Multivariate Probability Density and Regression Functions Estimation of Continuous-Time Stationary Processes from Discrete-Time Data

  • Elias Masry
Part of the Trends in Mathematics book series (TM)


Let be a real-valued continuous-time jointly stationary processes and let tj be a renewal point process on [0,00], with finite mean rate independent of (Y,X). Given the observations and a measurable function, we estimate the multivariate probability density and the regression function of given X(0) = xo, X(T) = XI, …, X(r m ) = x m for arbitrary lags m. We present consistency and asymptotic normality results for appropriate estimates of f and r.


Regression Function Asymptotic Normality Asymptotic Normality Result Local Polynomial Fitting Multivariate Probability Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. Hall and C.C. Heyde (1980), Martingale limit theory and its applications. New York: Academic Press.Google Scholar
  2. [2]
    G. Collomb and W. Härdle (1986), Strong uniform convergence rates in robust nonparametric time series analysis and prediction: Kernel regression estimation from dependent observations, Stochastic Processes and their Applies., vol. 23, pp. 77–89.zbMATHCrossRefGoogle Scholar
  3. [3]
    J. Fan (1992), Design-adaptive nonparametric regression, Jour. Amer. Statist. Assoc, vol. 87, pp. 998–1004.zbMATHCrossRefGoogle Scholar
  4. [4]
    J. Fan (1993), Local linear regression smoothers and their minimax efficiency, Ann. Statist., vol. 21, pp. 196–216.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    J. Fan and I. Gijbels (1992), Variable bandwidth and local linear regression smoothers, Ann. Statist., vol. 20, pp. 2008–2036.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    J. Fan and E. Masry (1992), Multivariate Regression Estimation With Errors-in-Variables: Asymptotic Normality for Mixing Processes, J. Multivariate Analysis, vol. 43, pp. 237–271.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    W. Härdle (1990), Applied Nonparametric Regression, Boston, Cambridge University Press.zbMATHGoogle Scholar
  8. [8]
    A.N. Kolmogorov, and Yu. A. Rozanov (1960), On strong mixing conditions for stationary Gaussian processes, Theory Prob. Appl., vol. 52, pp. 204–207.CrossRefGoogle Scholar
  9. [9]
    E. Masry (1988), Random sampling of continuous-time stationary processes: statistical properties of joint density estimators, J. Multivariate Analysis, vol. 26, pp. 133–165.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    E. Masry (1993), Multivariate Regression Estimation with Errors-in-Variables for Stationary Processes, Nonparametric Statistics, vol. 3, pp. 13–36.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    E. Masry (1996a), Multivariate local polynomial regression for time series: uniform strong consistency and rates, J. Time Series Analysis, vol. 17, pp. 571–599.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    E. Masry (1996b), Multivariate regression estimation: Local polynomial fitting for time series, J. Stochastic Processes and their Applies., vol. 65, pp. 81–101.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    E. Masry (1997), Multivariate regression estimation of continuous-time processes from sampled-data: Local polynomial fitting approach. Manuscript.Google Scholar
  14. [14]
    M.I. Moore, P.J. Thompson, and T.G.L. Shirtcliffe (1988), Spectral analysis of ocean profiles from unequally spaced data, J. Geophys. Res. vol. 93, pp. 655–664.CrossRefGoogle Scholar
  15. [15]
    E. Parzen (1983), Time series analysis of irregularly observed data. Lecture Notes in Statistics, Vol. 25. New York: Springer-Verlag.Google Scholar
  16. [16]
    P. M. Robinson (1983), Nonparametric estimators for time series, J. Time Series Anal, vol. 4, pp. 185–297.zbMATHCrossRefGoogle Scholar
  17. [17]
    M. Rosenblatt (1956), A central limit theorem and strong mixing conditions, Proc. Nat. Acad. Sci, vol. 4, pp. 43–47.MathSciNetCrossRefGoogle Scholar
  18. [18]
    G. G. Roussas (1990), Nonparametric regression estimation under mixing conditions, Stochastic Processes and their Applies., vol. 36, pp. 107–116.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    D. Ruppert and M.P. Wand (1994), Multivariate weighted least squares regression, Ann. Statist., vol. 22, pp. 1346–1370.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    D. Tjostheim (1994), Non-linear time series: A selective review, Scandinavian J. of Statistics, vol. 21, pp. 97–130.MathSciNetGoogle Scholar
  21. [21]
    L. T. Tran (1993), Nonparametric function estimation for time series by local average estimators, Ann. Statist., vol. 21, pp. 1040–1057.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    V.A. Volkonskii and Yu. A. Rozanov (1959), Some limit theorems for random functions, Theory Prob. Appl, vol. 4, pp. 178–197.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Elias Masry
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of California, San DiegoLa JollaUSA

Personalised recommendations