detection in misspecified diffusion processes

We consider the problem of the non-sequential detection of a change in the drift coefficient of a stochastic differential equation, when a misspecified model is used. We formulate the generalized likelihood ratio (GLR) test for this problem, and we study the behaviour of the associated error probabilities (false alarm and nodetection) in the small noise asymptotics. We obtain the following robustness result: even though a wrong model is used, the error probabilities go to zero with exponential rate, and the maximum likelihood estimator (MLE) of the change time is consistent, provided the change to be detected is larger (in some sense) than the misspecification error. We give also computable bounds for selecting the threshold of the test so as to achieve these exponential rates.