Abstract
We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of the kernel density estimators with the order of bandwidths fixed as h = cn −1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations. In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties.
Similar content being viewed by others
References
Bryk A., Mielniczuk J. (2005). Asymptotic properties of density estimates for linear processes: application of projection method. Journal of Nonparametric Statistics 17, 121–133
Cheng B., Robinson P.M. (1991). Density estimation in strongly dependent non-linear time series. Statistica Sinica 1, 335–359
Chow Y.S., Teicher H. (1988). Probability theory (2nd ed). New York, Springer
Csörgő S., Mielniczuk J. (1995). Density estimation under long-range dependence. The Annals of Statistics 28, 990–999
Doukhan P. (1994). Mixing: properties and examples Lecture Notes in Statistics, Vol. 85. New York, Springer
Fan J., Yao Q. (2003). Nonlinear time series. New York, Springer
Giraitis L., Koul H.L., Surgailis D. (1996). Asymptotic normality of regression estimators with long memory errors. Statistics & Probability Letters 29, 317–335
Giraitis L., Surgailis D. (1986). Multivariate Appell polynomials and the central limit theorem. In: Eberlein E., Taqqu M.S. (eds). Dependence in probability and statistics. Boston, Birkhäuser, pp. 21–71
Hall P., Hart J.D. (1990). Convergence rates in density estimation for data from infinite-order moving average process. Probability Theory and Related Fields 87, 253–274
Hallin M., Tran L.T. (1996). Kernel density estimation for linear processes: asymptotic normality and optimal bandwidth derivation. The Annals of the Institute of Statistical Mathematics 48, 429–449
Hidalgo J. (1997). Non-parametric estimation with strongly dependent multivariate time series. Journal of Time Series Analysis, 18, 95–122
Ho H.-C. (1996). On central and non-central limit theorems in density estimation for sequences of long-range dependence. Stochastic Processes and their Applications 63, 153–174
Ho H.-C., Hsing T. (1996). On the asymptotic expansion of the empirical process of long-memory moving averages. The Annals of Statistics 24, 992–1024
Ho H.-C., Hsing T. (1997). Limit theorems for functionals of moving averages. The Annals of Probability 25, 1636–1669
Honda T. (2000). Nonparametric density estimation for a long-range dependent linear process. The Annals of the Institute of Statistical Mathematics 52, 599–611
Hsing T. (1999). On the asymptotic distribution of partial sum of functionals of infinite-variance moving averages. The Annals of Probability 27, 1579–1599
Koul H.L., Surgailis D. (2001). Asymptotics of empirical processes of long memory moving averages with infinite variance. Stochastic Processes and their Applications 91, 309–336
Koul H.L., Surgailis D. (2002). Asymptotic expansion of the empirical process of long memory moving averages. In: Dehling H., Mikosch T., Sørensen M. (eds). Empirical process techniques for dependent data. Boston, Birkhäuser, pp. 213–239
Peng L., Yao Q. (2004). Nonparametric regression under dependent errors with infinite variance. The Annals of the Institute of Statistical Mathematics 56, 73–86
Pipiras V., Taqqu M.S. (2003). Central limit theorems for partial sums of bounded functionals of infinite-variance moving averages. Bernoulli 9, 833–855
Samorodnitsky G., Taqqu M.S. (1994). Stable non-Gaussian processes: stochastic models with infinite variance. London, Chapman & Hall
Schick A., Wefelmeyer W. (2006). Pointwise convergence rates for kernel density estimators in linear processes. Statistics & Probability Letters 76, 1756–1760
Surgailis D. (2002). Stable limits of empirical processes of moving averages with infinite variance. Stochastic Processes and their Applications 100, 255–274
Surgailis D. (2004). Stable limits of sums of bounded functions of long-memory moving averages with finie variance. Bernoulli 10, 327–355
Taniguchi M. Kakizawa Y. (2000). Asymptotic theory of statistical inference for time series. New York, Springer
Wu W.B., Mielniczuk J. (2002). Kernel density estimation for linear processes. The Annals of Statistics 30, 1441–1459
Wuertz, D. and many others and see the SOURCE file (2006). fBasics: Rmetrics -Marketes and Basic Statistics, R package version 221.10065. http://www.rmetrics.org.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Honda, T. Nonparametric density estimation for linear processes with infinite variance. Ann Inst Stat Math 61, 413–439 (2009). https://doi.org/10.1007/s10463-007-0149-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-007-0149-x