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Smoothness in Bayesian Non-parametric Regression with Wavelets

Author

Listed:
  • Marco Di Zio

    (ISTAT)

  • Arnoldo Frigessi

    (Norwegian Computing Center)

Abstract

Consider the classical nonparametric regression problem yi = f(ti) + ɛii = 1,...,n where ti = i/n, and ɛi are i.i.d. zero mean normal with variance σ2. The aim is to estimate the true function f which is assumed to belong to the smoothness class described by the Besov space B pq q . These are functions belonging to Lp with derivatives up to order s, in Lp sense. The parameter q controls a further finer degree of smoothness. In a Bayesian setting, a prior on B pq q is chosen following Abramovich, Sapatinas and Silverman (1998). We show that the optimal Bayesian estimator of f is then also a.s. in B pq q if the loss function is chosen to be the Besov norm of B pq q . Because it is impossible to compute this optimal Bayesian estimator analytically, we propose a stochastic algorithm based on an approximation of the Bayesian risk and simulated annealing. Some simulations are presented to show that the algorithm performs well and that the new estimator is competitive when compared to the more standard posterior mean.

Suggested Citation

  • Marco Di Zio & Arnoldo Frigessi, 1999. "Smoothness in Bayesian Non-parametric Regression with Wavelets," Methodology and Computing in Applied Probability, Springer, vol. 1(4), pages 391-405, December.
    • Handle: RePEc:spr:metcap:v:1:y:1999:i:4:d:10.1023_a:1010038117836
    DOI: 10.1023/A:1010038117836
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    References listed on IDEAS

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    1. F. Abramovich & T. Sapatinas & B. W. Silverman, 1998. "Wavelet thresholding via a Bayesian approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 725-749.
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