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Wavelet linear density estimator for a discrete-time stochastic process: Lp-losses

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  • Leblanc, Frédérique

Abstract

We establish that the Lp'-loss (2 [less-than-or-equals, slant] p' > [infinity]) of the linear wavelet density estimator for a stochastic process converges at the rate N[-s'/(2s' + 1)] (s' = 1 - 1/p + 1/p') when the density f belongs to the Besov space Bp,qs. This estimator is optimal when p' = p. We suppose that the process is strongly mixing and we show that the rate of convergence essentially depends on the behavior of a special quadratic characteristic. After a discussion about the assumptions of the main result, we present some examples of Markov processes which satisfy these assumptions.

Suggested Citation

  • Leblanc, Frédérique, 1996. "Wavelet linear density estimator for a discrete-time stochastic process: Lp-losses," Statistics & Probability Letters, Elsevier, vol. 27(1), pages 71-84, March.
    • Handle: RePEc:eee:stapro:v:27:y:1996:i:1:p:71-84
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    1. 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.
    2. Kerkyacharian, G. & Picard, D., 1992. "Density estimation in Besov spaces," Statistics & Probability Letters, Elsevier, vol. 13(1), pages 15-24, January.
    3. Nummelin, Esa & Tuominen, Pekka, 1982. "Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 12(2), pages 187-202, March.
    4. Lanh Tran, 1990. "Recursive kernel density estimators under a weak dependence condition," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(2), pages 305-329, June.
    5. Bosq, Denis, 1995. "Optimal asymptotic quadratic error of density estimators for strong mixing or chaotic data," Statistics & Probability Letters, Elsevier, vol. 22(4), pages 339-347, March.
    6. Tran, Lanh Tat, 1990. "Kernel density estimation under dependence," Statistics & Probability Letters, Elsevier, vol. 10(3), pages 193-201, August.
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    Cited by:

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    2. Masry, Elias, 1997. "Multivariate probability density estimation by wavelet methods: Strong consistency and rates for stationary time series," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 177-193, May.
    3. B. L. S. Prakasa Rao, 2010. "Wavelet linear estimation for derivatives of a density from observations of mixtures with varying mixing proportions," Indian Journal of Pure and Applied Mathematics, Springer, vol. 41(1), pages 275-291, February.

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