Wavelet-based estimation of power densities of size-biased data

The Annals of Statistics, 1996

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 coecients. 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 < 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 > 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 to attain exactly optimal rates of convergence for quadratic error (p 0 = 2).

Journal of the Royal Statistical Society: Series D (The Statistician), 2000

In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this paper gives a relatively accessible introduction to standard wavelet analysis and provides a review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be effective, rather than to researchers who are already familiar with the ®eld. Given that objective, we do not emphasize mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in greater detail and generality if required. The paper ®rst establishes some necessary basic mathematical background and terminology relating to wavelets. It then reviews the more well-established applications of wavelets in statistics including their use in nonparametric regression, density estimation, inverse problems, changepoint problems and in some specialized aspects of time series analysis. Possible extensions to the uses of wavelets in statistics are then considered. The paper concludes with a brief reference to readily available software packages for wavelet analysis.

2012

The density estimation problem under bias and multiplicative censoring is considered. Adopting the wavelet approach, we construct a linear nonadaptive estimator and a nonlinear adaptive estimator. The adaptive one belongs to the family of the hard thresholding estimators. We evaluate their performances by determining upper bounds of the mean integrated squared error over a wide range of functions. Sharp upper bounds are obtained.

Journal of statistical theory and practice, 2018

This paper considers estimation of the density function in the context of biased data using thresholded wavelet estimator. We adapt the method of estimating the square root of the density function as advocated in Pinheiro and Vidakovic (Comput Stat Data Anal 25:399-415, 1997) in the iid case. The density estimator is obtained by squaring the resulting estimator that as a result guarantees non-negativity. It is shown that the resulting estimator achieves the optimal 2 convergence in Besov spaces B s pq (M) for p ≥ 2 whereas for p ∈ [1, 2) there is a logarithmic penalty attached to the optimal order. Finally, a simulation study shows that the resulting estimator may be favored over the usual wavelet estimator, with a proper choice of the preliminary estimator of the biased density.

1997

In these notes we present the main uses of wavelets in statistics. Advances in the area are occuring rapidly, with applications in several elds. There are several books about the mathematical aspects of wavelets, like those by Daubechies(1992) and Chui(1992), and others in areas like signal processing and image compression. The content of these notes borrows heavily from recent works, most of them referenced in the bibliography section. We decided to not include the proofs of the theorems, which the interested reader may nd in the original papers. I would like to thank the organizers of the Third International Conference on Statistical Data Analysis Based on the L 1 ?Norm and Related Methods for the invitation to give a tutorial on wavelets. This gave me the opportunity to prepare this text. Comments and suggestions are welcome and can be sent to pam@ime.usp.br. I would like to thank Chang Chiann for several conversations and David Brillinger for the continuous support. A version in portuguese was presented as a short course in the Seventh Time Series and Econometrics School, in Canela, RS, Brazil and is part of a book to be published by the

2001

Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatially-variable objects. We discuss in detail wavelet methods in nonparametric regression, where the data are modelled as observations of a signal contaminated with additive Gaussian noise, and provide an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimators developed to denoise such data. These estimators arise from a wide range of classical and empirical Bayes methods treating either individual or blocks of wavelet coefficients. We compare various estimators in an extensive simulation study on a variety of sample sizes, test functions, signal-to-noise ratios and wavelet filters. Because there is no single criterion that can adequately summarise the behaviour of an estimator, we use various criteria to measure performance in finite sample situations. Insight into the performance of these estimators is obtained from graphical outputs and numerical tables. In order to provide some hints of how these estimators should be used to analyse real data sets, a detailed practical step-by-step illustration of a wavelet denoising analysis on electrical consumption is provided. Matlab codes are provided so that all figures and tables in this paper can be reproduced.

Quantitative evidence is presented in order to study the performance of a wavelet-based, second order estimator of relevance to mul-tiscaling phenomena such as telecommunications traffic. Special attention is given to the behavior of the confidence intervals of the scaling exponent under Gaussian assumptions with synthetic long-range dependent trajectories, allowing for a straightforward contrast with the theoretical results presented in the bibliography. The bias-variance tradeoff decision behind the choice of the onset of the scaling region is also re-visited, along with its influence on the confidence of the intervals of the exponent.

IEEE Transactions on Information Theory, 2002

Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are studied extensively. These estimators mainly include the (bi)orthogonal wavelet estimators and the Wavelet Transform Modulus Maxima (WTMM) estimator. This study focuses both on short and long time-series. In the framework of Fractional Auto-Regressive Integrated Moving Average (FARIMA) processes, we advocate the use of approximately adapted wavelet estimators. For these "ideal" processes, the scaling behavior actually extends down to the smallest scale, i.e., the sampling period of the time series, if an adapted decomposition is used. But in practical situations, there generally exists a cut-off scale below which the scaling behavior no longer holds. We test the robustness of the set of wavelet-based estimators with respect to that cut-off scale as well as to the specific density of the underlying law of the process. In all situations the WTMM estimator is shown to be the best or among the best estimators in terms of the mean square error. We also compare the wavelet estimators with the Detrended Fluctuation Analysis (DFA) estimator which was recently proved to be among the best estimators which are not wavelet-based estimators. The WTMM estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior.

Statistical Methodology, 2014

We consider the estimation of a two dimensional continuous-discrete density function with applications to competing risks. We construct two new wavelet estimators (non-adaptive and adaptive) for the joint density function taking into account this special continuous-discrete structure. The rates of convergence of the proposed estimators are established under the L 2 risk over Besov balls. Our main result proves that our adaptive wavelet estimator (based on hard thresholding) attains a sharp rate of convergence. A simulation study illustrates the usefulness of the proposed estimators.

2000

In this paper we describe theoretical properties of wavelet-based random densities and give algorithms for their generation. We exhibit random densities subject to some standard constraints: smoothness, modality, and skewness. We also give a relevant example of use of random densities.