Large Sample Properties of Estimation in Time Series Observed at Unequally Spaced Times
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This paper surveys the current state of large sample theory for estimation in stationary discrete time series observed at unequally spaced times. The paper considers the nonparametric estimation of sample covariances, correlations and spectra and traces the development of consistency and asymptotic normality results for these quantities. The second part of the paper discusses the estimation of finite parameter models for stationary time series. Consistent, but inefficient, methods based on sample covariances and on spectra are discussed and compared. The final part of the paper reports on the recent results concerning a general central limit theorem (for the estimate obtained by a single iteration from a consistent estimate) for Gaussian data. The essential condition on the sampling times is that the finite sample information matrix, when divided by the sample size, has a limit which is non-singular and has finite norm. This condition will be illustrated by considering examples of periodic sampling, of random but time homogeneous sampling, of sparse early sampling, and of asymptotically stationary sampling. The effect on efficiency of the design of sampling pattern will be briefly discussed. Finally it will be indicated what extra complexity of proof is needed to handle the non-Gaussian case.
KeywordsCentral Limit Theorem Asymptotic Normality Asymptotic Variance ARMA Model Stationary Time Series
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- Alekseev, V.G, and Savitskii, Yu, A, (1973). “Estimation of the spectrum of a Gaussian Stochastic process on the basis of a realisation of the process with omissions”, Problems of Information Transmission. 91:50–54.Google Scholar
- Box, G.E.P. and Jenkins, G. (1976). Time series analysis — forecasting and control, 2nd edition, Holden-Day.Google Scholar
- Dunsmuir, W. (1981). “Estimation for stationary time series when data are irregularly spaced or missing”, in: D.F. Findley, ed., Applied Time Series II (Academic Press, New York, 1981).Google Scholar
- Dunsmuir, W. and Robinson, P.M. (1981c). “Estimation of time series models in the presence of missing data”, J. Amer. Stat. Assoc., 96, 560–568.Google Scholar
- Hinich, M.J. and Weber, W.E. (1981). “A method for estimating distributed lags when observations are randomly missing”. Pre-print.Google Scholar
- Marquardt, D.W. and Acuff, S.K. (1980). “Direct quadratic spectrum estimation for unequally spaced data”, proceedings of the Fifth International Time Series Meeting, Houston.Google Scholar
- Parzen, E. (1962). “Spectral analysis of asymptotically stationary time series”, Bull. I.S.I., 33rd Session, Paris.Google Scholar
- Robinson, P.M. (1980). “Estimation and forecasting for time series containing censored or missing observations”, Time Series, O.D. Anderson, ed., North-Holland Publishing Company.Google Scholar
- Tan, Suan-Boon. (1979). Maximum likelihood estimation in autoregressive processes with missing data, Ph.D. Thesis, University of Pittsburgh.Google Scholar
- Thrall, A.D. (1980). “A spectral analysis of a time series in which probabilities of observations are periodic”, Time Series, O.D. Anderson, ed., North Holland Publishing Company.Google Scholar