Serguei Dachian

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.

Estimation of Cusp Location by Poisson Observations

We consider an inhomogeneous Poisson process X on [0, T]. The intensity function of X is supposed to be regular on [0, T] except at the point θ, in which it has a singularity (a cusp) of order p. We suppose that we know the shape of the intensity function, but not the location (given by the parameter θ) of the point of cusp. We consider the problem of estimation of this location (shift) parameter θ based on n observations of the process X. We study the maximum likelihood estimator and the Bayesian estimators. We show that these estimators are consistent, their rate of convergence is n1/(2p+1), they have different limit distributions, and the Bayesian estimators are asymptotically efficient.

On compound Poisson processes arising in change-point type statistical models as limiting likelihood ratios

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In a previous paper of one of the authors it was established that one of these likelihood ratios, which is an exponential functional of a two-sided Poisson process driven by some parameter, can be approximated (for sufficiently small values of the parameter) by another one, which is an exponential functional of a two-sided Brownian motion. In this paper we consider yet another likelihood ratio, which is the exponent of a two-sided compound Poisson process driven by some parameter. We establish, that similarly to the Poisson type one, the compound Poisson type likelihood ratio can be approximated by the Brownian type one for sufficiently small values of the parameter. We equally discuss the asymptotics for large values of the parameter and illustrate the results by numerical simulations.

On parameter estimation for cusp-type signals

We consider the problem of parameter estimation by continuous time observations of a deterministic signal in white Gaussian noise. It is supposed that the signal has a cusp-type singularity. The properties of the maximum-likelihood and Bayesian estimators are described in the asymptotics of small noise. Special attention is paid to the problem of parameter estimation in the situation of misspecification in regularity, i.e., when the statistician supposes that the observed signal has this singularity, but the real signal is smooth. The rate and the asymptotic distribution of the maximum-likelihood estimator in this situation are described.

On Gaussian Compound Poisson Type Limiting Likelihood Ratio Process

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. Recently it was established that one of these likelihood ratios, which is an exponential functional of a two-sided Poisson process driven by some parameter, can be approximated (for sufficiently small values of the parameter) by another one, which is an exponential functional of a two-sided Brownian motion. In this chapter we consider yet another likelihood ratio, which is the exponent of a two-sided compound Poisson process driven by some parameter. We establish that the compound Poisson type likelihood ratio can also be approximated by the Brownian type one for sufficiently small values of the parameter. We equally discuss the asymptotics for large values of the parameter.

Estimation of the location of a 0-type or ∞-type singularity by Poisson observations

We consider an inhomogeneous Poisson process X on [0, T]. The intensity function of X is supposed to be strictly positive and smooth on [0, T] except at the point θ, in which it has either a 0-type singularity (tends to 0 like |x| p , p∈(0, 1)), or an ∞-type singularity (tends to ∞ like |x| p , p∈(−1, 0)). We suppose that we know the shape of the intensity function, but not the location of the singularity. We consider the problem of estimation of this location (shift) parameter θ based on n observations of the process X. We study the Bayesian estimators and, in the case p>0, the maximum-likelihood estimator. We show that these estimators are consistent, their rate of convergence is n 1/(p+1), they have different limit distributions, and the Bayesian estimators are asymptotically efficient.

On hypothesis testing for Poisson processes. Singular cases

ABSTRACT We consider the problem of hypothesis testing in the situation where the first hypothesis is simple and the second one is local one-sided composite. Wedescribe the choice of the thresholds and the power functions of different tests when the intensity function of the observed inhomogeneous Poisson process has two different types of singularity: cusp and discontinuity. The asymptotic results are illustrated by numerical simulations.

Statistical analysis of fast fluctuating random signals with arbitrary-function envelope under breach of consistency of discontinuous information parameter estimate

We carried out the synthesis and analysis of quasi-likelihood detector and measurer of a highfrequency random pulse with arbitrary-function envelope, unknown appearance time and inaccurately known duration. We introduced the asymptotically exact method of theoretical calculation of detection and estimation characteristics including anomalous errors effect, if conditions of the decision statistics regularity and consistency of discontinuous parameter estimate are not fulfilled. It has allowed us to define the efficiency of presented receivers analytically. By methods of statistical computer modeling we corroborated the adequacy of the considered analytical approach of the statistical analysis of discontinuous random pulse signals, and also we established its applicability borders for considered tasks.

Estimation of cusp location of stochastic processes: a survey

We present a review of some recent results on estimation of location parameter for several models of observations with cusp-type singularity at the change point. We suppose that the cusp-type models fit better to the real phenomena described usually by change point models. The list of models includes Gaussian, inhomogeneous Poisson, ergodic diffusion processes, time series and the classical case of i.i.d. observations. We describe the properties of the maximum likelihood and Bayes estimators under some asymptotic assumptions. The asymptotic efficiency of estimators are discussed as well and the results of some numerical simulations are presented. We provide some heuristic arguments which demonstrate the convergence of log-likelihood ratios in the models under consideration to the fractional Brownian motion.

On hypothesis testing for Poisson processes: Regular case

ABSTRACT We consider the problem of hypothesis testing in the situation when the firsthypothesis is simple and the second one is local one-sided composite. We describe the choice of the thresholds and the power functions of the Score Function test, of the General Likelihood Ratio test, of the Wald test, and of two Bayes tests in the situation when the intensity function of the observed inhomogeneous Poisson process is smooth with respect to the parameter. It is shown that almost all these tests are asymptotically uniformly most powerful. The results of numerical simulations are presented.