Nonparametric estimation of a two dimensional continuous–discrete density function by wavelets

We consider the estimation of a two dimensional continuous–discrete density function. A new methodology based on wavelets is proposed. We construct a linear wavelet estimator and a non-linear wavelet estimator based on a term-by-term thresholding. Their rates of convergence are established under the mean integrated squared error over Besov balls. In particular, we prove that our adaptive wavelet estimator attains a fast rate of convergence. A simulation study illustrates the usefulness of the proposed estimators.     

Nonparametric Regression with Sample Design Following a Random Process

Nonparametric regression is considered where the sample point placement is not fixed and equispaced, but generated by a random process with rate n. Conditions are found for the random processes that result in optimal rates of convergence for nonparametric regression when using a block thresholded wavelet estimator. Previous results on nonparametric regression via wavelets on both fixed and random sample point placement are shown to be special cases of the general result given here. The estimator is adaptive over a large range of Hölder function spaces and the convergence rate exhibited is an improvement over term-by-term wavelet estimators. Threshold selection is implemented in a data-adaptive fashion, rather than using a fixed threshold as is usually done in block thresholding. This estimator, BlockSure, is compared against fixed-threshold block estimators and the more traditional term-by-term threshold wavelet estimators on several random design schemes via simulations.     

A trimmed translation-invariant denoising estimator

A popular wavelet method for estimating jumps in functions is through the use of the translation-invariant (TI) estimator. The TI estimator addresses a particular problem, the susceptibility of the wavelet estimates to the location of the features in a function with respect to the support of the wavelet basis functions. The TI estimator reduces this reliance by cycling the data through a set of shifts, thus changing the relation between the wavelet support and the jump location. However, a drawback of the TI estimator is that it includes every shifted analysis in the reconstruction, even those that may reduce, rather than improve, the effectiveness of the method. In this paper, we propose a method that modifies the TI estimator to improve the jump reconstruction in terms of the mean squared errors of the reconstructions and visual performance. Information from the set of shifted data sets is used to mimic the performance of an oracle which knows exactly which are the best TI shifts to retain in the reconstruction. The TI estimate is a special case of the proposed method. A simulation study comparing this proposed method to the existing wavelet estimators and the oracle is provided.     

Wavelet estimation of density for censored data with censoring indicator missing at random

In this paper, we define the nonlinear wavelet estimator of density for the right censoring model with the censoring indicator missing at random (MAR), and develop its asymptotic expression for mean integrated squared error (MISE). Unlike for kernel estimator, the MISE expression of the estimator is not affected by the presence of discontinuities in the curve. Meanwhile, asymptotic normality of the estimator is established. The proposed estimator can reduce to the estimator defined by Li [Non-linear wavelet-based density estimators under random censorship. J Statist Plann Inference. 2003;117(1):35–58] when the censoring indicator MAR does not occur and a bandwidth in non-parametric estimation is close to zero. Also, we define another two nonlinear wavelet estimators of the density. A simulation is done to show the performance of the three proposed estimators.     

A non linear wavelet based estimator for long memory processes

Two wavelet based estimators are considered in this paper for the two parameters that characterize long range dependence processes. The first one is linear and is based on the statistical properties of the coefficients of a discrete wavelet transform of long range dependence processes. The estimator consists in measuring the slope (related to the long memory parameter) and the intercept (related to the variance of the process) of a linear regression after a discrete wavelet transform is performed (Veitch and Abry, 1999). In this paper its properties are reviewed, and analytic evidence is produced that the linear estimator is applicable only when the second parameter is unknown. To overcome this limitation a non linear wavelet based estimator - that takes into account that the intercept depends on the long memory parameter - is proposed here for the cases in which the second parameter is known or the only parameter of interest is the long memory parameter. Under the same hypothesis assumed for the linear estimator, the non linear estimator is shown to be asymptotically more efficient for the long memory parameter. Numerical simulations show that, even for small data sets, the bias is very small and the variance close to optimal. An application to ATM based Internet traffic is presented.Financial support from the Italian Ministry of University and Scientific Research (MIUR), also in the context of the COFIN 2002 ALINWEB (Algorithms for the Internet and the Web) Project, is gratefully acknowledged.     

Nonparametric regression estimates with censored data based on block thresholding method

Here we consider wavelet-based identification and estimation of a censored nonparametric regression model via block thresholding methods and investigate their asymptotic convergence rates. We show that these estimators, based on block thresholding of empirical wavelet coefficients, achieve optimal convergence rates over a large range of Besov function classes, and in particular enjoy those rates without the extraneous logarithmic penalties that are usually suffered by term-by-term thresholding methods. This work is extension of results in Li et al. (2008). The performance of proposed estimator is investigated by a numerical study.     

Wavelet estimation in time-varying coefficient time series models with measurement errors

The article studies a time-varying coefficient time series model in which some of the covariates are measured with additive errors. In order to overcome the bias of estimator of the coefficient functions when measurement errors are ignored, we propose a modified least squares estimator based on wavelet procedures. The advantage of the wavelet method is to avoid the restrictive smoothness requirement for varying-coefficient functions of the traditional smoothing approaches, such as kernel and local polynomial methods. The asymptotic properties of the proposed wavelet estimators are established under the α-mixing conditions and without specifying the error distribution. These results can be used to make asymptotically valid statistical inference.     

Wavelets for Nonparametric Stochastic Regression with Mixing Stochastic Process

We propose a wavelet based stochastic regression function estimator for the estimation of the regression function for a sequence of mixing stochastic process with a common one-dimensional probability density function. Some asymptotic properties of the proposed estimator are investigated. It is found that the estimators have similar properties to their counterparts studied earlier in literature.     

Wavelet Threshold Estimators for Data with Correlated Noise

Wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level-dependent soft threshold to the coefficients in the wavelet transform. A variety of threshold choices is proposed, including one based on an unbiased estimate of mean-squared error. The practical performance of the method is demonstrated on examples, including data from a neurophysiological context. The theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable `bench-mark', that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity, and more generally is obtained by the use of an `oracle' that provides information that is not actually available in the data. It is shown that the level-dependent threshold estimator performs well relative to the bench-mark risk, and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator. The wavelet domain structure of both short- and long-range dependent noise is considered, and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes, adapting automatically to the regularity properties of the underlying model. The proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem.     

Adaptive wavelet estimation of a function from an m-dependent process with possibly unbounded m

The estimation of a multivariate function from a stationary m-dependent process is investigated, with a special focus on the case where m is large or unbounded. We develop an adaptive estimator based on wavelet methods. Under flexible assumptions on the nonparametric model, we prove the good performances of our estimator by determining sharp rates of convergence under two kinds of errors: the pointwise mean squared error and the mean integrated squared error. We illustrate our theoretical result by considering the multivariate density estimation problem, the derivatives density estimation problem, the density estimation problem in a GARCH-type model and the multivariate regression function estimation problem. The performance of proposed estimator has been shown by a numerical study for a simulated and real data sets.     

A new Bayesian wavelet thresholding estimator of nonparametric regression

The methods of estimation of nonparametric regression function are quite common in statistical application. In this paper, the new Bayesian wavelet thresholding estimation is considered. The new mixture prior distributions for the estimation of nonparametric regression function by applying wavelet transformation are investigated. The reversible jump algorithm to obtain the appropriate prior distributions and value of thresholding is used. The performance of the proposed estimator is assessed with simulated data from well-known test functions by comparing the convergence rate of the proposed estimator with respect to another by evaluating the average mean square error and standard deviations. Finally by applying the developed method, density function of galaxy data is estimated.     

On wavelet analysis of the nth order fractional Brownian motion

In this paper, we investigate the use of wavelet techniques in the study of the nth order fractional Brownian motion (n-fBm). First, we exploit the continuous wavelet transform??s capabilities in derivative calculation to construct a two-step estimator of the scaling exponent of the n-fBm process. We show, via simulation, that the proposed method improves the estimation performance of the n-fBm signals contaminated by large-scale noise. Second, we analyze the statistical properties of the n-fBm process in the time-scale plan. We demonstrate that, for a convenient choice of the wavelet basis, the discrete wavelet detail coefficients of the n-fBm process are stationary at each resolution level whereas their variance exhibits a power-law behavior. Using the latter property, we discuss a weighted least squares regression based-estimator for this class of stochastic process. Experiments carried out on simulated and real-world datasets prove the relevance of the proposed method.     

Simultaneous Wavelet Deconvolution in Periodic Setting

Abstract.  The paper proposes a method of deconvolution in a periodic setting which combines two important ideas, the fast wavelet and Fourier transform-based estimation procedure of Johnstone et al . [ J. Roy. Statist. Soc. Ser. B 66 (2004) 547] and the multichannel system technique proposed by Casey and Walnut [ SIAM Rev . 36 (1994) 537]. An unknown function is estimated by a wavelet series where the empirical wavelet coefficients are filtered in an adapting non-linear fashion. It is shown theoretically that the estimator achieves optimal convergence rate in a wide range of Besov spaces. The procedure allows to reduce the ill-posedness of the problem especially in the case of non-smooth blurring functions such as boxcar functions: it is proved that additions of extra channels improve convergence rate of the estimator. Theoretical study is supplemented by an extensive set of small-sample simulation experiments demonstrating high-quality performance of the proposed method.     

Wavelet Estimator in Nonparametric Regression Model with Dependent Error’s Structure

In this article, we consider a nonparametric regression model with replicated observations based on the dependent error’s structure, for exhibiting dependence among the units. The wavelet procedures are developed to estimate the regression function. The moment consistency, the strong consistency, strong convergence rate and asymptotic normality of wavelet estimator are established under suitable conditions. A simulation study is undertaken to assess the finite sample performance of the proposed method.     

Dilation-Invariant Wavelet Estimation

A wavelet method is proposed that reduces function estimation error and provides smooth reconstructions, while still estimating jumps in the function well. It is based on analyzing multiple dilated versions of the sampled function. In simulation studies, the estimator exhibits low mean squared errors without sacrificing smoothness or jump detection ability when compared to other wavelet methods.     

Wavelet-Based Density Estimation in a Heteroscedastic Convolution Model

We consider a heteroscedastic convolution density model under the “ordinary smooth assumption.” We introduce a new adaptive wavelet estimator based on term-by-term hard thresholding rule. Its asymptotic properties are explored via the minimax approach under the mean integrated squared error over Besov balls. We prove that our estimator attains near optimal rates of convergence (lower bounds are determined). Simulation results are reported to support our theoretical findings.     

A Non‐Parametric Estimator of the Spectral Density of a Continuous‐Time Gaussian Process Observed at Random Times

Abstract. In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a non‐parametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between successive observations. In the case of stationary Gaussian processes, one can compare this estimator with estimators based on the empirical periodogram. Both estimators reach the same optimal rate of convergence, but the estimator based on wavelet analysis converges for a different class of random times. Simulation examples and an application to biological data are also provided.     

Performance of coefficient of variation estimators in ranked set sampling

In this paper, we propose and evaluate the performance of different parametric and nonparametric estimators for the population coefficient of variation considering Ranked Set Sampling (RSS) under normal distribution. The performance of the proposed estimators was assessed based on the bias and relative efficiency provided by a Monte Carlo simulation study. An application in anthropometric measurements data from a human population is also presented. The results showed that the proposed estimators via RSS present an expressively lower mean squared error when compared to the usual estimator, obtained via Simple Random Sampling. Also, it was verified the superiority of the maximum likelihood estimator, given the necessary assumptions of normality and perfect ranking are met.     

Robust estimation for boundary correction in wavelet regression

For boundary problems present in wavelet regression, two common methods are usually considered: polynomial wavelet regression (PWR) and hybrid local polynomial wavelet regression (LPWR). Normality assumption played a key role for making such choices for the order of the low-order polynomial, the wavelet thresholding value and other calculations involved in LPWR. However, in practice, the normality assumption may not be valid. In this paper, for PWR, we propose three automatic robust methods based on: MM-estimator, bootstrap and robust threshold procedure. For LPWR, the use of a robust local polynomial (RLP) estimator with a robust threshold procedure has been investigated. The proposed methods do not require any knowledge of noise distribution, are easy to implement and achieve high performances when only a small amount of data is in hand. A simulation study is conducted to assess the numerical performance of the proposed methods.     

Robust estimation and wavelet thresholding in partially linear models

This paper is concerned with a semiparametric partially linear regression model with unknown regression coefficients, an unknown nonparametric function for the non-linear component, and unobservable Gaussian distributed random errors. We present a wavelet thresholding based estimation procedure to estimate the components of the partial linear model by establishing a connection between an l ₁-penalty based wavelet estimator of the nonparametric component and Huber’s M-estimation of a standard linear model with outliers. Some general results on the large sample properties of the estimates of both the parametric and the nonparametric part of the model are established. Simulations are used to illustrate the general results and to compare the proposed methodology with other methods available in the recent literature.