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In this paper, we had constructed the goodness-of-fit tests incorporating several components, like expectation and covariance function, for identification of a non-centered stationary Gaussian stochastic process. For the construction of the corresponding estimators and investigation of their properties we had utilized the theory of Square Gaussian random variables.

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In this paper we had made an attempt to incorporate the results from the theory of square Gaussian random variables in order to construct the goodness of fits test for random sequences (time series). We considered two versions of such tests. The first one was designed for testing the adequacy of the hypotheses on expectation and covariance function of univariate non-centered sequence, the other one was constructed for testing the hypotheses on covariance of the multivariate centered sequence. The simulation results illustrate the behavior of these tests in some particular cases.

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 $\alpha $ and discuss the behaviour of the power function under local alternatives. The results of numerical simulations of the tests are presented.

In this paper, a new criterion is constructed for testing hypothesis about covariance function of Gaussian stationary stochastic process with an unknown mean. This criterion is based on the fact that we can estimate the deviation of covariance function from its estimator with a given accuracy and reliability in Lp metric.

Simulation has now become an integral part of research and development across many fields of study. Despite the large amounts of literature in the field of simulation and modeling, one recurring problem is the issue of accuracy and confidence level of constructed models. By outlining the new approaches and modern methods of simulation of stochastic processes, this book provides methods and tools in measuring accuracy and reliability in functional spaces. The authors explore analysis of the theory of Sub-Gaussian (including Gaussian one) and Square Gaussian random variables and processes and Cox processes. Methods of simulation of stochastic processes and fields with given accuracy and reliability in some Banach spaces are also considered. Provides an analysis of the theory of Sub-Gaussian (including Gaussian one) and Square Gaussian random variables and processes Contains information on the study of the issue of accuracy and confidence level of constructed models not found in other books on the topic Provides methods and tools in measuring accuracy and reliability in functional spaces

In this paper we have constructed the goodness-of-fit tests incorporating several components, like expectation and covariance function for identification of a non-centered univariate random sequence or auto-covariances and cross-covariances for identification of a centered multivariate random sequence. For the construction of the corresponding estimators and investigation of their properties we utilized the theory of square Gaussian random variables.

An Lp-criterion for testing a hypothesis about the covariance function for a centered stationary Gaussian sequence is constructed in this paper. The criterion is analyzed for some particular cases by using the simulated data.

Estimates are obtained for P{sup t∈T |η(t)|>x} in the case of a pre-Gaussian random process η(t). These results are applied to the investigation of conditions for convergence and of the rate of convergence in C(T) of estimators of the covariance function of a stationary Gaussian random process.

New upper and lower bounds for distributions of quadratic forms of Gaussian random variables as well as those for the limits of quadratic forms are found in this paper. Based on these estimates, a criterion is proposed to test a hypothesis about the covariance function ρ (τ) of a Gaussian stochastic process.

In this paper inequalities for distributions of quadratic forms from square-Gaussian random variables and distributions of suprema of quadratic forms from square-Gaussian random processes are proved. These inequalities enable us to investigate the jointly distributions of estimators of covariance functions of Gaussian processes.

Estimation of covariance functions of Gaussian stochastic fields and their simulation

- Y V Kozachenko
- T V Hudyvok
- V B Troshki
- N V Troshki

Applied methods of statistical modelling. Leningrad: Mashinostroenie

- A S Shalygin
- Y I Palagin