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 \((p'=2)\).


62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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[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 · doi:10.1007/BF01199026
[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 Unpublished manuscript.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 · doi:10.1214/aos/1176347758
[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 · doi:10.1016/0167-7152(92)90231-S
[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 · doi:10.1016/0167-7152(93)90024-D
[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 · doi:10.1007/BF02771562
[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 · doi:10.1016/0021-9045(92)90123-6
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