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Parametric estimators for stationary time series with missing observations

Published online by Cambridge University Press:  01 July 2016

W. Dunsmuir  and
P. M. Robinson
W. Dunsmuir
Affiliation:
Massachusetts Institute of Technology
P. M. Robinson*
Affiliation:
University of Surrey
*
∗∗Postal address: Department of Mathematics, University of Surrey, Guildford, GU2 5XH, U.K.
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Abstract

Three related estimators are considered for the parametrized spectral density of a discrete-time process X(n), n = 1, 2, · · ·, when observations are not available for all the values n = 1(1)N. Each of the estimators is obtained by maximizing a frequency domain approximation to a Gaussian likelihood, although they do not appear to be the most efficient estimators available because they do not fully utilize the information in the process a(n) which determines whether X(n) is observed or missed. One estimator, called M3, assumes that the second-order properties of a(n) are known; another, M2, lets these be known only up to an unknown parameter vector; the third, M1, requires no model for a(n). Under representative sets of conditions, which allow for both deterministic and stochastic a(n), the strong consistency and asymptotic normality of M1, M2, and M3 are established. The conditions needed for consistency when X(n) is an autoregressive moving-average process are discussed in more detail. It is also shown that in general M1 and M3 are equally efficient asymptotically and M2 is never more efficient, and may be less efficient, than M1 and M3.

Keywords

PARAMETRIC SPECTRAL DENSITYMISSING DATASTRONG CONSISTENCYASYMPTOTIC NORMALITYASYMPTOTIC EFFICIENCYARMA MODELS
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 1 , March 1981 , pp. 129 - 146
Copyright
Copyright © Applied Probability Trust 1981 

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Footnotes

a

Present address: Department of Statistics, SGS, The Australian National University, P.O. Box 4, Canberra, A.C.T. 2600, Australia.

Research supported by grants SOC75-13436 and SOC78-05803 from the National Science Foundation.

Research supported by grants MCS76-07211 and MCS78-1118 from the National Science Foundation.

References

[1] Bloomfield, P. (1970) Spectral analysis with randomly missing observations. J.R. Statist. Soc. B32, 369–380.Google Scholar
[2] Caines, P. E. (1976) Prediction error identification methods for stationary stochastic processes. IEEE Trans. Automatic Control AC-21, 500–505.CrossRefGoogle Scholar
[3] Davies, R. B. (1973) Asymptotic inference in stationary Gaussian time series. Adv. Appl. Prob. 5, 469–497.CrossRefGoogle Scholar
[4] Deistler, M., Dunsmuir, W. and Hannan, E. J. (1978) Vector linear time series models: corrections and extensions. Adv. Appl. Prob. 10, 360–372.CrossRefGoogle Scholar
[5] Dunsmuir, W. (1979) A central limit theorem for parameter estimation in stationary vector time series and its application to models for a signal observed with noise. Ann. Statist. 7, 490–506.Google Scholar
[6] Dunsmuir, W. and Hannan, E. J. (1976) Vector linear time series models. Adv. Appl. Prob. 8, 339–364.Google Scholar
[7] Dunsmuir, W. and Robinson, P. M. (1981) Asymptotic theory for time series containing missing and amplitude modulated observations. Sankhyā A. To appear.Google Scholar
[8] Dunsmuir, W. and Robinson, P. M. (1979) Estimation of time series models in the presence of missing data. Technical Report No. 9, Dept. of Mathematics, MIT.Google Scholar
[9] Hannan, E. J. (1970) Multiple Time Series, Wiley, New York.CrossRefGoogle Scholar
[10] Hannan, E. J. (1973) The asymptotic theory of linear time series models. J. Appl. Prob. 10, 130–145.Google Scholar
[11] Jones, R. M. (1962) Spectral analysis with regularly missed observations. Ann. Math. Statist. 33, 455–461.Google Scholar
[12] Jones, R. H. (1971) Spectrum estimation with missing observations. Ann. Inst. Statist. Math. 23, 387–398.CrossRefGoogle Scholar
[13] Jones, R. H. (1978) Fitting rational spectra for unequally spaced data. Preprint, Sixth Conference on Probability and Statistics in Atmospheric Sciences, American Meteorology Society, Boston.Google Scholar
[14] Ljung, L. and Caines, P. E. (1979) Asymptotic normality of prediction error estimators for approximate system models. Stochastics 3, 29–46.Google Scholar
[15] Mann, H. B. and Wald, A. (1943) On the statistical treatment of linear stochastic difference equations. Econometrica 11, 173–220.Google Scholar
[16] Parzen, E. (1963) On spectral analysis with missing observations and amplitude modulation. Sankhyā A 25, 383–392.Google Scholar
[17] Robinson, P. M. (1977) Estimation of a time series model from unequally spaced data. Stoch. Proc. Appl. 6, 9–24.Google Scholar
[18] Robinson, P. M. (1978) Alternative models for stationary stochastic processes. Stoch. Proc. Appl. 8, 141–152.Google Scholar
[19] Robinson, P. M. (1980) Estimation and forecasting for time series containing censored observations. In Time Series ed. Anderson, O. D., North-Holland, Amsterdam, 167–182.Google Scholar
[20] Sargan, J. D. and Drettakis, E. G. (1976) Missing data in an autoregressive model. Internat. Econ. Rev. 15 (1), 39–58.CrossRefGoogle Scholar
[21] Scheinok, P. A. (1965) Spectral analysis with randomly missed observations: the binomial case. Ann. Math. Statist. 36, 971–977.Google Scholar
[22] Stout, W. F. (1974) Almost Sure Convergence. Academic Press, New York.Google Scholar
[23] Walker, A. M. (1964) Asymptotic properties of least squares estimates of parameters of the spectrum of a stationary nondeterministic time series. J. Austral. Math. Soc. 4, 363–384.CrossRefGoogle Scholar
[24] Whittle, P. (1951) Hypothesis Testing in Time Series Analyses. Almqvist and Wiksell, Uppsala.Google Scholar
[25] Whittle, P. (1953) The analysis of multiple stationary time series. J. R. Statist. Soc. B15, 125–139.Google Scholar
[26] Whittle, P. (1961) Gaussian estimation in stationary time series. Bull. I.S.I. 33, 1–26.Google Scholar
[27] Zygmund, A. (1959) Trigonometric Series, Volume I, Cambridge University Press.Google Scholar