Estimation of Variance Function in Heteroscedastic Regression Models by Generalized Coiflets

A wavelet approach is presented to estimate the variance function in heteroscedastic nonparametric regression model. The initial variance estimates are obtained as squared weighted sums of neighboring observations. The initial estimator of a smooth variance function is improved by means of wavelet smoothers under the situation that the samples at the dyadic points are not available. Since the traditional wavelet system for the variance function estimation is not appropriate in this situation, we demonstrate that the choice of the wavelet system is significant to have better performance. This is accomplished by choosing a suitable wavelet system known as the generalized coiflets. We conduct extensive simulations to evaluate finite sample performance of our method. We also illustrate our method using a real dataset.     

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.     

Full Bayesian wavelet inference with a nonparametric prior

In this paper, we introduce a new Bayesian nonparametric model for estimating an unknown function in the presence of Gaussian noise. The proposed model involves a mixture of a point mass and an arbitrary (nonparametric) symmetric and unimodal distribution for modeling wavelet coefficients. Posterior simulation uses slice sampling ideas and the consistency under the proposed model is discussed. In particular, the method is shown to be computationally competitive with some of best Empirical wavelet estimation methods.     

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.     

Detection and Estimation of Jump Points in Non parametric Regression Function with AR(1) Noise

In this paper, we propose a method based on wavelet analysis to detect and estimate jump points in non parametric regression function. This method is applied to AR(1) noise process under random design. First, the test statistics are constructed on the empirical wavelet coefficients. Then, under the null hypothesis, the critical values of test statistics are obtained. Under the alternative, the consistency of the test is proved. Afterward, the rate of convergence, the estimators of the number, and locations of change points are given theoretically. Finally, the excellent performance of our method is demonstrated through simulations using artificial and real datasets.     

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.     

Posterior probability intervals for wavelet thresholding

Summary. We use cumulants to derive Bayesian credible intervals for wavelet regression estimates. The first four cumulants of the posterior distribution of the estimates are expressed in terms of the observed data and integer powers of the mother wavelet functions. These powers are closely approximated by linear combinations of wavelet scaling functions at an appropriate finer scale. Hence, a suitable modification of the discrete wavelet transform allows the posterior cumulants to be found efficiently for any given data set. Johnson transformations then yield the credible intervals themselves. Simulations show that these intervals have good coverage rates, even when the underlying function is inhomogeneous, where standard methods fail. In the case where the curve is smooth, the performance of our intervals remains competitive with established nonparametric regression methods.     

Wavelet Analysis of Change Points in Nonparametric Hazard Rate Models Under Random Censorship

In this article, we consider detection and estimation of change points in nonparametric hazard rate models. Wavelet methods are utilized to develop a testing procedure for change points detection. The asymptotic properties of the test statistic are explored. When there exist change points in hazard function, we also propose estimators for the number, the locations, and the jump sizes of the change points. The asymptotic properties of these estimators are systematically derived. Some simulation examples are conducted to assess the finite sample performance of the proposed approach and to make comparisons with some existing methods. A real data analysis is provided to illustrate the new approach.     

Semiparametric method for detecting multiple change points model in financial time series

The problem of multiple change points has been discussed in these years on the background of financial shocks. In order to decrease the damage, it is worthy to find a more available model for the problem as precise as possible by the information from data set. This paper proposes the problem of detecting the change points by semiparametric test. The change points estimations are obtained by empirical likelihood method. Then some asymptotic results for multiple change points are obtained by loglikelihood ratio test and law of large numbers. Furthermore, the consistency of change points estimations is presented. Indeed, the method and steps to find the change points are derived. The simulation experiments prove that the semiparametric test is more efficient than nonparametric test. The diagnosis with simulation and the applications for multiple change points also illustrates the proposed model well.     

Wavelets in statistics: A review

The field of nonparametric function estimation has broadened its appeal in recent years with an array of new tools for statistical analysis. In particular, theoretical and applied research on the field of wavelets has had noticeable influence on statistical topics such as nonparametric regression, nonparametric density estimation, nonparametric discrimination and many other related topics. This is a survey article that attempts to synthetize a broad variety of work on wavelets in statistics and includes some recent developments in nonparametric curve estimation that have been omitted from review articles and books on the subject. After a short introduction to wavelet theory, wavelets are treated in the familiar context of estimation of «smooth» functions. Both «linear» and «nonlinear» wavelet estimation methods are discussed and cross-validation methods for choosing the smoothing parameters are addressed. Finally, some areas of related research are mentioned, such as hypothesis testing, model selection, hazard rate estimation for censored data, and nonparametric change-point problems. The closing section formulates some promising research directions relating to wavelets in statistics.     

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.     

Density and hazard rate estimation for right-censored data by using wavelet methods

This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right-censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers, and then the hazard rate function estimators are obtained by taking the ratio. We prove that the estimators have pointwise and global mean-square consistency, obtain the best possible asymptotic mean integrated squared error convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show that these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The first uses survival time data for patients with liver metastases from a colorectal primary tumour without other distant metastases. The second is concerned with times of unemployment for women and the wavelet estimate, through its flexibility, provides a new and interesting interpretation.     

Real nonparametric regression using complex wavelets

Summary.  Wavelet shrinkage is an effective nonparametric regression technique, especially when the underlying curve has irregular features such as spikes or discontinuities. The basic idea is simple: take the discrete wavelet transform of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; then invert the discrete wavelet transform to form an estimate of the true underlying curve. Various researchers have proposed increasingly sophisticated methods of doing this by using real-valued wavelets. Complex-valued wavelets exist but are rarely used. We propose two new complex-valued wavelet shrinkage techniques: one based on multiwavelet style shrinkage and the other using Bayesian methods. Extensive simulations show that our methods almost always give significantly more accurate estimates than methods based on real-valued wavelets. Further, our multiwavelet style shrinkage method is both simpler and dramatically faster than its competitors. To understand the excellent performance of this method we present a new risk bound on its hard thresholded coefficients.     

A robust and efficient estimation method for nonparametric models with jump points

Nonparametric models with jump points have been considered by many researchers. However, most existing methods based on least squares or likelihood are sensitive when there are outliers or the error distribution is heavy tailed. In this article, a local piecewise-modal method is proposed to estimate the regression function with jump points in nonparametric models, and a piecewise-modal EM algorithm is introduced to estimate the proposed estimator. Under some regular conditions, the large-sample theory is established for the proposed estimators. Several simulations are presented to evaluate the performances of the proposed method, which shows that the proposed estimator is more efficient than the local piecewise-polynomial regression estimator in the presence of outliers or heavy tail error distribution. What is more, the proposed procedure is asymptotically equivalent to the local piecewise-polynomial regression estimator under the assumption that the error distribution is a Gaussian distribution. The proposed method is further illustrated via the sea-level pressures.     

Robust signed-rank estimation and variable selection for semi-parametric additive partial linear models

A fully nonparametric model may not perform well or when the researcher wants to use a parametric model but the functional form with respect to a subset of the regressors or the density of the errors is not known. This becomes even more challenging when the data contain gross outliers or unusual observations. However, in practice the true covariates are not known in advance, nor is the smoothness of the functional form. A robust model selection approach through which we can choose the relevant covariates components and estimate the smoothing function may represent an appealing tool to the solution. A weighted signed-rank estimation and variable selection under the adaptive lasso for semi-parametric partial additive models is considered in this paper. B-spline is used to estimate the unknown additive nonparametric function. It is shown that despite using B-spline to estimate the unknown additive nonparametric function, the proposed estimator has an oracle property. The robustness of the weighted signed-rank approach for data with heavy-tail, contaminated errors, and data containing high-leverage points are validated via finite sample simulations. A practical application to an economic study is provided using an updated Canadian household gasoline consumption data.     

Nonparametric product partition models for multiple change-points analysis

ABSTRACT

We propose an extension of parametric product partition models. We name our proposal nonparametric product partition models because we associate a random measure instead of a parametric kernel to each set within a random partition. Our methodology does not impose any specific form on the marginal distribution of the observations, allowing us to detect shifts of behaviour even when dealing with heavy-tailed or skewed distributions. We propose a suitable loss function and find the partition of the data having minimum expected loss. We then apply our nonparametric procedure to multiple change-point analysis and compare it with PPMs and with other methodologies that have recently appeared in the literature. Also, in the context of missing data, we exploit the product partition structure in order to estimate the distribution function of each missing value, allowing us to detect change points using the loss function mentioned above. Finally, we present applications to financial as well as genetic data.     

Wavelet thresholding via a Bayesian approach

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion that is 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 relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov space parameters. Moreover, the relationship established 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 difficult to elicit; with this in mind, we propose a standard choice 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 that was collected in an anaesthesiological study.     

A fast algorithm for univariate log‐concave density estimation

A new fast algorithm for computing the nonparametric maximum likelihood estimate of a univariate log‐concave density is proposed and studied. It is an extension of the constrained Newton method for nonparametric mixture estimation. In each iteration, the newly extended algorithm includes, if necessary, new knots that are located via a special directional derivative function. The algorithm renews the changes of slope at all knots via a quadratically convergent method and removes the knots at which the changes of slope become zero. Theoretically, the characterisation of the nonparametric maximum likelihood estimate is studied and the algorithm is guaranteed to converge to the unique maximum likelihood estimate. Numerical studies show that it outperforms other algorithms that are available in the literature. Applications to some real‐world financial data are also given.     

Partially linear models and their applications to change point detection of chemical process data

In many chemical data sets, the amount of radiation absorbed (absorbance) is related to the concentration of the element in the sample by Lambert–Beer's law. However, this relation changes abruptly when the variable concentration reaches an unknown threshold level, the so-called change point. In the context of analytical chemistry, there are many methods that describe the relationship between absorbance and concentration, but none of them provide inferential procedures to detect change points. In this paper, we propose partially linear models with a change point separating the parametric and nonparametric components. The Schwarz information criterion is used to locate a change point. A back-fitting algorithm is presented to obtain parameter estimates and the penalized Fisher information matrix is obtained to calculate the standard errors of the parameter estimates. To examine the proposed method, we present a simulation study. Finally, we apply the method to data sets from the chemistry area. The partially linear models with a change point developed in this paper are useful supplements to other methods of absorbance–concentration analysis in chemical studies, for example, and in many other practical applications.     

Estimation of partially linear single-index additive hazards model with current status data

In this paper, we introduce a partially linear single-index additive hazards model with current status data. Both the unknown link function of the single-index term and the cumulative baseline hazard function are approximated by B-splines under a monotonicity constraint on the latter. The sieve method is applied to estimate the nonparametric and parametric components simultaneously. We show that, when the nonparametric link function is an exact B-spline, the resultant estimator of regression parameter vector is asymptotically normal and achieves the semiparametric information bound and the rate of convergence of the estimator for the cumulative baseline hazard function is optimal. Simulation studies are presented to examine the finite sample performance of the proposed estimation method. For illustration, we apply the method to a clinical dataset with current status outcome.