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Estimating of Main Characteristics of Processes with Non-Regular Observations

Estimating of Main Characteristics of Processes with Non-Regular Observations. Tatiana Varatnitskaya Bel а russian State University, Minsk. The Amplitude Modulated Version of Process. (1). (2). - mathematical expectation of process X(t). - covariance function of process X(t).

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Estimating of Main Characteristics of Processes with Non-Regular Observations

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

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

  3. - 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) - 4-th moment of process X(t) The Main Designation

  4. - 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) - 4-th moment of process d(t) I. d(t) is a stationary random process

  5. The Estimation of Mathematical Expectation (3)

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

  7. The Estimation of Covariance Function (4)

  8. Theorem. If spectral densities and are continuous in П, semi-invariant spectral densities of fourth order and are continuous onП3andthen statistics is defined as (4) is mean-square consistent estimate. (5)

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

  10. where

  11. 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

  12. The Estimation of Spectral Density (8)

  13. 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)

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

  15. Theorem.If semi-invariant spectral density of fourth order is continuous onП3, spectral density is continuous on П and then statistics defined as (9) is mean-square consistent estimate. (11)

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