A criterion for testing hypotheses on covariance functions of vector-valued Gaussian stationary stochastic process defined on a interval. (English. Ukrainian original) Zbl 0947.62053

Stus’, O. V.

Theory Probab. Math. Stat. 60, 175-179 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 158-161 (1999).

The author studies random processes which may be presented as quadratic forms of simultaneously Gaussian random processes or as mean-square limits of such quadratic forms, that is the square-Gaussian random processes. He presents an inequality for the distribution of the supremum of square-Gaussian random process. This inequality is used to study the simultaneous distribution of estimates of covariance functions of simultaneously Gaussian stationary processes. The author constructs a criterion of verification of hypotheses on the type of the covariance functions of these Gaussian processes on a segment. He considers an example of verification of the hypothesis that two simultaneously Gaussian stationary processes \(\xi_1(t)\), \(\xi_2(t)\), \(E\xi_1(t)= E\xi_2(t)=0,\) have the following covariance functions: \[ E\xi_1(t+\tau)\xi_1(t)= A_1\exp\{ -c|\tau|^{\alpha}\}=E\xi_2(t+\tau)\xi_2(t)= A_2\exp\{ -c|\tau|^{\alpha}\}, \]
\[ E\xi_1(t+\tau)\xi_2(t)= A_1\exp\{ -c|\tau|^{\alpha}\},\quad A_1>0,\;A_2>0,\;c>0,\;1\leq\alpha\leq 2. \]

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

62M07	Non-Markovian processes: hypothesis testing
62M99	Inference from stochastic processes
62E15	Exact distribution theory in statistics
60G10	Stationary stochastic processes

Keywords:

covariance functions; square-Gaussian random processes; quadratic forms; Gaussian process