Tatiana Varatnitskaya Belаrussian State University, Minsk

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Presentation transcript:

Tatiana Varatnitskaya Belаrussian State University, Minsk Estimating of Main Characteristics of Processes with Non-Regular Observations Tatiana Varatnitskaya Belаrussian State University, Minsk

The Amplitude Modulated Version of Process (1) (2)

The Main Designation - mathematical expectation of process X(t) - 4-th moment of process X(t) - mathematical expectation of process X(t) - covariance function of process X(t) - spectral density of process X(t) - semi-invariant spectral density of fourth order of process X(t)

I. d(t) is a stationary random process - 4-th moment of process d(t) - mathematical expectation of process d(t) - covariance function of process d(t) - spectral density of process d(t) - semi-invariant spectral density of fourth order of process d(t)

The Estimation of Mathematical Expectation (3)

Theorem. The statistics (3) is asymptotically unbiased estimate and the limit of dispersion of this statistics is defined as on the understanding that spectral density is continuous at point and bounded in

The Estimation of Covariance Function (4)

Theorem. If spectral densities and are continuous in П, semi-invariant spectral densities of fourth order and are continuous on П3 and then statistics is defined as (4) is mean-square consistent estimate. (5)

II. d(t) is a Poisson sequence The Estimation of Covariance Function (6) - is a parameter of distribution

where

Theorem. The estimate of covariance function (6) is asymptotically unbiased estimate. On the understanding that (7) the statistics (6) is mean-square consistent estimate of covariance function of process X(t). That is

The Estimation of Spectral Density (8)

Theorem. Let semi-invariant spectral density of fourth order to be continuous on П3 and spectral density be continuous on П, then statistics defined as (8) is asymptotically unbiased estimate for and (9)

To get the consistent estimate of spectral density it is necessary to smooth this estimate using spectral windows . (10) - is integer part of number

Theorem. If semi-invariant spectral density of fourth order is continuous on П3, spectral density is continuous on П and then statistics defined as (9) is mean-square consistent estimate. (11)