On the Choice of Smoothing Parameter, Threshold and Truncation in Nonparametric Regression by Non-Linear Wavelet Methods

Abstract

Concise asymptotic theory is developed for non-linear wavelet estimators of regression means, in the context of general error distributions, general designs, general normalizations in the case of stochastic design, and non-structural assumptions about the mean. The influence of the tail weight of the error distribution is addressed in the setting of choosing threshold and truncation parameters. Mainly, the tail weight is described in an extremely simple way, by a moment condition; previous work on this topic has generally imposed the much more stringent assumption that the error distribution be normal. Different approaches to correction for stochastic design are suggested. These include conventional kernel estimation of the design density, in which case the interaction between the smoothing parameters of the non-linear wavelet estimator and the linear kernel method is described.HALL AND PATIL [No.2, = n-Ip JiffV + r"1\;2(1 -2-2r )-1J f-2(g~r) -gfr)iw + op(n-2r/(2r+I»).The same asymptotic argument applies to Jiin, the linear wavelet estimator. In the case where f is a kernel estimator the first step in the argument changes tõwhere and so J(ig)2 w = n-Ip J{t r(hp) + if}fV +p-2r1\;2(1 -2-2r )-1J{gr) -,(hp)gf r)}2 V + op(n-2r/(2r+I»). ACKNOWLEDGEMENTWe are grateful to two reviewers, whose constructive comments encouraged this succinct version of the paper.

Nonlinear regression modeling via regularized wavelets and smoothing parameter selection

“…, 2 J , > 0 is a smoothing parameter and d j = 2 (j −j 0 +1) is a level-dependent constant. Note that the above shrinkage method of coefficients differs from nonlinear methods such as hard thresholding c * jk =c jk (|c jk | > ) and soft thresholdingc * jk = sgn(c jk )(|c jk | − ) (|c jk | > ), both of which have been mainly used in wavelet-based estimates ( [6,7,9] among others).…”

“…, 2 J } with 2 J L, by using the wavelet interpolation. Hall and Patil [9] and Antoniadis and Pham [3] approached this problem instead by assuming that the design points are independent random variables each with identical density function w(t). The estimator given by Hall and Patil [9] isĥ(t) =ĝ(t)/ŵ(t) in which the density w(t) and g(t) = h(t)w(t) are estimated separately by a nonlinear wavelet estimate.…”

“…Wavelets form an orthonormal basis and enable multiresolution analysis by localizing a function in different phases of both time and frequency domains simultaneously, and thus offer some advantages over traditional Fourier expansions. Theoretical and practical developments of their use in statistics have been made by Donoho et al [6,7], Hall and Patil [9], among others. These papers focused on density estimation and regression estimation with the use of nonlinear thresholding, and demonstrated remarkable local adaptivity for large classes of irregular functions.…”

“…For the case that the design points are not equally spaced, the corresponding design matrix is no longer orthogonal, and wavelet-based decomposition/reconstruction procedure cannot be directly applied. Several different approaches for this case of unequally spaced design points have been made by Hall and Patil [9], Hall and Turlach [10], Antoniadis and Fan [2] and Pensky and Vidakovic [15] among others.…”

Wavelets in statistics: A review

“…Hall and PatiI ( [48], [49], [50]) studied asymptotic wavelet shrinkage methods in non-parametric curve estimation from the different viewpoint of a fixed target function, as opposed to the minimax approach of Donoho et al In the case of functions that are smooth or piecewise smooth in the classical sense, using wavelet decompositions which allow non-integer resolution levels, already described in Section 3, they derive necessary and sufficient conditions on the asymptotic form of the threshold and smoothing parameters for their resulting curve estimator to achieve optimal mean square convergence rates.…”