Abstract
The problem of estimating the marginal density of a linear process by kernel methods is considered. Under general conditions, kernel density estimators are shown to be asymptotically normal. Their limiting covariance matrix is computed. We also find the optimal bandwidth in the sense that it asymptotically minimizes the mean square error of the estimators. The assumptions involved are easily verifiable.
Similar content being viewed by others
References
Chan N. H. and Tran L. T. (1991). Nonparametric tests for serial dependence, J. Time Ser. Anal., 13, 19–28.
Chanda K. C. (1983). Density estimation for linear processes, Ann. Inst. Statist. Math., 35, 439–446.
Devroye L. and Györfi L. (1985). Nonparametric Density Estimation: The L1 View, Wiley, New York.
Gorodetskii V. V. (1977). On the strong mixing properties for linear sequences, Theory Probab. Appl., 22, 411–413.
Györfi L., Härdle W. Sarda P. and Vieu P. (1989). Nonparametric curve estimation from time series, Lecture Notes in Statist., 60, Springer, New York.
Helson H. and Sarason D. (1990). Past and future, Math. Scand., 21, 5–16.
Ibragimov I. A. and Rozanov Yu. V. (1978). Gaussian Random Processes, Springer, New York.
Ioannides D. and Roussas G. G. (1987). Note on uniform convergence for mixing random variables, Statist. Probab. Lett., 5, 279–285.
Masry E. (1986). Recursive probability density estimation for weakly dependent stationary processes, IEEE Trans. Inform. Theory, IT 18, 254–267.
Masry E. (1987). Almost sure convergence of recursive density estimators for stationary mixing processes, Statist. Probab. Lett., 5, 249–254.
Masry E. and Györfi L. (1987). Strong consistency and rates for recursive density estimators for stationary mixing processes, J. Multivariate Anal., 22, 79–93.
Parzen E. (1962). On estimation of a probability density function and mode, Ann. Math. Statist., 33, 1065–1076.
Robinson P. M. (1983). Nonparametric estimators for time series. J. Time Ser. Anal., 4, 185–297.
Robinson P. M. (1986). On the consistency and finite sample properties of nonparametric kernel time series regression, autoregression and density estimators, Ann. Inst. Statist. Math., 38, 539–549.
Robinson P. M. (1987). Time series residuals with application to probability density estimation, J. Time Ser. Anal., 3, 329–344.
Rosenblatt M. (1956). Remarks on some nonparametric estimates of a density function. Ann. Math. Statist., 27, 832–837.
Roussas G. G. (1988). Nonparametric estimation in mixing sequences of random variables, J. Statist. Plann. Inference, 18, 135–149.
Roussas G. G. (1990). Nonparametric regression estimation under mixing conditions, Stochastic Process. Appl., 36, 107–116.
Sarason D. (1972). An addendum to “Past and future”, Math. Scand., 30, 62–64.
Silverman B. W. (1986). Density Estimation for Statistics and Data Analysis, Chapman and Hall, London.
Tran L. T. (1989). Recursive density estimation under dependence, IEEE Trans. Inform. Theory, 35, 1103–1108.
Tran L. T. (1992). Kernel density estimation for linear processes, Stochastic Process. Appl. 41, 281–296.
Withers C. S. (1981). Conditions for linear processes to be storng mixing, Z. Wahrschr. Verw. Gebiete, 57, 477–480.
Yakowitz S. (1985). Nonparametric density estimation, prediction and regression for Markov sequences, J. Amer. Statist. Assoc., 80, 215–221.
Yakowitz S. (1987). Nearest-neighbor methods for time series analysis, J. Time Ser. Anal., 8, 235–247.
Author information
Authors and Affiliations
Additional information
Supported in part by NSF grant DMS-9403718.
About this article
Cite this article
Hallin, M., Tran, L.T. Kernel density estimation for linear processes: Asymptotic normality and optimal bandwidth derivation. Ann Inst Stat Math 48, 429–449 (1996). https://doi.org/10.1007/BF00050847
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00050847