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Nonparametric Density Estimation for Linear Processes with Infinite Variance

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  • Honda, Toshio
  • 本田, 敏雄

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.
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Suggested Citation

  • Honda, Toshio & 本田, 敏雄, 2006. "Nonparametric Density Estimation for Linear Processes with Infinite Variance," Discussion Papers 2005-13, Graduate School of Economics, Hitotsubashi University.
    • Handle: RePEc:hit:econdp:2005-13
    Note: February 2006; August 2006 (Revised)
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    References listed on IDEAS

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    1. Toshio Honda, 2000. "Nonparametric Density Estimation for a Long-Range Dependent Linear Process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 599-611, December.
    2. Hwai-Chung, Ho, 1996. "On central and non-central limit theorems in density estimation for sequences of long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 63(2), pages 153-174, November.
    3. Marc Hallin & Lanh T. Tran, 1996. "Kernel density estimation for linear processes: asymptotic normality and bandwidth selection," ULB Institutional Repository 2013/2055, ULB -- Universite Libre de Bruxelles.
    4. Giraitis, Liudas & Koul, Hira L. & Surgailis, Donatas, 1996. "Asymptotic normality of regression estimators with long memory errors," Statistics & Probability Letters, Elsevier, vol. 29(4), pages 317-335, September.
    5. Peng, Liang & Yao, Qiwei, 2004. "Nonparametric regression under dependent errors with infinite variance," LSE Research Online Documents on Economics 22874, London School of Economics and Political Science, LSE Library.
    6. Marc Hallin & Lanh Tran, 1996. "Kernel density estimation for linear processes: Asymptotic normality and optimal bandwidth derivation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(3), pages 429-449, September.
    7. Koul, Hira L. & Surgailis, Donatas, 2001. "Asymptotics of empirical processes of long memory moving averages with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 309-336, February.
    8. Javier Hidalgo, 1997. "Non‐Parametric Estimation With Strongly Dependent Multivariate Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 18(2), pages 95-122, March.
    9. Liang Peng & Qiwei Yao, 2004. "Nonparametric regression under dependent errors with infinite variance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(1), pages 73-86, March.
    10. Surgailis, Donatas, 0. "Stable limits of empirical processes of moving averages with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 255-274, July.
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    Cited by:

    1. Toshio Honda, 2010. "Nonparametric estimation of conditional medians for linear and related processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(6), pages 995-1021, December.
    2. Toshio Honda, 2013. "Nonparametric quantile regression with heavy-tailed and strongly dependent errors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 23-47, February.
    3. Chang, Yoosoon & Kim, Chang Sik & Park, Joon Y., 2016. "Nonstationarity in time series of state densities," Journal of Econometrics, Elsevier, vol. 192(1), pages 152-167.

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