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 qpq . 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 qpq is chosen following Abramovich, Sapatinas and Silverman (1998). We show that the optimal Bayesian estimator of f is then also a.s. in B qpq if the loss function is chosen to be the Besov norm of B qpq . 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.
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References
F. Abramovich, T. Sapatinas, and B.W. Silverman, “Wavelet thresholding via a Bayesian approach, ” J. Roy. Stat. Soc. vol. B 60 pp. 725–749, 1998.
J. Bergh and J. Löfström, Interpolation spaces, Springer-Verlag: Berlin, 1976.
H. A. Chipman, E. D. Kolaczyk, and R. E. McCullooch, “Adaptive Bayesian wavelet shrinkage, ” Technical report 421, Department of Statistics, Chicago University, USA, 1996.
C. K. Chui, An Introduction to Wavelets, Academic Press: New York, 1992.
M. Clyde, G. Parmigiani, and B. Vidakovic, “Multiple shrinkage and subset selection in wavelets, ” Technical report 95–37, ISDS, Duke University, USA, 1995.
I. Daubechies, Ten lectures on Wavelets, SIAM, 1992.
I. Daubechies, “Orthonormal bases of compactly supported wavelets II: Variations on a theme, ” J. Math. Anal. vol. 24 SIAM, pp. 499–519, 1993.
R. A. DeVore and V. A. Popov, “Interpolation of Besov spaces, ” Trans. Am. Math. Soc. vol. 305 pp. 397–414, 1988.
R. A. DeVore, B. Jawerth, and V. A. Popov, “Compression of wavelet decomposition, ” Amer. Journ. of Mathematics vol. 114 pp. 737–785, 1992.
D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage, ” Biometrika vol. 81 (3) pp. 425–455, 1994.
D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet shrinkage: asymptopia?” J. Roy. Stat. Soc. vol. B 57(2) pp. 301–369, 1995.
D. L. Donoho and I. M. Johnstone, “Minimax estimation via wavelet shrinkage, ” Ann. Statis., vol. 26(3), 1998.
D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage, ” J. of American Stat. Assoc. vol. 90 pp. 1200–1224, 1995.
W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation and Statistical Applications, Lecture Notes in statistics 103, Springer, Heidelberg, 1998.
S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation, ” IEEE Trans. Pattern Analysis Machine Intelligence vol. 11 pp. 674–693, 1989.
Y. Meyer, Ondelettes et Opérateurs, Hermann: Paris, 1990.
J. Peetre, “New thoughts on Besov spaces, ” Duke Univ. Math. Series 1, 1976.
H. Rue, “New loss functions in Bayesian imaging, ” J. of American Stat. Assoc. vol. 90 pp. 900–908, 1995.
H. Triebel, Theory of function spaces II, Birkhäuser: Basel, 1990.
B. Vidakovic, “Non-linear wavelet shrinkage with Bayes rules and Bayes factors, ” J. of American Stat. Assoc.
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Zio, M.D., Frigessi, A. Smoothness in Bayesian Non-parametric Regression with Wavelets. Methodology and Computing in Applied Probability 1, 391–405 (1999). https://doi.org/10.1023/A:1010038117836
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DOI: https://doi.org/10.1023/A:1010038117836