A Smirnov–Bickel–Rosenblatt Theorem for Compactly-Supported Wavelets

Abstract

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, . . . , 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases.

Adaptive Confidence Bands for Nonparametric Regression Functions

“…Therefore, the convergence is uniform for all the function f that we are interested in. Following the lines of the proof in Bull (2011), one sees that the convergence is also uniform for all sequences $false\{j_n^{\prime }false\}$ such that $j_n^{\prime }\ge j_n$ for all n .…”

“…Giné and Nickl (2010) and Giné et al (2011) verified that the unique maximum assumption on ${\sigma }_{\psi }^{2}false(tfalse)$ is satisfied by spline bases, and Bull (2011a) showed numerically that it is also satisfied by the Daubechies and Symmlet classes. Thus, assumption (W) is satisfied by the Daubechies and Symmlet bases, whose mother wavelets are compactly supported.…”

“…We use a result in Bull (2011a) to bound the stochastic error, which builds on the extreme value theory for cyclostationary Gaussian processes (Piterbarg and Seleznjev, 1994; Husler, 1999). It provides an extension of Theorem 2 of Giné and Nickl (2010), both of which improve earlier works of Smirnov (1950) and Bickel and Rosenblatt (1973).…”

Anti-concentration and honest, adaptive confidence bands

“…In the next subsection, we will show that as long the bias ∆ n,f (l n ) can be controlled, our theorem applies whenf n (·) is defined using either convolution or projection kernels under mild conditions, and, as far as wavelet projection kernels are concerned, it covers estimators based on compactly supported wavelets, Battle-Lemarié wavelets of any order as well as other non-wavelet projection kernels such as those based on Legendre polynomials and Fourier series. When L n is a singleton, the SBR condition for compactly supported wavelets was obtained in [5] under certain assumptions that can be verified numerically for any given wavelet, for Battle-Lemarié wavelets of degree up-to 4 in [19], and for Battle-Lemarié wavelets of degree higher than 4 in [15]. To the best of our knowledge, the SBR condition for non-wavelet projection kernel functions (such as those based on Legendre polynomials and Fourier series) has not been obtained in the literature.…”

“…If, for some normalizing constants A n and B n , A n ( G n,f Vn − B n ) has a continuous limit distribution, the validity of the confidence band would follow via the continuity of the limit distribution. For the density estimation problem, if L n is a singleton, i.e., the smoothing parameter is chosen deterministically, the existence of such a continuous limit distribution, which is typically a Gumbel distribution, has been established for convolution kernel density estimators and some wavelet projection kernel density estimators [see 37,1,16,19,4,5,15]. We refer to the existence of the limit distribution as the Smirnov-Bickel-Rosenblatt (SBR) condition.…”