Above are papers related to the following paper: [Paper_ID] Paper #884094. 	[Title] ESTIMATING THE SQUARE ROOT OF A DENSITY VIA COMPACTLY SUPPORTED WAVELETS [Abstract] A large body of nonparametric statistical literature is devoted to density estimation. Overviews are given in Silverman (1986) and Izenman (1991). This paper addresses the problem of univariate density estimation in a novel way. Our approach falls in the class of so called projection estimators, introduced by Cencov (1962). The orthonor-mal basis used is a basis of compactly supported wavelets from Daubechies' family. Kerkyacharian and Picard (1992, 1993), Donoho et al. (1996), and Delyon and Judit-sky (1993), among others, applied wavelets in density estimation. The local nature of wavelet functions makes the wavelet estimator superior to projection estimators that use classical orthonormal bases (Fourier, Hermite, etc.) Instead of estimating the unknown density directly, we estimate the square root of the density, which enables us to control the positiveness and the L 1 norm of the density estimate. However, in that approach one needs a pre-estimator of the density to calculate sample wavelet coefficients. We describe VISUSTOP, a data-driven procedure for determining the maximum number of levels in the wavelet density estimator. Coefficients in the selected levels are thresholded to make the estimator parsimonious. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #31489. 	[Title] Minimax Estimation via Wavelet Shrinkage  a pleasure to acknowledge friendly conversations with Gerard Kerkyacharian, [Abstract] We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel- and Besov-type smoothness constraints, and asymptotically minimax over Besov bodies with p q. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with p &lt; 2, so our method can significantly outperform every linear method (kernel, smoothing spline, sieve, : : : ) in a minimax sense. Variants of our method based on simple threshold nonlinearities are nearly minimax. Our method possesses the interpretation of spatial adaptivity: it reconstructs using a kernel which may vary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper discuss practical implementation, spatial adaptation properties and applications to inverse problems. Acknowledgements. This work was completed while the first author was on leave from U.C. Berkeley, where his research was supported by NSF DMS 88-10192, by NASA Contract NCA2-488, and by a grant from ATT Foundation. The second author was supported in part by NSF grants DMS 84-51750, 86-00235, and NIH PHS grant GM21215-12. Supersedes an earlier version, titled "Wavelets and Optimal Function Estimation", dated November 10, 1990, and issued as Technical reports by the Departments of Statistics at both Stanford and at U.C. Berkeley. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #633081. 	[Title] Minimax Risk over l p -Balls for l q -error Key Words. Minimax Decision Theory. [Abstract] Consider estimating the mean vector from data N n (; 2 I) with l q norm loss, q 1, when is known to lie in an n-dimensional l p ball, p 2 (0; 1). For large n, the ratio of minimax linear risk to minimax risk can be arbitrarily large if p &lt; q. Obvious exceptions aside, the limiting ratio equals 1 only if p = q = 2. Our arguments are mostly indirect, involving a reduction to a univariate Bayes minimax problem. When p &lt; q, simple non-linear co-ordinatewise threshold rules are asymptotically minimax at small signal-to-noise ratios, and within a bounded factor of asymptotic minimaxity in general. Our results are basic to a theory of estimation in Besov spaces Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #40583. 	[Title] Wavelet Thresholding via a Bayesian Approach [Abstract] We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in non-parametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might be hard to elicit; with this in mind, we propose a standard choise of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set collected in an anaesthesiological study. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #9581. 	[Title] DE-NOISING BY reconstruction f n is defined in the wavelet domain by translating all the [Abstract] p n. We prove two results about that estimator. [Smooth]: With high probability ^ f fl n is at least as smooth as f , in any of a wide variety of smoothness measures. [Adapt]: The estimator comes nearly as close in mean square to f as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes. These two properties are unprecedented in several ways. Our proof of these results develops new facts about abstract statistical inference and its connection with Acknowledgements. These results were described at the Symposium on Wavelet Theory, held in connection with the Shanks Lectures at Van-derbilt University, April 3-4 1992. The author would like to thank Professor L.L. Schumaker for hospitality at the conference, and R.A. DeVore, Iain Johnstone, Gerard Kerkyacharian, Bradley Lucier, A.S. Nemirovskii, Ingram Olkin, and Dominique Picard for interesting discussions and correspondence on related topics. The author is also at the University of California, Berkeley Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #633585. 	[Title] Choice of Thresholds for Wavelet Shrinkage Estimate of the Spectrum fff j g are level-dependent [Abstract] We study the problem of estimating the log spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds t j;n depending on sample size n, wavelet basis and resolution level j. At fine resolution levels (j = 1; 2; :::), we propose The purpose of this thresholding level is to make the reconstructed log-spectrum as nearly noise-free as possible. In addition to being pleasant from a visual point of view, the noise-free character leads to attractive theoretical properties over a wide range of smoothness assumptions. Previous proposals set much smaller thresholds and did not enjoy these properties. t j;n = ff j log n; Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #633721. 	[Title] The Stationary Wavelet Transform and some Statistical Applications [Abstract] Wavelets are of wide potential use in statistical contexts. The basics of the discrete wavelet transform are reviewed using a filter notation that is useful subsequently in the paper. A `stationary wavelet transform', where the coefficient sequences are not decimated at each stage, is described. Two different approaches to the construction of an inverse of the stationary wavelet transform are set out. The application of the stationary wavelet transform as an exploratory statistical method is discussed, together with its potential use in nonparametric regression. A method of local spectral density estimation is developed. This involves extensions to the wavelet context of standard time series ideas such as the periodogram and spectrum. The technique is illustrated by its application to data sets from astronomy and veterinary anatomy. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #632796. 	[Title] Density estimation by wavelet thresholding [Abstract] Density estimation is a commonly used test case for non-parametric 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 s;p;q and for a range of global L 0 p error measures, 1 p 0 &lt; 1. A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a range of spaces and error measures. In particular, when p 0 &gt; p, some form of non-linearity 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 Acknowledgements: We thank Alexandr Sakhanenko for helpful discussions and references to his work on Berry Esseen theorems used in Section 5. This work was supported in part by NSF DMS 92-09130. The second author would like to thank Universite de Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #40605. 	[Title] Nonlinear wavelet shrinkage with Bayes rules and Bayes factors 1 [Abstract] Wavelet shrinkage,the method proposed by seminal work of Donohoand Johnstone is a disarmingly simple and efficient way of de-noising data. Shrinking wavelet coefficients was proposed from several optimality criteria. The most notable are the asymptotic minimax and cross-validation criteria. In this paper a wavelet shrinkage by imposing natural properties of Bayesian models on data is proposed. The performance of methods are tested on standard Donoho-Johnstone test functions. Key Words and Phrases: Wavelets, Discrete Wavelet Transform, Thresholding, Bayes Model. 1991 AMS Subject Classification: 42A06, 62G07. Please summarise all the given information.

Paper #884094 has the title `ESTIMATING THE SQUARE ROOT OF A DENSITY VIA COMPACTLY SUPPORTED WAVELETS  
` and the abstract `A large body of nonparametric statistical literature is devoted to density estimation. Overviews are given in Silverman (1986) and Izenman (1991). This paper addresses the problem of univariate density estimation in a novel way. Our approach falls in the class of so called projection estimators, introduced by Cencov (1962). The orthonor-mal basis used is a basis of compactly supported wavelets from Daubechies' family. Kerkyacharian and Picard (1992, 1993), Donoho et al. (1996), and Delyon and Judit-sky (1993), among others, applied wavelets in density estimation. The local nature of wavelet functions makes the wavelet estimator superior to projection estimators that use classical orthonormal bases (Fourier, Hermite, etc.) Instead of estimating the unknown density directly, we estimate the square root of the density, which enables us to control the positiveness and the L 1 norm of the density estimate. However, in that approach one needs a pre-estimator of the density to calculate sample wavelet coefficients. We describe VISUSTOP, a data-driven procedure for determining the maximum number of levels in the wavelet density estimator. Coefficients in the selected levels are thresholded to make the estimator parsimonious. 
`.
Question: Which category should Paper #884094 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #884094 has the title `ESTIMATING THE SQUARE ROOT OF A DENSITY VIA COMPACTLY SUPPORTED WAVELETS  
` and the abstract `A large body of nonparametric statistical literature is devoted to density estimation. Overviews are given in Silverman (1986) and Izenman (1991). This paper addresses the problem of univariate density estimation in a novel way. Our approach falls in the class of so called projection estimators, introduced by Cencov (1962). The orthonor-mal basis used is a basis of compactly supported wavelets from Daubechies' family. Kerkyacharian and Picard (1992, 1993), Donoho et al. (1996), and Delyon and Judit-sky (1993), among others, applied wavelets in density estimation. The local nature of wavelet functions makes the wavelet estimator superior to projection estimators that use classical orthonormal bases (Fourier, Hermite, etc.) Instead of estimating the unknown density directly, we estimate the square root of the density, which enables us to control the positiveness and the L 1 norm of the density estimate. However, in that approach one needs a pre-estimator of the density to calculate sample wavelet coefficients. We describe VISUSTOP, a data-driven procedure for determining the maximum number of levels in the wavelet density estimator. Coefficients in the selected levels are thresholded to make the estimator parsimonious. 
`.
Question: Which category should Paper #884094 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Probabilistic_Methods