Abstract
This paper formulates a nonlinear time series model which encompasses several standard nonlinear models for time series as special cases. It also offers two methods for estimating missing observations, one using prediction and fixed point smoothing algorithms and the other using optimal estimating equation theory. Recursive estimation of missing observations in an autoregressive conditionally heteroscedastic (ARCH) model and the estimation of missing observations in a linear time series model are shown to be special cases. Construction of optimal estimates of missing observations using estimating equation theory is discussed and applied to some nonlinear models.
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References
Abraham, B. (1981). Missing observations in time series, Comm. Statist. A—Theory Methods, 10, 1645–1653.
Brockwell, P. J. and Davis, R. A. (1987). Time Series: Theory and Methods, Springer, New York.
Broemeling, L. D. (1985). Bayesian Analysis of Linear Models, Dekker, New York.
Charbonnier, R., Barlaud, M., Alengrin, G. and Menez, J. (1987). Results on AR-modelling of nonstationary signals, Signal Process., 12, 143–157.
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50, 987–1007.
Ferreiro, O. (1987). Methodologies for the estimation of missing observations in time series, Statist. Probab. Lett., 5, 65–69.
Godambe, V. P. (1985). The foundations of finite sample estimation in stochastic processes, Biometrika, 12, 419–428.
Harrison, P. J. and Stevens, C. F. (1976). Bayesian forecasting (with discussion), J. Roy. Statist. Soc. Ser. B, 88, 205–248.
Jones, R. H. (1985). Time series analysis with unequally spaced data, Handbook of Statistics, Vol. 5 (eds. E. J. Hannan, P. R. Krishnaiah and M. M. Rao), 157–177, North Holland, Amsterdam.
Miller, R. B. and Ferreiro, O. (1984). A strategy to complete a time series with missing observations, Lecture Notes in Statistics, 25, 251–275, Springer, New York.
Nicholls, D. F. and Quinn, B. G. (1982). Random coefficient autoregressive models: An introduction, Lecture Notes in Statistics, 11, Springer, New York.
Ozaki, T. (1985). Nonlinear time series models and dynamical systems, Handbook of Statistics, Vol. 5 (eds. E. J. Hannan, P. R. Krishnaiah and M. M. Rao), 25–83, North Holland, Amsterdam.
Priestley, M. B. (1980). State-dependent models: A general approach to time series analysis. J. Time Ser. Anal., 1, 47–71.
Ruskeepaa, H. (1985). Conditionally Gaussian distributions and an application to Kalman filtering with stochastic regressors, Comm. Statist. A—Theory Methods, 14, 2919–2942.
Shiryayev, A. N. (1984). Probability, Graduate Texts in Math., 95, Springer, New York.
Thavaneswaran, A. and Abraham, B. (1988). Estimation for nonlinear time series models using estimating equations, J. Time Ser. Anal., 9, 99–108.
Tjastheim, D. (1986). Estimation in nonlinear time series models, Stochastic. Process. Appl., 21, 251–273.
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Authors were supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.
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Abraham, B., Thavaneswaran, A. A nonlinear time series model and estimation of missing observations. Ann Inst Stat Math 43, 493–504 (1991). https://doi.org/10.1007/BF00053368
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DOI: https://doi.org/10.1007/BF00053368