Estimation of second-order properties from jittered time series

  • Peter J. Thomson
  • Peter M. Robinson
Time Series

Abstract

This paper considers spectral and autocovariance estimation for a zero-mean, band-limited, stationary process that has been sampled at time points jittered from a regular, equi-interval, sampling scheme. The case of interest is where the sampling scheme is near regular so that the jitter standard deviation is small compared to the sampling interval. Such situations occur with many time series collected in the physical sciences including, in particular, oceanographic profiles.

Spectral estimation procedures are developed for the case of independent jitter and autocovariance estimation procedures for both independent and dependent jitter. These are typically modifications of general estimation procedures proposed elsewhere, but tailored to the particular jittered sampling scheme considered. The theoretical properties of these estimators are developed and their relative efficiencies compared.

The properties of the jittered sampling point process are also developed. These lead to a better understanding, in this situation, of more general techniques available for processes sampled by stationary point processes.

Key words and phrases

Jittered sampling stationary processes spectral estimation autocovariance estimation kernel density estimation 

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References

  1. Akaike, H. (1960). The effect of timing error on the power spectrum of sampled data, Ann. Inst. Statist. Math., 11, 145–165.Google Scholar
  2. Akaike, H. and Ishiguro, M. (1980). Trend estimation with missing observations, Ann. Inst. Statist. Math., 32, 481–488.Google Scholar
  3. Balakrishnan, A. V. (1962). On the problem of time jitter in sampling, Institute of Radio Engineers Transactions on Information Theory, 8, 226–236.Google Scholar
  4. Brillinger, D. R. (1972). The spectral analysis of stationary interval functions, Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability (ed. L.LeCam, J.Neyman and E. L.Scott), 483–513, University of California Press, Berkeley.Google Scholar
  5. Brillinger, D. R. (1983). Statistical inference for irregularly observed processes, Time Series Analysis of Irregularly Observed Data, Lecture Notes in Statist., 25, 38–57, Springer, New York.Google Scholar
  6. Cox, D. R. and Lewis, P. A. W. (1966). The Statistical Analysis of Series of Events, Methuen, London.Google Scholar
  7. Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes, Springer, New YorkGoogle Scholar
  8. Ibragimov, I. A. and Rozanov, Y. A. (1978). Gaussian Random Processes, Springer, New York.Google Scholar
  9. Jones, R. H. (1971). Spectrum estimation with missing observations, Ann. Inst. Statist. Math., 23, 387–398.Google Scholar
  10. Lawrance, A. J. (1972). Some models for stationary series of univariate events, Stochastic Point Processes (ed. P. A. W.Lewis), 199–255, Wiley, New York.Google Scholar
  11. Lewis, T. (1961). The intervals between regular events displaced in time by independent random deviations of large dispersion, J. Roy. Statist. Soc. Ser. B, 23, 476–483.Google Scholar
  12. Lii, K.-S. and Masry, E. (1992). Model fitting for continuous-time stationary processes from discrete-time data, J. Multivariate Anal., 41, 56–79.Google Scholar
  13. Masry, E. (1978). Alias-free sampling: an alternative conceptualization and its applications, IEEE Trans. Inform. Theory, 24, 317–324.Google Scholar
  14. Masry, E. (1983a). Non-parametric covariance estimation from irregularly-spaced data, Advances in Applied Probability Theory, 15, 113–132.Google Scholar
  15. Masry, E. (1983b). Spectral and probability density estimation from irregularly observed data, Time Series Analysis of Irregularly Observed Data, Lecture Notes in Statist., 25, 225–250, Springer, New York.Google Scholar
  16. Moore, M. I. and Thomson, P. J. (1991). Impact of jittered sampling on conventional spectral estimates, Journal of Geophysical Research, 96, 1.519–18.526.Google Scholar
  17. Moore, M. I., Visser, A. W. and Shirtcliffe, T. G. L. (1987). Experiences with the Brillinger spectral estimator applied to simulated irregularly observed processes, J. Time Ser. Anal., 8, 433–442.Google Scholar
  18. Moore, M. I., Thomson, P. J. and Shirtcliffe, T. G. L. (1988). Spectral analysis of ocean profiles from unequally spaced data, Journal of Geophysical Research, 93, 655–664.Google Scholar
  19. Moran, P. A. P. (1950). Numerical integration by systematic sampling, Proceedings of the Cambridge Philosophical Society, 46, 111–115.Google Scholar
  20. Parzen, E. (1957). On consistent estimates of the spectrum of a stationary time series, Ann. Math. Statist., 28, 329–348.Google Scholar
  21. Parzen, E. (ed.) (1983). Time Series Analysis of Irregularly Observed Data, Lecture Notes in Statist., 25, Springer, New York.Google Scholar
  22. Robinson, P. M. (1980). Continuous model fitting from discrete data, Directions in Time Series (ed. D. R.Brillinger and G. C.Tiao), 263–278, Institute of Mathematical Statistics, California.Google Scholar
  23. Robinson, P. M. (1984). Kernel estimation and interpolation for time series containing missing observations, Ann. Inst. Statist. Math., 36, 403–417.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 1996

Authors and Affiliations

  • Peter J. Thomson
    • 1
  • Peter M. Robinson
    • 2
  1. 1.Institute of Statistics and Operations ResearchVictoria University of WellingtonWellingtonNew Zealand
  2. 2.Department of EconomicsLondon School of EconomicsLondonU.K.

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