Local Asymptotic Normality Property for Lacunar Wavelet Series multifractal model
ESAIM: Probability and Statistics, Tome 15 (2011), pp. 69-82.

We consider a lacunar wavelet series function observed with an additive Brownian motion. Such functions are statistically characterized by two parameters. The first parameter governs the lacunarity of the wavelet coefficients while the second one governs its intensity. In this paper, we establish the local and asymptotic normality (LAN) of the model, with respect to this couple of parameters. This enables to prove the optimality of an estimator for the lacunarity parameter, and to build optimal (in the Le Cam sense) tests on the intensity parameter.

DOI : 10.1051/ps/2009005
Classification : 60G17, 62G07
Mots clés : local asymptotic normality, lacunar wavelet series 
@article{PS_2011__15__69_0,
     author = {Loubes, Jean-Michel and Paindaveine, Davy},
     title = {Local {Asymptotic} {Normality} {Property} for {Lacunar} {Wavelet} {Series} multifractal model},
     journal = {ESAIM: Probability and Statistics},
     pages = {69--82},
     publisher = {EDP-Sciences},
     volume = {15},
     year = {2011},
     doi = {10.1051/ps/2009005},
     mrnumber = {2870506},
     zbl = {1270.60045},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2009005/}
}
TY  - JOUR
AU  - Loubes, Jean-Michel
AU  - Paindaveine, Davy
TI  - Local Asymptotic Normality Property for Lacunar Wavelet Series multifractal model
JO  - ESAIM: Probability and Statistics
PY  - 2011
SP  - 69
EP  - 82
VL  - 15
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2009005/
DO  - 10.1051/ps/2009005
LA  - en
ID  - PS_2011__15__69_0
ER  - 
%0 Journal Article
%A Loubes, Jean-Michel
%A Paindaveine, Davy
%T Local Asymptotic Normality Property for Lacunar Wavelet Series multifractal model
%J ESAIM: Probability and Statistics
%D 2011
%P 69-82
%V 15
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2009005/
%R 10.1051/ps/2009005
%G en
%F PS_2011__15__69_0
Loubes, Jean-Michel; Paindaveine, Davy. Local Asymptotic Normality Property for Lacunar Wavelet Series multifractal model. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 69-82. doi : 10.1051/ps/2009005. http://archive.numdam.org/articles/10.1051/ps/2009005/

[1] A. Arneodo, E. Bacry, S. Jaffard and J.F. Muzy, Singularity spectrum of multifractal functions involving oscillating singularities. The Journal of Fourier Analysis and Applications 4 (1998) 159-174. | MR | Zbl

[2] A. Arneodo, E. Bacry, S. Jaffard and J.F. Muzy, Oscillating singularities and fractal functions. In Spline functions and the theory of wavelets (Montreal, PQ, 1999) , Amer. Math. Soc. Providence, RI (1999) 315-329. | MR | Zbl

[3] J.M. Aubry and S. Jaffard, Random wavelet series. Comm. Math. Phys. 227 (2002) 483-514. | MR | Zbl

[4] E. Bacry, A. Arneodo, U. Frisch, Y. Gagne and E. Hopfinger, Wavelet analysis of fully developed turbulence data and measurement of scaling exponents. In Turbulence and coherent structures (Grenoble, 1989) , Kluwer Acad. Publ. Dordrecht (1989) 203-215. | MR

[5] Z. Chi, Construction of stationary self-similar generalized fields by random wavelet expansion. Probab. Theory Related Fields 121 (2001) 269-300. | MR | Zbl

[6] A. Durand, Random wavelet series based on a tree-indexed Markov chain. Comm. Math. Phys. 283 (2008) 451-477. | MR | Zbl

[7] P. Flandrin, Wavelet analysis and synthesis of fractional Brownian Motion. IEEE Trans. Inform. Theory 38 (1992) 910-917. | MR | Zbl

[8] F. Gamboa and J.-M. Loubes, Bayesian estimation of multifractal wavelet function. Bernoulli (2005) 34-57. | MR | Zbl

[9] F. Gamboa and J.-M. Loubes, Estimation of the parameters of a multifractal wavelet function. Test 16 (2007) 383-407. | MR | Zbl

[10] C. Genovese and L. Wasserman, Rates of convergence for the Gaussian mixture sieve. Ann. Statist. 28 (2000) 1105-1127. | MR | Zbl

[11] S. Jaffard, On lacunary wavelet series. The Annals of Applied Probability 10 (2000) 313-329. | MR | Zbl

[12] B. Lindsay, The geometry of mixture likelihoods: a general theory. Ann. Statist. 11 (1983) 86-94. | MR | Zbl

[13] G. Mclachlan and K. Basford, Mixture models. Inference and applications to clustering. Statistics: Textbooks and Monographs 84. Marcel Dekker, Inc., New York (1988). | MR | Zbl

[14] S.G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(r). Transactions of the American Mathematical Society 315 (1989) 69-87. | MR | Zbl

[15] D.L. Mcleish and C.G. Small, Likelihood methods for the discrimination problem. Biometrika 73 (1986) 397-403. | MR | Zbl

[16] Y. Meyer, Ondelettes et Opérateurs . Hermann (1990). | MR | Zbl

[17] R.H. Riedi, M.S. Crouse, V.J. Ribeiro and R.G. Baraniuk, A multifractal wavelet model with application to network traffic. Institute of Electrical and Electronics Engineers. Transactions on Information Theory 45 (1999) 992-1018. | MR | Zbl

[18] F. Roueff, Almost sure haussdorff dimensions of graphs of random wavelet series. J. Fourier Analysis and App. 9 (2003). | MR | Zbl

[19] A.R. Swensen, The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend. J. Multivariate Anal. 16 (1985) 54-70. | MR | Zbl

[20] S. Van De Geer, Rates of convergence for the maximum likelihood estimator in mixture models. J. Nonparametr. Statist. 6 (1996) 293-310. | MR | Zbl

[21] A.W. Van Der Vaart, Asymptotic statistics . Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1998). ISBN 0-521-49603-9; 0-521-78450-6. | MR | Zbl

Cité par Sources :