Skip to main content
SpringerLink
Log in

Estimation of frequency by random sampling

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Summary

We consider the estimation of frequency ω of a sinusoidal oscillation contaminated by a stationary noise under a random sampling scheme according to a stationary point processN. We prove the strong consistency and the asymptotic normality for a certain estimator of ω. Then we apply these results to the case whereN is a stationary delayed renewal process.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bloomfield, P. (1970). Spectral analysis with randomly missing observations,J. R. Statist. Soc., B,32, 369–380.

    MathSciNet  MATH  Google Scholar 

  2. Blum, J. R. and Rosenblatt, Judah (1964). On random sampling from a stochastic process,Ann. Math. Statist.,35, 1713–1717.

    Article  MathSciNet  Google Scholar 

  3. Brillinger, D. R. (1972). The spectral analysis of stationary interval functions,Proc. 6th Berkley Symp., University of California, Berkley,1, 483–513.

    MATH  Google Scholar 

  4. Brillinger, D. R. (1973). Estimation of the mean of a stationary time series by sampling.J. Appl. Prob.,10, 419–431.

    Article  MathSciNet  Google Scholar 

  5. Daley, D. J. and Vere Jones, D. (1972). A summary of the theory of point processes,Proc. Symp. Stochastic Point Process (ed. Lewis, P. A. W.), Wiley, New York, 299–383.

    Google Scholar 

  6. Feller, W. (1971).An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, New York.

    MATH  Google Scholar 

  7. Hall, P. and Heyde, C. C. (1980).Martingale Limit Theory and its Application, Academic Press, New York.

    MATH  Google Scholar 

  8. Hannan, E. J. (1973). The estimation of frequency,J. Appl. Prob.,10, 510–519.

    Article  MathSciNet  Google Scholar 

  9. Hannan, E. J. (1973). Central limit theorems for time series regression,Zeit. Wahrscheinlichkeitsth.,26, 157–170.

    Article  MathSciNet  Google Scholar 

  10. Ivanov, A. V. (1980). A solution of the problem of detecting hidden periodicities,Theory Prob. Math. Statist.,20, 51–68.

    MATH  Google Scholar 

  11. Marsy, E. (1978). Alias-free sampling; an alternative conceptualization and its applications,IEEE Trans. Inf. Theory,24, 317–324.

    Article  MathSciNet  Google Scholar 

  12. Rozanov, A. (1967).Stationary Random Processes, Holden-Day, New York.

    MATH  Google Scholar 

  13. Stone, C. (1966). On absolutely continuous components and renewal theory,Ann. Math. Statist.,37, 271–275.

    Article  MathSciNet  Google Scholar 

  14. Vere Jones, D. (1972). An elementary approach to the spectral theory of stationary random measure,Stochastic Geometry (ed. Kendall, D. G. and Harding, E. F.), Wiley, New York, 307–321.

    Google Scholar 

  15. Vere Jones, D. (1982). On the estimation of frequency in point-process data,Essays in Statistical Science (ed. Gani, J. and Hannan, E. J.), Moran Festschrift of J. Appl. Prob., 383–395.

  16. Walker, A. M. (1971). On the estimation of a harmonic component in a time series with stationary independent residuals,Biometrika,58, 21–36.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tokyo Institute of Technology, Tokyo, Japan

    Y. Isokawa

Authors
  1. Y. Isokawa
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Isokawa, Y. Estimation of frequency by random sampling. Ann Inst Stat Math 35, 201–213 (1983). https://doi.org/10.1007/BF02480976

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02480976

Key words

Search

Navigation