(a). Bumps (b). Wavelet Coefficients

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1 Incorporating Information on Neighboring Coecients into Wavelet Estimation T. Tony Cai Bernard W. Silverman Department of Statistics Department of Mathematics Purdue University University of Bristol West Lafayette, IN Bristol BS8 1TW U.S.A. U.K. Abstract In standard wavelet methods, the empirical wavelet coecients are thresholded term by term, on the basis of their individual magnitudes. Information on other coecients has no inuence on the treatment of particular coecients. We propose a wavelet shrinkage method that incorporates information on neighboring coecients into the decision making. The coecients are considered in overlapping blocks; the treatment of coecients in the middle of each block depends on the data in the whole block. The asymptotic and numerical performances of two particular versions of the estimator are investigated. We show that, asymptotically, one version of the estimator achieves the exact optimal rates of convergence over a range of Besov classes for global estimation, and attains adaptive minimax rate for estimating functions at a point. In numerical comparisons with various methods, both versions of the estimator perform excellently. Keywords: Adaptivity; Besov Space; Block Thresholding; James-Stein Estimator; Local Adaptivity; Nonparametric Regression; Wavelets; White Noise Model. 1

2 1 Introduction Consider the nonparametric regression model y i = f(t i )+z i (1) where t i = i=n for i = 1; 2;:::n; is the noise level, and the z i are i.i.d. N(0; 1). The function f() is an unknown function of interest. Wavelet methods have demonstrated success in nonparametric function estimation in terms of spatial adaptivity, computational eciency and asymptotic optimality. Standard wavelet methods achieve adaptivity through term-by-term thresholding of the empirical wavelet coecients. To obtain the wavelet coecients of the function estimate, each individual empirical wavelet coecient y is compared with a predetermined threshold, and is processed taking account solely of its own magnitude. Other coecients have no inuence on the estimate. Examples of shrinkage functions applied to individual coecients include the hard thresholding function h (y) = y I(jyj > ) and the soft thresholding function (y) s = sgn(y) (jyj,) +. For example, Donoho and Johnstone's (1995a) VisuShrink estimates the true wavelet coecients by soft thresholding with the universal threshold = (2 log n) 1=2. Hall et al. (1999) and Cai (1996, 1999a and 1999b) studied local block thresholding rules for wavelet function estimation. These threshold the empirical wavelet coecients in groups rather than individually, making simultaneous decisions to retain or to discard all the coecients within a block. The aim is to increase estimation accuracy by utilizing information about neighboring wavelet coecients. Estimators obtained by block thresholding enjoy a higher degree of spatial adaptivity than the standard term-by-term thresholding methods. The multiwavelet threshold estimators considered by Downie and Silverman (1998) also utilize block thresholding ideas. In the present paper, we propose a wavelet shrinkage method that incorporates into the thresholding decision information about neighboring coecients outside the block of current interest. The basic motivation is that if neighboring coecients contain some signal, then it is likely that the coecients of current direct interest also do, and so a lower threshold should be used. Two particular cases are considered. One method, which we call NeighBlock, estimates wavelet coecients simultaneously in groups, with the aim of gaining the advantages of the block thresholding method. In the other approach, called NeighCoe coecients are estimated individually, but with reference to their neighbors. Both methods are specied completely, with explicit denition of both the block size and the threshold level. After Section 2.1 in which basic notation and denitions are reviewed, the two estimators are dened in Section 2.2. We then investigate the two estimators both theoretically and by simulation. We show in Section 3 that the NeighBlock estimator enjoys a high degree of adaptivity and spatial adaptivity. Specically, we prove that the estimator simultaneously attains the exact optimal rate of convergence over a wide interval of the Besov classes with p 2 without prior knowledge of the smoothness of the underlying functions. Over the Besov classes with p < 2, the estimator simultaneously achieves the optimal convergence rate within a logarithmic factor. For estimating functions at a point, the estimator also 2

3 attains the local adaptive minimax rate. The theoretical properties of NeighCoe are discussed in Section 4. The estimator is within a logarithmic factor of being minimax over a range of Besov classes, and shares the pointwise optimality properties of NeighBlock. Technical details and proofs are given in Section 6. In Section 5, a simulation study of the two estimators is reported, together with a comparison on a data set collected in an anesthesiology study. Both estimators have excellent performance relative to conventional wavelet shrinkage methods. Our nite-sample comparison of the two estimators seems to rank the estimators in the opposite order to the theoretical discussion. If anything, the NeighCoe method is slightly superior. The estimators are appealing visually as well as quantitatively. The reconstructions jump where the target function jump; the reconstruction is smooth where the target function is smooth. They do not contain the spurious ne-scale structure contained in some wavelet estimators, but adapt well to subtle changes in the underlying functions. In comparison to the estimator of Hall et al. (1999), the NeighBlock estimator also enjoys a number of advantages. First, their estimator is not fully specied and no criterion is given for choosing the parameters objectively in nite sample cases. So the users need to select the block size and threshold level empirically. The NeighBlock estimator, on the other hand, is completely specied and easy to implement. Asymptotically, NeighBlock achieves the optimal local adaptivity while the estimator of Hall et al. does not. The web site Cai and Silverman (1999) contains SPlus scripts implementing both our estimators. It also describes additional simulation results not included in this paper. 2 The estimation method 2.1 Notation and conventions We shall assume that we are working within an orthonormal wavelet basis generated by dilation and translation of a compactly supported scaling function and a mother wavelet. We call a wavelet r-regular if has r vanishing moments and r continuous derivatives. For simplicity in exposition, we work with periodized wavelet bases on [0; 1], letting p jk(t) = jk (t, l); jk(t) p = jk (t, l); for t 2 [0; 1] l2z l2z where jk (t) =2 j=2 (2 j t, k); jk (t) =2 j=2 (2 j t, k): The collection f p j 0 k; k =1;:::;2 j 0 ; p jk; j j 0 0;k =1; :::; 2 j g is then an orthonormal basis of L 2 [0; 1], provided the primary resolution level j 0 is large enough to ensure that the support of the scaling functions and wavelets at level j 0 is not the whole of [0; 1]. The superscript \p" will be suppressed from the notation for convenience. An orthonormal wavelet basis has an associated exact orthogonal Discrete Wavelet Transform (DWT) that is norm-preserving and transforms sampled data into the wavelet coecient domain in O(n) steps. We use the standard device of transforming the problem 3

4 in the function domain into a problem, in the sequence domain, of estimating the wavelet coecients. See Daubechies (1992) and Strang (1992) for further details about the wavelets and the discrete wavelet transform. Wavelets are known for their excellent compression and localization properties. In very many cases of interest, information about a function is essentially contained in relatively small number of large coecients. Figure 1 displays the wavelet coecients of the wellknown test function Bumps (Donoho & Johnstone (1994)). It shows that large detail coecients come as groups; they cluster around the areas where the function changes signicantly. (a). Bumps (b). Wavelet Coefficients d1 d2 d3 d4 d5 d6 s Figure 1: Wavelet coecients. This example illustrates the motivation for our methods a coecient is more likely to contain information about a signal if neighboring coecients do also. Therefore when the observations are contaminated with noise, estimation accuracy might be improved by incorporating information on neighboring coecients. Indeed, as we shall see, our estimators show signicant numerical improvement over the conventional term-by-term thresholding estimators. Suppose we observe the data Y = fy i g as in (1). We shall assume that the noise level is known. Let ~ = W Y be the discrete wavelet transform of Y. Then ~ is an n-vector with elements ~ j0 k (k = 1; ; 2 j 0 ), which are the gross structure wavelet terms at the lowest resolution level, and ~ jk (j = 1; ;J, 1;k = 1; ; 2 j ), which are ne structure wavelet terms. Since the DWT is an orthogonal transform, the coecients are independently normally distributed with variance 2. For any particular estimation procedure based on the wavelet coecients, use the notation ^ for the estimate of the DWT of the values of f at the sample points. Up to the error involved in approximating f at the nest level by a wavelet series, the mean integrated square error of the estimation satises Ek ^f, fk 2 2 = n,1 Ek ^, k 2 : We therefore measure quality of recovery in terms of the mean square error in wavelet coecient space. 4

5 2.2 The NeighBlock and NeighCoe procedures We now dene the estimates studied in this paper. following stages: The NeighBlock estimator has the 1. Transform the data into the wavelet domain via the discrete wavelet transform: ~ = W Y. 2. At each resolution level j, group the empirical wavelet coecients into disjoint blocks b j i of length L 0 = [(log n)=2]. Each block b j i is extended by an amount L 1 = max(1; [L 0 =2]) in each direction to form overlapping larger blocks B j i of length L = L 0 +2L Within each block b j i, estimate the coecients simultaneously via a shrinkage rule ^ j;k = j i ~ j;k ; for all (j; k) 2 b j i : The shrinkage factor j i is chosen with reference to the coecients in the larger block B j i : j i =(1, L 2 =S 2 ji) + (2) where S 2 ji = (j;k)2b j i ~ 2 j;k (3) and = 4:5053. We can envision B j i as a sliding window which moves L 0 positions each time and, for each given window, only the half of the coecients in the center of the window are estimated. The choice of the block length L 0 and the threshold are discussed below. 4. Obtain the estimate of the function via the inverse discrete wavelet transform of the denoised wavelet coecients. The procedure is simple and easy to implement, at a computational cost of O(n). The NeighBlock procedure aims to combine the advantages previously found for block thresholding methods with those obtained by using information about neighboring coecients. In this case L log n, and the threshold is chosen by a James-Stein procedure; see Remark 2 below. Remark 1: If L 0 is not a power of two, then one or both of the b j i at the boundary is shortened to ensure all the b j i are nonoverlapping. In the periodic case, the corresponding B j i are kept of length L with b j i at the center. If periodic boundary conditions are not being used, then the b j i at the boundary are only extended in one direction to form B j i, again of length L. Remark 2: In the NeighBlock procedure, the thresholding constant is set to = 4:505:::, which is the solution of the equation, log =3. The value is derived from an oracle inequality introduced in Cai (1999). The estimator then enjoys superior numerical performance and asymptotic optimality, as our subsequent discussion shows. 5

6 Remark 3: The block length L 0 = [(log n)=2] depends on the sample size n only and the thresholding constant is an absolute constant. Therefore, in contrast to some other block thresholding methods, the various parameters are fully specied. Remark 4: The estimator can be modied by averaging over every possible position of the block centers. The resulting estimator sometimes has numerical advantages, at the cost of higher computational complexity. The NeighCoe procedure follows the same four steps as the NeighBlock estimator, but with L 0 = L 1 =1,L = 3, and =(2=3) log n. The eect is that each individual coecient is shrunk by an amount that may also depend on its immediate neighbors. See Section 4 for further details. Remark 5: NeighCoe uses lower threshold level than VisuShrink of Donoho and Johnstone (1995a). In NeighCoe, an coecient is estimated by zero only when the sum of squares of the empirical coecient and its immediate neighbors is less than (2 log n) 2, or the average of the squares is less than (2=3 log n) 2. We will investigate the asymptotic and numerical properties of the NeighBlock and NeighCoe estimators. Let us rst look at one example. The top-left panel of Figure 2 displays a noisy Doppler signal. We use NeighBlock and NeighCoe as well as three conventional methods, VisuShrink, SureShrink, and TI de-noising, to recover the true signal. The reconstructions are shown in Figure 2. Doppler + Noise NeighBlock VisuShrink NeighCoeff TI de-noising SureShrink Figure 2: The noisy Doppler signal and the reconstructions. The dotted line is the true signal. 6

7 All methods except SureShrink recover the smooth low frequency part reasonably well. SureShrink contains a signicant amount of local oscillations and is visually unpleasant. Both NeighBlock and NeighCoe automatically adapt to the changing frequency of the underlying signal. Both estimate the smooth and low frequency part accurately; at the same time, they also captures the more rapidly oscillating area between t = 0:1 and t = 0:4. In contrast, both VisuShrink and TI de-noising signicantly over-smooth in this region. SureShrink does better than VisuShrink and TI de-noising in recovering the high frequency part, but it contains a signicant amount of local oscillations and is visually unpleasant. None of the estimators do a particularly good job in the region t < 0:1 of very high frequency oscillation, partly because of the low sampling rate relative to the rate of oscillation; however, in contrast to any of the other estimators, both NeighBlock and NeighCoe do recover a little of the signal even in this region. Quantitatively, NeighBlock and NeighCoe are almost identical and both are signicantly better than the other methods. In this particular example, the ratios of the mean squared error of NeighBlock and NeighCoe to those of VisuShrink, SureShrink, and TI de-noising are 0.35, 0.72, and 0.45 respectively. See Section 5 for further numerical results. Inspection of wavelet coecients shows that NeighBlock NeighCoe, VisuShrink, and SureShrink use 33, 28, 15, and 61 detail coecients in the reconstruction, respectively. SureShrink retains many detail coecients in the low frequency area and as a result, the reconstruction contains spurious oscillations. VisuShrink keeps only 15 detail coecients and the reconstruction is over-smoothed. The additional smoothing inherent in the TIdenoising method has also led to over-smoothing. 3 Optimality of the NeighBlock procedure 3.1 Global properties As is traditional in the wavelet literature, we investigate the adaptivity of the NeighBlock procedure across Besov classes. Besov spaces are a very rich class of function spaces. They contain many traditional smoothness spaces such as Holder and Sobolev Spaces. We shall show that NeighBlock enjoys excellent adaptivity across a wide range of Besov classes. Full details of Besov spaces are given, for example, in DeVore and Popov (1988). For a given square-integrable function f on [0; 1], dene the scaling function and wavelet coecients jk = hf; jk i; jk = hf; jk i: The function f can be expanded into a wavelet series: f(x) = 2 j 0 k=1 j0 k j0 k(x)+ 1 2 j j=j 0 k=1 jk jk (x) (4) Let be the vector of the scaling function coecients, and for each j let j be the vector of the wavelet coecients at level j. Suppose > 0; 0 < p 1 and 0 < q 1. Then, roughly speaking, the Besov function norm of index (; p; q) quanties the size in an L p sense of the derivative of f 7

8 of order, with q giving a ner gradation; for a precise denition see DeVore and Popov (1988). Dene s = +1=2,1=p. For a given r-regular mother wavelet with r>, the Besov sequence norm of the wavelet coecients of a function f is then dened by where kk p + jj b s p;q ; jj q b s p;q = 1 j=j 0 2 jsq k j k q p: (5) It is an important fact (Meyer 1992) that the Besov function norm kfk B p;q is equivalent to the sequence norm of the wavelet coecients of f. The Besov class B p;q(m) is dened to be the set of all functions whose Besov norm is less than M. Denote the minimax risk over a function class F by R(F;n) = inf ^f n sup Ek ^f n, fk 2 2 f2f where ^f n are estimators based on n observed data points. Donoho and Johnstone (1998) showed that the minimax risk over a Besov class B p;q(m) is given by R(B p;q(m);n) n,2=(1+2) ; n!1 If attention is restricted to linear estimates, the corresponding minimax rate of convergence is n,0, with 0 +(1=p,, 1=p) = +1=2+(1=p,, 1=p) ; where p, = max(p; 2): (6) Therefore the traditional linear methods such as kernel, spline and orthogonal series estimates are suboptimal for estimation over the Besov bodies with p<2. We show in the following theorem that the NeighBlock method attains the exact optimal convergence rate over a wide range of the Besov scales. We denote by C a generic constant that may vary from place to place. Theorem 1 Suppose the wavelet is r-regular. Then the NeighBlock estimator satises sup f2b p;q (M ) Ek ^f n, fk 2 Cn,2=(1+2) for all M 2 (0; 1);2 (0;r);q 2 [1; 1] and p 2 [2; 1]. Thus, the NeighBlock estimator, without knowing the degree or amount of smoothness of the underlying function, attains the true optimal convergence rate that one could achieve knowing the regularity. The next theorem addresses the case of p<2, and shows that the NeighBlock method achieves advantages over linear methods even at the level of rates. Theorem 2 Assume that the wavelet is r-regular. Then the NeighBlock estimator is simultaneously within a logarithmic factor of minimax for p<2: sup f2b p;q (M ) Ek ^f n, fk 2 Cn,2=(1+2) (log n) (2,p)=p(1+2) for all M 2 (0; 1);2 [1=p; r);q 2 [1; 1] and p 2 [1; 2). The proofs of these theorems are given in Section 6. 8 (7) (8)

9 3.2 Local adaptation We now study the optimality of the NeighBlock procedure for estimating functions at a point. It is well known that for global estimation, it is possible to achieves complete adaptation without any cost in terms of convergence rate across a range of function classes. That is, one can achieve the same rate of convergence when the degree of smoothness is unknown as one could do if the degree of smoothness is known. But for estimation at a point, one must asymptotically pay a price (albeit only in log n terms) for not knowing the smoothness of the underlying function. Denote the minimax risk for estimating functions at a point t 0 over a function class F by R(F;n;t 0 ) = inf ^f n sup E( ^f n (t 0 ), f(t 0 )) 2 F Consider the Holder class (M). The optimal rate of convergence for estimating f(t 0 ) with known is n, where =2=(1+2). Brown and Low (1996b) and Lepski (1990) showed that even when is known to be one of two values, adaptation causes the rate of convergence to be worse by at least a logarithmic factor. They showed that the best one can do is (log n=n) when the smoothness parameter is unknown. We call (log n=n) the local adaptive minimax rate over the Holder class (M). The following theorem shows that NeighBlock achieves the local adaptive minimax rate over a wide range of Holder classes. Theorem 3 Suppose the wavelet is r-regular and has r vanishing moments with r. Let t 0 2 (0; 1) be xed. Then the NeighBlock estimator ^f n(t 0 ) of f(t 0 ) satises sup Ef ^f n(t 0 ), f(t 0 )g 2 C(n,1 log n) 2=(1+2) : (9) f2 (M ) Remark: As shown in Cai (1999b), the estimator of Hall, et al. (1999) does not achieve the optimal local adaptivity in terms of the rate of convergence; the block size L = C(log n) 1+ is too large. Therefore NeighBlock outperforms the estimator of Hall, et al. (1999) in terms of local adaptivity. 3.3 Denoising property In addition to the global and local estimation properties, the NeighBlock estimator enjoys an interesting smoothness property which should oer high visual quality of the reconstruction. The estimator, with high probability, removes pure noise completely. Theorem 4 If the target function is the zero function f 0, then, with probability tending to 1 as n!1, the NeighBlock estimator is also the zero function, i.e., there exist universal constants P n such that P ( ^f n 0) P n! 1; as n!1 (10) 9

10 4 Properties of the NeighCoe estimator The NeighCoe estimator is intuitively appealing and easy to implement. It can be shown that the estimator is locally adaptive and is within a logarithmic factor of being minimax over a wide range of Besov classes. This is the same asymptotic performance as VisuShrink. We summarize the results without proof in the following theorems. We shall see subsequently that, in most cases, the estimator enjoys superior numerical performance to many classical wavelet estimators, such as VisuShrink, SureShrink and TI-denoising estimators in most cases. It even outperforms NeighBlock, even though that estimator apparently has better asymptotic properties. The rst result shows that the global performance of the NeighCoe estimator is simultaneously within a logarithmic factor of minimax over a wide range of Besov classes, and the second result shows that NeighCoe has the same good pointwise behavior as NeighBlock. Theorem 5 Assume that the wavelet is r-regular. Then the NeighCoe estimator satises sup Ek ^f f2bp;q (M n, fk 2 C(n,1 log n) 2=(1+2) (11) ) for all M 2 (0; 1);2 (0;r);q 2 [1; 1] and p 2 [1; 1]. Theorem 6 Assume that the wavelet is r-regular and has r vanishing moments with r. Let t 0 2 (0; 1) be xed. Then the NeighCoe estimator satises sup E( ^f n(t 0 ), f(t 0 )) 2 C(n,1 log n) 2=(1+2) : (12) f2 (M ) 5 Numerical comparison A simulation study was conducted to compare the numerical performance of the NeighBlock and NeighCoe estimators with Donoho and Johnstone's VisuShrink and SureShrink as well as Coifman and Donoho's Translation-Invariant (TI) denoising method. SureShrink selects the threshold at each resolution level by minimizing Stein's (1981) unbiased estimate of risk. In the simulation, we use the hybrid method proposed in Donoho and Johnstone (1995b). The TI-denoising method was introduced by Coifman and Donoho (1995), and is equivalent to averaging over estimators based on all the shifts of the original data. This method has various advantages over the universal thresholding methods. For further details see the original papers. We implement the NeighBlock and NeighCoe estimators in the software package S+Wavelets. The programs are available from the web site Cai and Silverman (1999). We compare the numerical performance of the methods using eight test functions representing dierent level of spatial variability. The test functions are plotted in Figure 5. Sample sizes ranging from n = 512 to n = 8192 and root-signal-to-noise ratios (RSNR) from 3 to 7 were considered. The RSNR is the ratio of the standard deviation of the function values to the standard deviation of the noise. Several dierent wavelets were used. 10

11 For reasons of space, we only report in detail the results for one particular case, using Daubechies' compactly supported wavelet Symmlet 8 and RSNR equal to 3. Table 1 reports the average squared errors over 60 replications with sample sizes ranging from n = 512 to n = A graphical presentation is given in Figure 3. Dierent combinations of wavelets and signal-to-noise ratios yield basically the same results; for details see the web site Cai and Silverman (1999). NeighCoeff vs VisuShrink NeighCoeff vs SureShrink Doppler HeaviSine Bumps Blocks Spikes Blip Corner Wave Doppler HeaviSine Bumps Blocks Spikes Blip Corner Wave NeighCoeff vs TI NeighCoeff vs NeighBlock Doppler HeaviSine Bumps Blocks Spikes Blip Corner Wave Doppler HeaviSine Bumps Blocks Spikes Blip Corner Wave Figure 3: RSNR=3. The vertical bars represent the log ratios of the MSEs of various estimators to the corresponding MSE of the NeighCoe estimator. The higher the bar the better the relative performance of the NeighCoe estimator, and a value of zero means that the estimators have equal performance. For each signal the bars are ordered from left to right by the sample sizes (n = 512; 1024; 2048; 4096; 8192). The NeighBlock and NeighCoe methods both uniformly outperform VisuShrink in all examples. For ve of the eight test functions, Doppler, Bumps, Blocks, Spikes and Blip, our methods have better precision with sample size n than VisuShrink with sample size 2n for all sample sizes where the comparison is possible. The NeighCoe method is slightly better than NeighBlock in almost all cases, and outperforms the other methods as well. The NeighCoe method is also better than TI-denoising in most cases, especially when the underlying function is of signicant spatial variability. In terms of the mean square error criterion, the only conceivable competitor among the standard methods is SureShrink. Apart from being somewhat superior to SureShrink in mean square error, our methods yield noticeably better results visually. Our estimates do not contain the spurious ne-scale eects that are often contained in the SureShrink estimator. The curious behavior of some of the methods with the Waves signal calls for some explanation. Throughout, the primary resolution level j 0 = [log 2 log n]+1was used for all 11

12 methods. Thus, j 0 = 3 for n 2048, and j 0 = 4 for n = 4096 and This change in the value of j 0 aects whether or not the high frequency eect in the Waves signal is felt in the lowest level of wavelet coecients. For j 0 = 3, the standard methods all smooth out the high frequency eect to some extent, because of applying a soft threshold with xed threshold. An attractive feature of the NeighCoe and NeighBlock methods is that they are not sensitive to the choice of primary resolution level in this way, because the threshold adapts to the presence of signal in all the coecients. Figure 4 shows a typical segment of the result of the four methods applied to the inductance plethysmography data analyzed, for example, by Abramovich et al. (1998). It can be seen that the SureShrink estimator allows through high frequency eects that are almost certainly spurious while both the standard and the translation-invariant versions of VisuShrink smooth out features of the curve NeighBlock VisuShrink TI denoising NeighBlock NeighCoeff SureShrink Figure 4: A segment of the data and curve estimates for the inductance plethysmography data. Left gure: NeighBlock (solid), VisuShrink (dotted), TI denoising (dashed). Right gure: Neigh- Block (solid), NeighCoe (dotted), SureShrink (dashed). Both the VisuShrink and TI denoising estimates smooth out the broad features, while the SureShrink estimate contains high frequency eects near times 300 and 335, both of which are presumably spurious. It would be interesting to include practical comparisons with the block thresholding estimator of Hall et al. (1999). Their method requires the selection of smoothing parameters block length and threshold level neither of which is completely specied and no criterion is given for choosing the parameters objectively in nite sample cases, so any detailed comparison would have to be left for future work. However, simulation results by Hall et al. (1997) show that even the translation-averaged version of the estimator has little advantage over VisuShrink when the signal to noise ratio is high. Our simulation shows that NeighBlock uniformly outperforms VisuShrink in all examples, and indeed the relative performance of VisuShrink is even worse for values of RSNR higher than the one presented in detail. Therefore we expect our estimator to perform favorably over the estimator of Hall et al. in terms of mean squared error, at least in the case of high signal-to-noise-ratio. 12

13 6 Proofs 6.1 Sequence space approximation We shall prove Theorems 1, 2 and 5 by using the sequence space approach introduced by Donoho and Johnstone (1998). A key step is to use the asymptotic equivalence results presented by Brown and Low (1996a) and to approximate the problem of estimating f from the noisy observations in (1) by the problem of estimating the wavelet coecient sequence of f contaminated with i.i.d. Gaussian noise. Donoho and Johnstone (1998) show a strong equivalence result on the nonparametric regression and the white noise model over the Besov classes B p;q(m). When the wavelet is r-regular with r > and p; q 1, then a simultaneously near-optimal estimator in the sequence estimation problem can be applied to the empirical wavelet coecients in the function estimation problem in (1), and will be a simultaneously near-optimal estimator in the function estimation problem. For further details about the equivalence and approximation arguments, the readers are referred to Donoho and Johnstone (1995b), (1998) and (1999) and Brown and Low (1996a). For approximation results, see also Chambolle et al. (1998). Under the correspondence between the estimation problem in function space and the estimation problem in sequence space, it suces to consider the following sequence estimation problem. 6.2 Estimation in sequence space by NeighBlock and NeighCoe Suppose we observe sequence data y jk = jk + n,1=2 z jk ; j 0; k =1; 2; ; 2 j (13) where z jk are i.i.d. N(0; 1). The mean array is the object that we wish to estimate. We assume that is in some Besov Body s p;q(m) =f : kk b s p;q Mg, where the norm is as dened in (5) above. Make the usual calibration s = +1=2, 1=p. Donoho and Johnstone (1998) show that the minimax rate of convergence for estimating over the Besov body s p;q(m) isn,2=(1+2) as n!1. The accuracy of estimation is measured by the expected squared error R(^; ) =E P j;k(^ j;k, j;k ) 2. We now approach this sequence estimation problem using a procedure corresponding to NeighBlock. Let J = [log 2 n]. Divide each resolution level j 0 j <J into nonoverlapping blocks of length L 0 = [(log n)=2]. Again denote by b j i the i-th block at level j and similarly dene B j i to be the larger block obtained by extending b j i by [(log n)=4] elements in each direction. Dene Sji 2 to be the sum of the yjk 2 over B j i,by analogy with (3). Now estimate by ^ with ^ jk = 8 >< >: y jk for j j 0 (1, n,1 L 2 =Sji) 2 + y jk for (j; k) 2 b j i ; j 0 j <J 0 for j J (14) 13

14 Similarly, for each (j; i), dene ~B j i to be the large block containing (j; i, 1), (j; i), and (j; i + 1), and dene ~S ji 2 = y 2 j;i,1 + yj;i 2 + y 2 j;i+1 to be the sum of squares over ~B j i. W dene a corresponding procedure to NeighCoe: ^ jk = 8 >< >: y jk for j j 0 (1, 2 2 n,1 log n= ~S ji) 2 + y jk for j 0 j <J 0 for j J For the estimator (14), we have the following minimax results, demonstrating that the estimator attains the exact minimax rate over all the Besov Bodies s p;q(m) with p 2, and the exact minimax rate up to a logarithmic term for p<2. Theorem 7 Dene ^ as in (14). Then, as n!1, sup Ek^, k 2 s p;q (M 2 ) For the estimator (15), we have ( Cn,2=(1+2) for p 2 Cn,2=(1+2) (log n) (2,p)=fp(1+2)g for p<2 and p 1: Theorem 8 Dene ^ as in (15). Then, for p 1, as n!1, sup Ek^, k 2 s p;q (M 2 C(log n=n) 2=(1+2) : ) The results of Theorem 1 and 2 follow from Theorem 7 and the equivalence and the approximation arguments discussed in Section 6.1. And similarly, Theorem 5 follows from Theorem Proof of the main results We will prove Theorems 7 and 8 as well as Theorems 3 and 6. The proof of Theorem 4 is straightforward. The key results used in the proofs are the following oracle inequalities. Lemma 1 Assume that y j;k, ^ j;k and ^ j;k are given as in (13), (14), and (15) respectively. Then, dening > 1 by, log =3, for each i and j 0 j <J E(^ j;k, j;k ) 2 ( 2 n,1 log n ^ j;k)+2n 2,2 2 : (16) (j;k)2b j i (j;k)2b j i E(^ j;i, j;i ) 2 (2 2 n,1 log n) ^ (j;k)2 ~ B j i (15) 2 j;k +2 2 n,2 (log n) 1=2 : (17) The proof of this lemma is an extension of the proof of Theorem 1 of Cai (1999a). First consider (16). For j; k in B j i dene ^ y j;k =(1, n,1 L 2 =S 2 ji) + y jk : Since ^ y j;k = ^ j;k for (j; k) in b j i, extending the sum from b j i to B j i, and replacing ^ by ^ y, can only increase the left hand side of (16). The argument of Theorem 1 and Lemma 2 of Cai (1999a) shows that the inequality holds with these changes, completing the proof of (16). The proof of (17) follows from Theorem 1 in Cai (1999) and the following upper bound for the tail probability of the 2 distribution. 14

15 Lemma 2 Let >1. Then P ( 2 m >m),1=2 (, 1),1 m,1=2 e, m 2 (,log,1) : (18) We also recall two elementary inequalities between two dierent `p norms, and a bound for a certain sum. Lemma 3 Let x 2 IR m, and 0 <p 1 p 2 1. Then the following inequalities hold: kxk p2 kxk p1 m 1 p, 1 1 p 2 kxk p2 (19) P Lemma 4 Let 0 <a<1 and S = fx 2 IR k : ki=1 x a i B; x i 0; i =1; ;kg. Then for > 0, sup x2s k i=1 (x i ^ ) B 1,a : Proof of Theorems 7 and 8: First consider Theorem 7. Let y and ^ be given as in (13) and (14) respectively. Then, Ek^, k 2 2 = j<j 0 k E(^ jk, jk ) 2 + J,1 j=j 0 say. We bound the term S 2 by using Lemma 1. Let A j i = k E(^ jk, jk ) 2 + (j;k)2b j i 2 jk; 1 j=j k 2 jk S 1 + S 2 + S 3 ; (20) the sum of squared coecients within the block B j i. We then split up the sum dening S 2 into sums over the individual blocks b j i, and apply the oracle inequality (16). Since L = log n and the number of blocks is denitely less than n, this yields S 2 = J,1 j=j 0 k E(^ jk, jk ) 2 C J,1 j=j 0 i (A j i ^ 2 n,1 L)+2n,1 2 : (21) Note also that, since 2 s p;q(m), we have 2 js k j k p M for each j. We now complete the proof for the two cases separately. The case p 2 For 2 s p;q(m), Lemma 3 implies that k j k 2 2 (2 j ) 2( 1 2, 1 p ) k j k 2 p M 2 2 2j( 1 2, 1 p,s) = M 2 2,2j : (22) It follows that S 1 + S 3 2 j 0 n,1 2 + so that S 1 + S 3 can be neglected. 1 j=j M 2 2,2j = o(n,2=(1+2) ); (23) 15

16 We divide the sum in (21) into two parts. Choose J 1 such that 2 J 1 n 1=(1+2). Then, J 1,1 j=j 0 i (A j i ^ 2 n,1 L) and, making use of the bound (22), J,1 j=j 1 i (A j i ^ 2 n,1 L) J 1,1 j=j 0 i J,1 j=j 1 i 2 n,1 L C2 J 1 n,1 Cn,2=(1+2) ; (24) J,1 A j i 2 k j k 2 2 Cn,2=(1+2) : (25) j=j 1 Combining (24) and (25) demonstrates that S 2 Cn,2=(1+2), completing the proof for this case. The case p < 2 with p 1: For 2 s p;q(m), Lemma 3 now yields k j k 2 2 k j k 2 p M 2 2,2js. The assumption p 1 implies that s 1, so that 2 S 3 C 1 j=j 2,2js Cn,2s Cn,1 : Thus S 1 + S 3 = o(n,2=(1+2) )asbefore. Now let J 2 be an integer satisfying 2 J 2 n 1=(1+2) (log n),(2,p)=p(1+2). Then, by an argument analogous to that leading to (24), J 2,1 j=j 0 i (A j i ^ 2 n,1 L) J 2,1 j=j 0 i 2 n,1 L Cn,2=(1+2) (log n),(2,p)=p(1+2) : (26) Turning to the other part of S 2, it follows by convexity that, for each j, i (A j i ) p=2 i (j;k)2b j i ( 2 jk) p=2 2 k Applying Lemma 4 with a = p=2, we have, after some algebra, J,1 j=j 2 i ( 2 jk) p=2 2M p 2,jsp : (A j i ^ 2 n,1 L) Cn,2=(1+2) (log n),(2,p)=p(1+2) : (27) We complete the proof of Theorem 7 by combining the bounds (26) and (27), as in the case p 2: The proof of Theorem 8 is similar, using the oracle inequality (17) instead of (16). Proof of Theorems 3 and 6: The proofs of Theorems 3 and 6 are similar to each other. We prove Theorem 6 rst. Suppose we observe the data Y = fy i g as in (1). Let ~ =W n,1=2 Y be the discrete wavelet transform of n,1=2 Y. Write ~ =( ~ j0 1; ; ~ j0 2 j 0 ; ~ j0 1; ; ~ j0 2 j 0 ; ; ~ J,1;1 ; ; ~ J,1;2 J,1) T 16

17 where ~ j0 k are the gross structure terms at the lowest level, and the coecients ~ jk (j = 1; ;J, 1;k = 1; ; 2 j ) are ne structure wavelet terms. Denote j;k hf; j;k i and j;k hf; j;k i. One may write ~ jk = jk + a j;k + n,1=2 z jk (28) where a j;k are the approximation errors and z jk 's are i.i.d. N(0; 1). Lemma 5 below is a consequence of the vanishing moments conditions on and. Lemma 5 Under the conditions of Theorem 6, there exists a constant C > 0 such that for all f 2 (M), j j;k jc2,j(1=2+) and ja j;k jcn, 2,j=2 : (29) It follows from the simple inequality, E( P n i=1 i ) 2 ( P n i=1 (Ei 2 ) 1=2 ) 2, 2 4 2j j E( ^f n(t 0 ), f(t 0 )) 2 = E 2 4 2j 0 k=1 k=1 (E(^ j0 k, j0 k) 2 2 j 0 k(t 0 )) 1=2 + (Q 1 + Q 2 + Q 3 ) 2 : (^ j0 k, j0 k) j0 k(t 0 )+ J,1 2 j j=j 0 k=1 j=j 0 k=1 (^ jk, jk ) jk (t 0 ) (E(^ jk, jk ) 2 2 jk(t 0 )) 1= j j=j k=1 j jk jk (t 0 )j It is obvious that Q 1 = O(n,1 ). Note that there are at most N jk at level j that are nonvanishing at t 0, where N is the length of the support of. Denote K(t 0 ; j) = fk : j;k(t 0 ) 6= 0g. Then jk(t 0 ; j)j N: Now (29) yields Q 3 = 1 2 j j=j k=1 j jk jj jk (t 0 )j 1 j=j For Q 2, we use the oracle inequality (17) as well as (29). Q 2 J,1 j=j 0 k2k(t 0 ;j) J,1 Nk k 1 2 j=2 C2,j(1=2+) Cn, : (30) 2 j=2 k k 1 (E(^ jk, jk ) 2 ) 1=2 C 2 j=2 [(2,j(1+2) +2,j n,2 ) ^ (n,1 log n)+n,2 (log n) 1=2 ] 1=2 j=j 0 = C(log n=n) =(1+2) : (31) Theorem 6 is proved by putting together the three terms, Q 1, Q 2 and Q 3. To prove Theorem 3, we use the same bounds for Q 1 and Q 3, and use the oracle inequality (16) to bound the term Q 2. Acknowledgment Part of this work was carried out while Bernard Silverman was a Fellow at the Center for Advanced Study in the Behavioral Sciences, Stanford, supported by National Science Foundation Grant number SBR He is also grateful for the nancial support of the British Engineering and Physical Sciences Research Council

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20 Doppler HeaviSine Bumps Blocks Spikes Blip Corner Wave Figure 5: Test functions. Doppler, HeaviSine, Bumps and Blocks are from Donoho and Johnstone (1995a). Blip and Wave are from Marron et al. (1995). The test functions are normalized so that every function has standard deviation 10. Formulae for Spikes and Corner are given in Cai (1999a). 20

21 Table 1: Mean Squared Error From 60 Replications (RSNR=3) n NeighCoe NeighBlock SureShrink TI-denoising VisuShrink Doppler HeaviSine Bumps Blocks Spikes Blip Corner Wave