Kernel estimation and interpolation for time series containing missing observations

  • P. M. Robinson


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.


Conditional Expectation Nonparametric Estimation Kernel Estimation Stationary Time Series Mean Integrate Square Error 


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  1. [1]
    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.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Akaike, H. and Ishiguro, M. (1980). Trend estimation with missing observations,Ann. Inst. Statist. Math.,32, 481–488.CrossRefGoogle Scholar
  3. [3]
    Collomb, G. (1982). Proprietes de convergence presque complete du predicteur a noyau (preprint).Google Scholar
  4. [4]
    Deo, C. M. (1973) A note on empirical processes of strong mixing sequences,Ann. Prob. 5, 870–875.MathSciNetCrossRefGoogle Scholar
  5. [5]
    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.CrossRefGoogle Scholar
  6. [6]
    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.CrossRefGoogle Scholar
  7. [7]
    Hall, P. (1983) Large sample optimality of least squares cross-validation in density estimation,Ann. Statist.,11, 1156–1174.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Ibragimov, I. A. and Linnik, Yu. V. (1971).Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen.zbMATHGoogle Scholar
  9. [9]
    Masry, E. (1983). Probability density estimation from sampled data,IEEE Trans. Inf. Theory,29, 696–709.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Parzen, E. M. (1963) On spectral analysis with missing observations and amplitude modulation,Sankhyã A,25, 383–392.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Pham, T. D. (1981). Nonparametric estimation of the drift coefficient in the diffusion equation,Math. Oper.,12, 61–73.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Rao, C. R. (1973)Linear Statistical Inference and its Applications, 2nd ed., Wiley, New York.CrossRefGoogle Scholar
  13. [13]
    Robinson, P. M. (1983). Nonparametric estimators for time series,J. Time Series Anal.,4, 185–207.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Rosenblatt, M. (1970). Density estimators and Markov sequences, inNonparametric Techniques in Statistical Inference (ed. Puri, M. C.), Cambridge University Press, Cambridge, 199–210.Google Scholar
  15. [15]
    Roussas, G. (1967). Nonparametric estimation in Markov processes,Ann. Inst. Statist. Math.,21, 73–87.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Roussas, G. (1969). Nonparametric estimation of the transition distribution function of a Markov process,Ann. Math. Statist.,40, 1386–1400.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Sakai, H. (1980). Fitting autoregressions with regularly missed observations,Ann. Inst. Statist. Math.,32, 393–400.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Silverman, B. (1978). Choosing the window width when estimating a density,Biometrika,65, 1–11.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Stein, E. M. (1970).Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J.zbMATHGoogle Scholar
  20. [20]
    Takahata, H. (1977). Limiting behaviour of density estimates for stationary asymptotically uncorrelated processes,Bull. Tokyo Gakugei Univ., IV,29, 1–9.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Takahata, H. (1980). Almost sure convergence of density estimators for weakly dependent stationary processes,Bull. Tokyo Gakugei Univ. IV,32, 11–32.MathSciNetzbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1984

Authors and Affiliations

  • P. M. Robinson
    • 1
  1. 1.London School of EconomicsLondonUK

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