Summary
Kernel estimators of conditional expectations are adapted for use in the analysis of stationary time series containing missing observations. Estimators of conditional expectations at fixed points are shown to have an asymptotic distribution with a relatively simple variance-covariance structure. The kernel method is also used to interpolate missing observations, and is shown to converge in probability to the least squares predictor. The results are established under the strong mixing condition and moment conditions, and the methods are applied to a real data set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmad, I. A. (1979) Strong consistency of density estimation by orthogonal series methods for dependent variables with applications,Ann. Inst. Statist. Math.,31, 279–288.
Akaike, H. and Ishiguro, M. (1980). Trend estimation with missing observations,Ann. Inst. Statist. Math.,32, 481–488.
Collomb, G. (1982). Proprietes de convergence presque complete du predicteur a noyau (preprint).
Deo, C. M. (1973) A note on empirical processes of strong mixing sequences,Ann. Prob. 5, 870–875.
Devroye, L. P. and Wagner, T. J. (1980) On theL 1 convergence of kernel estimators of regression with applications in discrimination,Zeit. Wahrscheinlichkeitsth.,51, 15–25.
Dunsmuir, W. and Robinson, P. M. (1982) Estimation of time series models in the presence of missing data,J. Amer. Statist. Ass.,76, 560–568.
Hall, P. (1983) Large sample optimality of least squares cross-validation in density estimation,Ann. Statist.,11, 1156–1174.
Ibragimov, I. A. and Linnik, Yu. V. (1971).Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen.
Masry, E. (1983). Probability density estimation from sampled data,IEEE Trans. Inf. Theory,29, 696–709.
Parzen, E. M. (1963) On spectral analysis with missing observations and amplitude modulation,Sankhyã A,25, 383–392.
Pham, T. D. (1981). Nonparametric estimation of the drift coefficient in the diffusion equation,Math. Oper.,12, 61–73.
Rao, C. R. (1973)Linear Statistical Inference and its Applications, 2nd ed., Wiley, New York.
Robinson, P. M. (1983). Nonparametric estimators for time series,J. Time Series Anal.,4, 185–207.
Rosenblatt, M. (1970). Density estimators and Markov sequences, inNonparametric Techniques in Statistical Inference (ed. Puri, M. C.), Cambridge University Press, Cambridge, 199–210.
Roussas, G. (1967). Nonparametric estimation in Markov processes,Ann. Inst. Statist. Math.,21, 73–87.
Roussas, G. (1969). Nonparametric estimation of the transition distribution function of a Markov process,Ann. Math. Statist.,40, 1386–1400.
Sakai, H. (1980). Fitting autoregressions with regularly missed observations,Ann. Inst. Statist. Math.,32, 393–400.
Silverman, B. (1978). Choosing the window width when estimating a density,Biometrika,65, 1–11.
Stein, E. M. (1970).Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J.
Takahata, H. (1977). Limiting behaviour of density estimates for stationary asymptotically uncorrelated processes,Bull. Tokyo Gakugei Univ., IV,29, 1–9.
Takahata, H. (1980). Almost sure convergence of density estimators for weakly dependent stationary processes,Bull. Tokyo Gakugei Univ. IV,32, 11–32.
Author information
Authors and Affiliations
About this article
Cite this article
Robinson, P.M. Kernel estimation and interpolation for time series containing missing observations. Ann Inst Stat Math 36, 403–417 (1984). https://doi.org/10.1007/BF02481979
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02481979