Adaptive confidence bands in the nonparametric fixed design regression model

In this note, we consider the problem of the existence of adaptive confidence bands in the fixed design regression model, adapting ideas in Hoffmann and Nickl [(2011), ‘On Adaptive Inference and Confidence Bands’, Annals of Statistics, 39, 2383–2409] to the present case. In the course of the proof, we show that sup-norm adaptive estimators exist as well in the regression setting.     

Optimal simultaneous confidence bands in simple linear regression

A simultaneous confidence band provides useful information on the plausible range of an unknown regression model. For simple linear regression models, the most frequently quoted bands in the statistical literature include the hyperbolic band and the three-segment bands. One interesting question is whether one can construct confidence bands better than the hyperbolic and three-segment bands. The optimality criteria for confidence bands include the average width criterion considered by Gafarian (1964) and Naiman (1984) among others, and the minimum area confidence set (MACS) criterion of Liu and Hayter (2007). In this paper, two families of exact 1−α$1-\alpha$ confidence bands, the inner-hyperbolic bands and the outer-hyperbolic bands, which include the hyperbolic and three-segment bands as special cases, are introduced in simple linear regression. Under the MACS criterion, the best confidence band within each family is found by numerical search and compared with the hyperbolic band, the best three-segment band and with each other. The methodologies are illustrated with a numerical example and the Matlab programs used are available upon request.     

Easy-to-construct confidence bands for comparing two simple linear regression lines

This paper shows how to construct confidence bands for the difference between two simple linear regression lines. These confidence bands provide directly the information on the magnitude of the difference between the regression lines over an interval of interest and, as a by-product, can be used as a formal test of the difference between the two regression lines. Various different shapes of confidence bands are illustrated, and particular attention is paid towards confidence bands whose construction only involves critical points from standard distributions so that they are consequently easy to construct.     

Slope modified confidence bands for a simple linear regression model

Considerable attention has been directed in the statistical literature towards the construction of confidence bands for a simple linear regression model. These confidence bands allow the experimenter to make inferences about the model over a particular region of interest. However, in practice an experimenter will usually first check the significance of the regression line before proceeding with any further inferences such as those provided by the confidence bands. From a theoretical point of view, this raises the question of what the conditional confidence level of the confidence bands might be, and from a practical point of view it is unsatisfactory if the confidence bands contain lines that are inconsistent with the directional decision on the slope. In this paper it is shown how confidence bands can be modified to alleviate these two problems.     

Adjusted Confidence Bands in Nonparametric Regression

Suppose we have {(x _i , y _i )} i = 1, 2,…, n, a sequence of independent observations. We wish to find approximate 1 ? α simultaneous confidence bands for the regression curve. Many previous confidence bands in the literature have practical difficulties. In this article, the local linear smoother is used to estimate the regression curve. The bias of the estimator is considered. Different methods of constructing confidence bands are discussed. Finally, a possible method incorporating logistic regression in an innovative way is proposed to construct the bands for random designs. Simulations are used to study the performance or properties of the methods. The procedure for constructing confidence bands is entirely data-driven. The advantage of the proposed method is that it is simple to use and can be applied to random designs. It can be considered as a practically useful and efficient method.     

A ray method of confidence band construction for multiple linear regression models

This paper addresses the problem of confidence band construction for a standard multiple linear regression model. A “ray” method of construction is developed which generalizes the method of Graybill and Bowden [1967. Linear segment confidence bands for simple linear regression models. J. Amer. Statist. Assoc. 62, 403–408] for a simple linear regression model to a multiple linear regression model. By choosing suitable directions for the rays this method requires only critical points from t-distributions so that the confidence bands are easy to construct. Both one-sided and two-sided confidence bands can be constructed using this method. An illustration of the new method is provided.     

On the Uniform Strong Consistency of Local Polynomial Regression Under Dependence Conditions

Abstract

In this article, nonparametric estimators of the regression function, and its derivatives, obtained by means of weighted local polynomial fitting are studied. Consider the fixed regression model where the error random variables are coming from a stationary stochastic process satisfying a mixing condition. Uniform strong consistency, along with rates, are established for these estimators. Furthermore, when the errors follow an AR(1) correlation structure, strong consistency properties are also derived for a modified version of the local polynomial estimators proposed by Vilar-Fernández and Francisco-Fernández (Vilar-Fernández, J. M., Francisco-Fernández, M. (2002 Vilar-Fernández, J. M. and Francisco-Fernández, M. 2002. Local polynomial regression smoothers with AR-error structure. TEST, 11(2): 439–464.  [Google Scholar]). Local polynomial regression smoothers with AR-error structure. TEST 11(2):439–464).     

Empirical likelihood confidence intervals for nonparametric functional data analysis

We consider the problem of constructing confidence intervals for nonparametric functional data analysis using empirical likelihood. In this doubly infinite-dimensional context, we demonstrate the Wilk's phenomenon and propose a bias-corrected construction that requires neither undersmoothing nor direct bias estimation. We also extend our results to partially linear regression models involving functional data. Our numerical results demonstrate improved performance of the empirical likelihood methods over normal approximation-based methods.     

Simultaneous confidence bands for nonlinear regression models

The maximization and minimization procedure for constructing confidence bands about general regression models is explained. Then, using an existing confidence region about the parameters of a nonlinear regression model and the maximization and minimization procedure, a generally conservative simultaneous confidence band is constructed about the model. Two examples are given, and some problems with the procedure are discussed     

Testing heteroscedasticity in nonparametric regression

The importance of being able to detect heteroscedasticity in regression is widely recognized because efficient inference for the regression function requires that heteroscedasticity is taken into account. In this paper a simple consistent test for heteroscedasticity is proposed in a nonparametric regression set-up. The test is based on an estimator for the best L ²-approximation of the variance function by a constant. Under mild assumptions asymptotic normality of the corresponding test statistic is established even under arbitrary fixed alternatives. Confidence intervals are obtained for a corresponding measure of heteroscedasticity. The finite sample performance and robustness of these procedures are investigated in a simulation study and Box-type corrections are suggested for small sample sizes.     

Bias-corrected random forests in regression

It is well known that random forests reduce the variance of the regression predictors compared to a single tree, while leaving the bias unchanged. In many situations, the dominating component in the risk turns out to be the squared bias, which leads to the necessity of bias correction. In this paper, random forests are used to estimate the regression function. Five different methods for estimating bias are proposed and discussed. Simulated and real data are used to study the performance of these methods. Our proposed methods are significantly effective in reducing bias in regression context.     

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.     

On difference-based variance estimation in nonparametric regression when the covariate is high dimensional

Summary.  We consider the problem of estimating the noise variance in homoscedastic nonparametric regression models. For low dimensional covariates t  ∈  R ^d ,  d =1, 2, difference-based estimators have been investigated in a series of papers. For a given length of such an estimator, difference schemes which minimize the asymptotic mean-squared error can be computed for d =1 and d =2. However, from numerical studies it is known that for finite sample sizes the performance of these estimators may be deficient owing to a large finite sample bias. We provide theoretical support for these findings. In particular, we show that with increasing dimension d this becomes more drastic. If d 4, these estimators even fail to be consistent. A different class of estimators is discussed which allow better control of the bias and remain consistent when d 4. These estimators are compared numerically with kernel-type estimators (which are asymptotically efficient), and some guidance is given about when their use becomes necessary.     

Detecting discontinuities in nonparametric regression curves and surfaces

The existence of a discontinuity in a regression function can be inferred by comparing regression estimates based on the data lying on different sides of a point of interest. This idea has been used in earlier research by Hall and Titterington (1992), Müller (1992) and later authors. The use of nonparametric regression allows this to be done without assuming linear or other parametric forms for the continuous part of the underlying regression function. The focus of the present paper is on assessing the evidence for the presence of a discontinuity within a regression function through examination of the standardised differences of ‘left’ and ‘right’ estimators at a variety of covariate values. The calculations for the test are carried out through distributional results on quadratic forms. A graphical method in the form of a reference band to highlight the sources of the evidence for discontinuities is proposed. The methods are also developed for the two covariate case where there are additional issues associated with the presence of a jump location curve. Methods for estimating this curve are also developed. All the techniques, for the one and two covariate situations, are illustrated through applications.     

Automatic bandwidth selection in robust nonparametric regression

Nonparametric regression techniques have been studied extensively in the literature in recent years due to their flexibility.In addition robust versions of these techniques have become popular and have been incorporated into some of the standard statistical analysis packages.With new techniques available comes the responsibility of using them properly and in appropriate situations. Often, as in the case presented here, model-fitting diagnostics, such as cross-validation statistics,are not available as tools to determine if the smoothing parameter value being used is preferable to some other arbitrarily chosen value.     

Adaptive lifting for nonparametric regression

Many wavelet shrinkage methods assume that the data are observed on an equally spaced grid of length of the form 2^J for some J. These methods require serious modification or preprocessed data to cope with irregularly spaced data. The lifting scheme is a recent mathematical innovation that obtains a multiscale analysis for irregularly spaced data. A key lifting component is the “predict” step where a prediction of a data point is made. The residual from the prediction is stored and can be thought of as a wavelet coefficient. This article exploits the flexibility of lifting by adaptively choosing the kind of prediction according to a criterion. In this way the smoothness of the underlying ‘wavelet’ can be adapted to the local properties of the function. Multiple observations at a point can readily be handled by lifting through a suitable choice of prediction. We adapt existing shrinkage rules to work with our adaptive lifting methods. We use simulation to demonstrate the improved sparsity of our techniques and improved regression performance when compared to both wavelet and non-wavelet methods suitable for irregular data. We also exhibit the benefits of our adaptive lifting on the real inductance plethysmography and motorcycle data.     

More efficient local polynomial regression with random-effects panel data models

We propose a modification on the local polynomial estimation procedure to account for the “within-subject” correlation presented in panel data. The proposed procedure is rather simple to compute and has a closed-form expression. We study the asymptotic bias and variance of the proposed procedure and show that it outperforms the working independence estimator uniformly up to the first order. Simulation study shows that the gains in efficiency with the proposed method in the presence of “within-subject” correlation can be significant in small samples. For illustration purposes, the procedure is applied to explore the impact of market concentration on airfare.