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Consistency results for the kernel density estimate on continuous time stationary and dependent data

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  • Didi, Sultana
  • Louani, Djamal

Abstract

The aim of this paper is to study the consistency of the kernel density estimator pertaining to a continuous time stationary process X=(Xt)t≥0, with an underlying density f. More precisely, in a rather general dependency setting, where we use a martingale difference device and a technique based on a sequence of projections on σ-fields, we establish the almost sure pointwise and uniform consistencies with rates of the estimate fT of f built upon the part (Xt)0≤t≤T of the process X.

Suggested Citation

  • Didi, Sultana & Louani, Djamal, 2013. "Consistency results for the kernel density estimate on continuous time stationary and dependent data," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1262-1270.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:4:p:1262-1270
    DOI: 10.1016/j.spl.2013.01.024
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    References listed on IDEAS

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    1. Wu, Wei Biao & Huang, Yinxiao & Huang, Yibi, 2010. "Kernel estimation for time series: An asymptotic theory," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2412-2431, December.
    2. Castellana, J. V. & Leadbetter, M. R., 1986. "On smoothed probability density estimation for stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 21(2), pages 179-193, February.
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    Cited by:

    1. El Heda, Khadijetou & Louani, Djamal, 2018. "Optimal bandwidth selection in kernel density estimation for continuous time dependent processes," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 9-19.
    2. Laïb, Naâmane & Louani, Djamal, 2019. "Asymptotic normality of kernel density function estimator from continuous time stationary and dependent processes," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 187-196.
    3. Chaouch, Mohamed & Laïb, Naâmane, 2019. "Optimal asymptotic MSE of kernel regression estimate for continuous time processes with missing at random response," Statistics & Probability Letters, Elsevier, vol. 154(C), pages 1-1.

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