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Asymptotic normality of the mixture density estimator in a disaggregation scheme

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  • Dmitrij Celov
  • Remigijus Leipus
  • Anne Philippe

Abstract

The paper concerns the asymptotic distribution of the mixture density estimator, proposed by Leipus et al. [Leipus, R., Oppenheim, G., Philippe, A., and Viano, M.-C. (2006), ‘Orthogonal Series Density Estimation in a Disaggregation Scheme’, Journal of Statistical Planning and Inference, 136, 2547–2571], in the aggregation/disaggregation problem of random parameter AR(1) process. We prove that, under mild conditions on the (semiparametric) form of the mixture density, the estimator is asymptotically normal. The proof is based on the limit theory for the quadratic form in linear random variables developed by Bhansali et al. [Bhansali, R.J., Giraitis, L., and Kokoszka, P.S. (2007), Approximations and Limit Theory for Quadratic Forms of Linear Processes’, Stochastic Processes and their Applications, 117, 71–95]. The moving average representation of the aggregated process is investigated. A simulation study illustrates the result.

Suggested Citation

  • Dmitrij Celov & Remigijus Leipus & Anne Philippe, 2010. "Asymptotic normality of the mixture density estimator in a disaggregation scheme," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(4), pages 425-442.
    • Handle: RePEc:taf:gnstxx:v:22:y:2010:i:4:p:425-442
    DOI: 10.1080/10485250903045528
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    References listed on IDEAS

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    1. Paolo Zaffaroni, 2007. "Contemporaneous aggregation of GARCH processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(4), pages 521-544, July.
    2. Terence Tai-Leung Chong, 2006. "The polynomial aggregated AR(1) model," Econometrics Journal, Royal Economic Society, vol. 9(1), pages 98-122, March.
    3. Granger, C. W. J., 1980. "Long memory relationships and the aggregation of dynamic models," Journal of Econometrics, Elsevier, vol. 14(2), pages 227-238, October.
    4. Zaffaroni, Paolo, 2004. "Contemporaneous aggregation of linear dynamic models in large economies," Journal of Econometrics, Elsevier, vol. 120(1), pages 75-102, May.
    5. Bhansali, R.J. & Giraitis, L. & Kokoszka, P.S., 2007. "Approximations and limit theory for quadratic forms of linear processes," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 71-95, January.
    6. Georges Oppenheim & Marie‐Claude Viano, 2004. "Aggregation of random parameters Ornstein‐Uhlenbeck or AR processes: some convergence results," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(3), pages 335-350, May.
    7. Hosking, Jonathan R. M., 1996. "Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series," Journal of Econometrics, Elsevier, vol. 73(1), pages 261-284, July.
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    Cited by:

    1. Remigijus Leipus & Anne Philippe & Vytautė Pilipauskaitė & Donatas Surgailis, 2020. "Estimating Long Memory in Panel Random‐Coefficient AR(1) Data," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(4), pages 520-535, July.
    2. Leipus, Remigijus & Philippe, Anne & Pilipauskaitė, Vytautė & Surgailis, Donatas, 2017. "Nonparametric estimation of the distribution of the autoregressive coefficient from panel random-coefficient AR(1) data," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 121-135.
    3. Anne Philippe & Donata Puplinskaite & Donatas Surgailis, 2014. "Contemporaneous Aggregation Of Triangular Array Of Random-Coefficient Ar(1) Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(1), pages 16-39, January.

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