×

Density estimation by wavelet thresholding. (English) Zbl 0860.62032

Summary: Density estimation is a commonly used test case for nonparametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients. Minimax rates of convergence are studied over a large range of Besov function classes \(B_{\sigma pq}\) and for a range of global \(L'_p\) error measures, \(1\leq p'<\infty\). A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a range of spaces and error measures. In particular, when \(p'>p\), some form of nonlinearity is essential, since the minimax linear estimators are suboptimal by polynomial powers of \(n\). A second approach, using an approximation of a Gaussian white-noise model in a Mallows metric, is used to attain exactly optimal rates of convergence for quadratic error \((p'=2)\).

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bergh, J. and L öfstr öm, J. (1976). Interpolation Spaces-An Introduction. Springer, New York. · Zbl 0344.46071
[2] Birgé, L. and Massart, P. (1996). From model selection to adaptive estimation. In Festschrift for Lucien Le Cam (D. Pollard and G. Yang, eds.). Springer, New York. · Zbl 0920.62042
[3] Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia. · Zbl 0776.42018
[4] Dely on, B. and Juditsky, A. (1993). Wavelet estimators, global error measures: revisited. Technical Report 782, Institut de Recherche en Informatique et Sy st emes Aléatoires, Campus de Beaulieu.
[5] Devroy e, L. and Gy örf, L. (1985). Nonparametric Density Estimation, The L1 View. Wiley, New York. · Zbl 0546.62015
[6] Donoho, D. L. and Johnstone, I. M. (1994). Minimax risk over p-balls for q-error. Probab. Theory Related Fields 99 277-303. · Zbl 0802.62006
[7] Donoho, D. L. and Johnstone, I. M. (1996). Minimax estimation via wavelet shrinkage. Unpublished manuscript. · Zbl 0935.62041
[8] Donoho, D. L., Johnstone, I. M., Kerky acharian, G. and Picard, D. (1996). Universal near minimaxity of wavelet shrinkage. In Festschrift for Lucien Le Cam (D. Pollard and G. Yang, eds.). Springer, New York. · Zbl 0891.62025
[9] Donoho, D. L., Johnstone, I. M., Kerky acharian, G. and Picard, D. (1995). Wavelet shrinkage: asy mptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301-369. JSTOR: · Zbl 0827.62035
[10] Donoho, D. L., Liu, R. C. and MacGibbon, K. B. (1990). Minimax risk over hy perrectangles, and implications. Ann. Statist. 18 1416-1437. · Zbl 0705.62018
[11] Doukhan, P. and Leon, J. (1990). Déviation quadratique d’estimateurs d’une densité par projection orthogonale. C. R. Acad. Sci. Paris Sér. I Math. 310 425-430. · Zbl 0702.60035
[12] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York. · Zbl 0219.60003
[13] Fix, G. and Strang, G. (1969). A Fourier analysis of the finite element method. Stud. Appl. Math. 48 265-273. · Zbl 0179.22501
[14] Frazier, M., Jawerth, B. and Weiss, G. (1991). Littlewood-Paley Theory and the Study of Function Spaces. Amer. Math. Soc., Providence, RI. · Zbl 0757.42006
[15] Härdle, W., Kerky acharian, G., Picard, D. and Tsy bakov, A. (1996). Wavelets and econometric applications. Technical report, Humboldt Univ., Berlin.
[16] Johnstone, I., Kerky acharian, G. and Picard, D. (1992). Estimation d’une densité de probabilité par méthode d’ondelettes. C. R. Acad. Sci. Paris Sér. I Math. 315 211-216. · Zbl 0755.62036
[17] Kerky acharian, G. and Picard, D. (1992). Density estimation in Besov spaces. Statist. Probab. Lett. 13 15-24. · Zbl 0749.62026
[18] Kerky acharian, G. and Picard, D. (1993). Density estimation by kernel and wavelet methods: optimality of Besov spaces. Statist. Probab. Lett. 18 327-336. · Zbl 0793.62019
[19] Meyer, Y. (1990). Ondelettes et Opérateurs, I: Ondelettes, II: Opérateurs de Calderón-Zy gmund, III: (with R. Coifman), Opérateurs Multilinéaires. Hermann, Paris. (English translation of first volume published by Cambridge Univ. Press.)
[20] Nemirovskii, A. (1985). Nonparametric estimation of smooth regression function. Izv. Akad. Nauk. SSSR Tekhn. Kibernet. 3 50-60 (in Russian); Soviet J. Comput. Sy stems Sci. 23 1-11 (1986) (in English). · Zbl 0604.62033
[21] Nemirovskii, A., Poly ak, B. and Tsy bakov, A. (1985). Rate of convergence of nonparametric estimates of maximum-likelihood ty pe. Problems Inform. Transmission 21 258-272. · Zbl 0616.62048
[22] Nussbaum, M. (1995). Personal communication.
[23] Peetre, J. (1975). New Thoughts on Besov Spaces. Dept. Mathematics, Duke Univ. · Zbl 0356.46038
[24] Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York. · Zbl 0322.60043
[25] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045
[26] Rosenthal, H. P. (1972). On the span in lp of sequences of independent random variables. Israel J. Math. 8 273-303. · Zbl 0213.19303
[27] Sakhanenko, A. I. (1991). Berry-Esseen ty pe estimates for large deviation probabilities. Siberian Math. J. 32 647-656. · Zbl 0778.60018
[28] Scott, D. (1992). Multivariate Density Estimation. Wiley, New York. · Zbl 0850.62006
[29] Silverman, B. W. (1986). Density Estimation for Statistics and Data Analy sis. Chapman and Hall, London. · Zbl 0617.62042
[30] Triebel, H. (1992). Theory of Function Spaces 2. Birkhäuser, Basel. · Zbl 0763.46025
[31] Walter, G. (1992). Approximation of the delta function by wavelets. J. Approx. Theory 71 329-343. · Zbl 0766.41020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.