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Asymptotic Behavior of the Variance of the Best Linear Unbiased Estimator for the Mean of a Discrete-time Singular Stationary Process

  • Probability Theory and Mathematical Statistics
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Abstract

It is known that for a wide class of discrete-time stationary processes possessing spectral densities f, the variance σ2n(f) of the best linear unbiased estimator for the mean depends asymptotically only on the behavior of the spectral density f near the origin, and behaves hyperbolically as n → ∞. In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of σ2n(f) as n → ∞. In particular, we show that a necessary condition for σ2n(f) to decrease to zero exponentially is that the spectral density f vanishes on a set of positive measure in any vicinity of zero, and if f vanishes only at the origin, then it is impossible to obtain exponential decay of σ2n(f), no mater how high the order of the zero of f at the origin.

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References

  1. R. K. Adenstedt, “On large-sample estimation for the mean of a stationary random sequence”, Ann. Stat., 2, 1095–1107, 1974.

    Article  MathSciNet  Google Scholar 

  2. R. K. Adenstedt, B. Eisenberg, “Linear estimation of regression coefficients”, Quart. Appl. Math., 32(3), 317–327, 1974.

    Article  MathSciNet  Google Scholar 

  3. N. M. Babayan, “On the asymptotic behavior of prediction error”, J. of Soviet Mathematics, 27(6), 3170–3181, 1984.

    Article  Google Scholar 

  4. N. M. Babayan, “On asymptotic behavior of the prediction error in the singular case”, Theory Probab. Appl., 29(1), 147–150, 1985.

    Article  MathSciNet  Google Scholar 

  5. J. Beran, “Recent developments in location estimation and regression for long-memory processes”, In “New directions in time series analysis, part II” (D. Brillingeret al. eds.) 46, 1–10, 1993.

    Google Scholar 

  6. J. Beran, H. Künsch, “Location estimates for processes with long range dependence”, Research Report No. 40, Seminar für Statistik, Eidenössische Technische Hochschule, Zürich, 1985.

    Google Scholar 

  7. J. Beran, Y. Feng, S. Ghosh, R. Kulik, Long-Memory Processes: Probabilistic Properties and Statistical Methods (Springer, Berlin, 2013).

    Book  Google Scholar 

  8. Ya. L. Geronimus, “On certain asymptotic properties of polynomials”, Mat. Sb., 23(65) (1), 77–88, 1948.

    MathSciNet  MATH  Google Scholar 

  9. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable (Amer. Math. Soc., Providence, 1969).

    Book  Google Scholar 

  10. U. Grenander, “Stochastic processes and statistical inference”, Ark. Mat., 1(17), 195–277, 1950.

    Article  MathSciNet  Google Scholar 

  11. U. Grenander, “On Toeplitz forms and stationary processes”, Ark. Mat., 1(37), 555–571, 1952.

    Article  MathSciNet  Google Scholar 

  12. U. Grenander, “On the estimation of regression coefficients in the case of an autocorrelated disturbance”, Ann. Math. Stat., 25, 252–272, 1954.

    Article  MathSciNet  Google Scholar 

  13. U. Grenander, M. Rosenblatt, Statistical Analysis of Stationary Time Series (Wiley, New York, 1957).

    Book  Google Scholar 

  14. U. Grenander, G. Szegö, Toeplitz Forms and Their Applications (University of California Press, Berkeley, 1958).

    Book  Google Scholar 

  15. S. Mazurkievicz, “Un theoreme sur les polynomes”, Ann. Soc. Polon. Math., 18, 113–118, 1945.

    MathSciNet  Google Scholar 

  16. E. M. Nikishin, V. N. Sorokin, Rational Approximations and Orthogonality (American Mathematical Society, Providence, 1991).

    Book  Google Scholar 

  17. M. Rosenblatt “Some Purely Deterministic Processes”, J. of Math. and Mech., 6(6), 801–810, 1957.

    MathSciNet  MATH  Google Scholar 

  18. A. Samarov, M.S. Taqqu, “On the efficiency of the sample mean in long-memory noise”, J. Time Series Analysis, 9, 191–200, 1988.

    Article  MathSciNet  Google Scholar 

  19. R.A. Vitale, “An asymptotically efficient estimate in time series analysis”, Quart. Appl. Math., 30, 421–440, 1973.

    Article  MathSciNet  Google Scholar 

  20. Y. Yajima, “On estimation of a regression model with long-memory stationary errors”, Ann. Stat., 16, 791–807, 1988.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Russian-Armenian University, Yerevan, Armenia

    N. M. Babayan

  2. Institute of Mathematics, Yerevan, Armenia

    M. S. Ginovyan

  3. Boston University, Boston, USA

    M. S. Ginovyan

Authors
  1. N. M. Babayan
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  2. M. S. Ginovyan
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Corresponding authors

Correspondence to N. M. Babayan or M. S. Ginovyan.

Additional information

Russian Text © The Author(s), 2019, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2019, No. 6, pp. 28–41.

The research of M. S. Ginovyan was partially supported by National Science Foundation Grant #DMS-1309009 at Boston University.

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Babayan, N.M., Ginovyan, M.S. Asymptotic Behavior of the Variance of the Best Linear Unbiased Estimator for the Mean of a Discrete-time Singular Stationary Process. J. Contemp. Mathemat. Anal. 54, 371–380 (2019). https://doi.org/10.3103/S1068362319060074

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  • DOI: https://doi.org/10.3103/S1068362319060074

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