Estimation of the Spectral Density with Assigned Risk

It is well known that adaptive sequential nonparametric estimation of differentiable functions with assigned mean integrated squared error and minimax expected stopping time is impossible. In other words, no sequential estimator can compete with an oracle estimator that knows how many derivatives an estimated curve has. Differentiable functions are typical in probability density and regression models but not in spectral density models, where considered functions are typically smoother. This paper shows that for a large class of spectral densities, which includes spectral densities of classical autoregressive moving average processes, an adaptive minimax sequential estimation with assigned mean integrated squared error is possible. Furthermore, a two‐stage sequential procedure is proposed, which is minimax and adaptive to smoothness of an underlying spectral density.     

A Necessary Condition for the Strong Oracle Property

A problem of using a non‐convex penalty for sparse regression is that there are multiple local minima of the penalized sum of squared residuals, and it is not known which one is a good estimator. The aim of this paper is to give a guide to design a non‐convex penalty that has the strong oracle property. Here, the strong oracle property means that the oracle estimator is the unique local minimum of the objective function. We summarize three definitions of the oracle property – the global, weak and strong oracle properties. Then, we give sufficient conditions for the weak oracle property, which means that the oracle estimator becomes a local minimum. We give an example of non‐convex penalties that possess the weak oracle property but not the strong oracle property. Finally, we give a necessary condition for the strong oracle property.     

Modal non‐linear regression in the presence of Laplace measurement error

In this paper, we propose a robust estimation procedure for a class of non‐linear regression models when the covariates are contaminated with Laplace measurement error, aiming at constructing an estimation procedure for the regression parameters which are less affected by the possible outliers, and heavy‐tailed underlying distribution, as well as reducing the bias introduced by the measurement error. Starting with the modal regression procedure developed for the measurement error‐free case, a non‐trivial modification is made so that the modified version can effectively correct the potential bias caused by measurement error. Large sample properties of the proposed estimate, such as the convergence rate and the asymptotic normality, are thoroughly investigated. A simulation study and real data application are conducted to illustrate the satisfying finite sample performance of the proposed estimation procedure.     

Kernel Density-Based Linear Regression Estimate

For linear regression models with non normally distributed errors, the least squares estimate (LSE) will lose some efficiency compared to the maximum likelihood estimate (MLE). In this article, we propose a kernel density-based regression estimate (KDRE) that is adaptive to the unknown error distribution. The key idea is to approximate the likelihood function by using a nonparametric kernel density estimate of the error density based on some initial parameter estimate. The proposed estimate is shown to be asymptotically as efficient as the oracle MLE which assumes the error density were known. In addition, we propose an EM type algorithm to maximize the estimated likelihood function and show that the KDRE can be considered as an iterated weighted least squares estimate, which provides us some insights on the adaptiveness of KDRE to the unknown error distribution. Our Monte Carlo simulation studies show that, while comparable to the traditional LSE for normal errors, the proposed estimation procedure can have substantial efficiency gain for non normal errors. Moreover, the efficiency gain can be achieved even for a small sample size.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

Small area estimation via heteroscedastic nested‐error regression

We show that the maximum likelihood estimators (MLEs) of the fixed effects and within‐cluster correlation are consistent in a heteroscedastic nested‐error regression (HNER) model with completely unknown within‐cluster variances under mild conditions. The result implies that the empirical best linear unbiased prediction (EBLUP) method for small area estimation is valid in such a case. We also show that ignoring the heteroscedasticity can lead to inconsistent estimation of the within‐cluster correlation and inferior predictive performance. A jackknife measure of uncertainty for the EBLUP is developed under the HNER model. Simulation studies are carried out to investigate the finite‐sample performance of the EBLUP and MLE under the HNER model, with comparisons to those under the nested‐error regression model in various situations, as well as that of the jackknife measure of uncertainty. The well‐known Iowa crops data is used for illustration. The Canadian Journal of Statistics 40: 588–603; 2012 © 2012 Statistical Society of Canada     

Estimating nonlinear additive models with nonstationarities and correlated errors

In this paper, we study a nonparametric additive regression model suitable for a wide range of time series applications. Our model includes a periodic component, a deterministic time trend, various component functions of stochastic explanatory variables, and an AR(p) error process that accounts for serial correlation in the regression error. We propose an estimation procedure for the nonparametric component functions and the parameters of the error process based on smooth backfitting and quasimaximum likelihood methods. Our theory establishes convergence rates and the asymptotic normality of our estimators. Moreover, we are able to derive an oracle‐type result for the estimators of the AR parameters: Under fairly mild conditions, the limiting distribution of our parameter estimators is the same as when the nonparametric component functions are known. Finally, we illustrate our estimation procedure by applying it to a sample of climate and ozone data collected on the Antarctic Peninsula.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Re-weighting estimation of the coefficients in the varying coefficient model with heteroscedastic errors

ABSTRACT

This article explores the estimation problem of the coefficients in the varying coefficient model with heteroscedastic errors. Specifically, we first present a method for estimating the variance function of the error term and the resulting estimator is proved to be consistent. Then, motivated by the generalized least-squares procedure for dealing with heteroscedasticity in the linear regression literature, we re-weight each squared residual term in the local linear smoother with the inverse of the corresponding estimated error variance to construct estimates of the coefficients. Simulation experiments and practical data analysis conducted demonstrate that the re-weighting approach can improve the accuracy of the coefficient estimates under a finite sample size, especially when the error heteroscedasticity is strong.     

Variable selection in partially linear additive models for modal regression

Based on B-spline basis functions and smoothly clipped absolute deviation (SCAD) penalty, we present a new estimation and variable selection procedure based on modal regression for partially linear additive models. The outstanding merit of the new method is that it is robust against outliers or heavy-tail error distributions and performs no worse than the least-square-based estimation for normal error case. The main difference is that the standard quadratic loss is replaced by a kernel function depending on a bandwidth that can be automatically selected based on the observed data. With appropriate selection of the regularization parameters, the new method possesses the consistency in variable selection and oracle property in estimation. Finally, both simulation study and real data analysis are performed to examine the performance of our approach.     

Robust designs for one-point extrapolation

We consider the construction of designs for the extrapolation of a regression response to one point outside of the design space. The response function is an only approximately known function of a specified linear function. As well, we allow for variance heterogeneity. We find minimax designs and corresponding optimal regression weights in the context of the following problems: (P1) for nonlinear least squares estimation with homoscedasticity, determine a design to minimize the maximum value of the mean squared extrapolation error (MSEE), with the maximum being evaluated over the possible departures from the response function; (P2) for nonlinear least squares estimation with heteroscedasticity, determine a design to minimize the maximum value of MSEE, with the maximum being evaluated over both types of departures; (P3) for nonlinear weighted least squares estimation, determine both weights and a design to minimize the maximum MSEE; (P4) choose weights and design points to minimize the maximum MSEE, subject to a side condition of unbiasedness. Solutions to (P1)–(P4) are given in complete generality. Numerical comparisons indicate that our designs and weights perform well in combining robustness and efficiency. Applications to accelerated life testing are highlighted.     

Robust extrapolation designs and weights for biased regression models with heteroscedastic errors

We consider the construction of designs for the extrapolation of regression responses, allowing both for possible heteroscedasticity in the errors and for imprecision in the specification of the response function. We find minimax designs and correspondingly optimal estimation weights in the context of the following problems: (1) for ordinary least squares estimation, determine a design to minimize the maximum value of the integrated mean squared prediction error (IMSPE), with the maximum being evaluated over both types of departure; (2) for weighted least squares estimation, determine both weights and a design to minimize the maximum IMSPE; (3) choose weights and design points to minimize the maximum IMSPE, subject to a side condition of unbiasedness. Solutions to (1) and (2) are given for multiple linear regression with no interactions, a spherical design space and an annular extrapolation space. For (3) the solution is given in complete generality; as one example we consider polynomial regression. Applications to a dose-response problem for bioassays are discussed. Numerical comparisons, including a simulation study, indicate that, as well as being easily implemented, the designs and weights for (3) perform as well as those for (1) and (2) and outperform some common competitors for moderate but undetectable amounts of model bias.     

Adaptive estimation of error density in nonparametric regression with small sample size

It has been established recently in Efromovich [2005. Estimation of the density of regression errors. Ann. Statist. 33, 2194–2227] that, under a mild assumption, the error density in a nonparametric regression can be asymptotically estimated with the accuracy of an oracle that knows underlying regression errors. The asymptotic nature of the result, and in particular the used methodology of splitting data for estimating nuisance functions and the error density, does not make an asymptotic estimator, suggested in that article, feasible for practically interesting cases of small sample sizes. This article continues the research and solves two important issues. First, it shows that the asymptotic holds without splitting the data. Second, a data-driven estimator, based on the new asymptotic, is suggested and then tested on real and simulated examples.     

ROBUST ESTIMATION OF SMALL‐AREA MEANS AND QUANTILES

Small‐area estimation techniques have typically relied on plug‐in estimation based on models containing random area effects. More recently, regression M‐quantiles have been suggested for this purpose, thus avoiding conventional Gaussian assumptions, as well as problems associated with the specification of random effects. However, the plug‐in M‐quantile estimator for the small‐area mean can be shown to be the expected value of this mean with respect to a generally biased estimator of the small‐area cumulative distribution function of the characteristic of interest. To correct this problem, we propose a general framework for robust small‐area estimation, based on representing a small‐area estimator as a functional of a predictor of this small‐area cumulative distribution function. Key advantages of this framework are that it naturally leads to integrated estimation of small‐area means and quantiles and is not restricted to M‐quantile models. We also discuss mean squared error estimation for the resulting estimators, and demonstrate the advantages of our approach through model‐based and design‐based simulations, with the latter using economic data collected in an Australian farm survey.     

Sieve Empirical Likelihood and Extensions of the Generalized Least Squares

The empirical likelihood cannot be used directly sometimes when an infinite dimensional parameter of interest is involved. To overcome this difficulty, the sieve empirical likelihoods are introduced in this paper. Based on the sieve empirical likelihoods, a unified procedure is developed for estimation of constrained parametric or non-parametric regression models with unspecified error distributions. It shows some interesting connections with certain extensions of the generalized least squares approach. A general asymptotic theory is provided. In the parametric regression setting it is shown that under certain regularity conditions the proposed estimators are asymptotically efficient even if the restriction functions are discontinuous. In the non-parametric regression setting the convergence rate of the maximum estimator based on the sieve empirical likelihood is given. In both settings, it is shown that the estimator is adaptive for the inhomogeneity of conditional error distributions with respect to predictor, especially for heteroscedasticity.     

Heteroscedasticity diagnostics in varying-coefficient partially linear regression models and applications in analyzing Boston housing data

It is important to detect the variance heterogeneity in regression model because efficient inference requires that heteroscedasticity is taken into consideration if it really exists. For the varying-coefficient partially linear regression models, however, the problem of detecting heteroscedasticity has received very little attention. In this paper, we present two classes of tests of heteroscedasticity for varying-coefficient partially linear regression models. The first test statistic is constructed based on the residuals, in which the error term is from a normal distribution. The second one is motivated by the idea that testing heteroscedasticity is equivalent to testing pseudo-residuals for a constant mean. Asymptotic normality is established with different rates corresponding to the null hypothesis of homoscedasticity and the alternative. Some Monte Carlo simulations are conducted to investigate the finite sample performance of the proposed tests. The test methodologies are illustrated with a real data set example.     

Bandwidth selection for local linear regression smoothers

Summary. The paper presents a general strategy for selecting the bandwidth of nonparametric regression estimators and specializes it to local linear regression smoothers. The procedure requires the sample to be divided into a training sample and a testing sample. Using the training sample we first compute a family of regression smoothers indexed by their bandwidths. Next we select the bandwidth by minimizing the empirical quadratic prediction error on the testing sample. The resulting bandwidth satisfies a finite sample oracle inequality which holds for all bounded regression functions. This permits asymptotically optimal estimation for nearly any regression function. The practical performance of the method is illustrated by a simulation study which shows good finite sample behaviour of our method compared with other bandwidth selection procedures.     

Efficient Estimation and Robust Inference of Linear Regression Models in the Presence of Heteroscedastic Errors and High Leverage Points

It is common for linear regression models that the error variances are not the same for all observations and there are some high leverage data points. In such situations, the available literature advocates the use of heteroscedasticity consistent covariance matrix estimators (HCCME) for the testing of regression coefficients. Primarily, such estimators are based on the residuals derived from the ordinary least squares (OLS) estimator that itself can be seriously inefficient in the presence of heteroscedasticity. To get efficient estimation, many efficient estimators, namely the adaptive estimators are available but their performance has not been evaluated yet when the problem of heteroscedasticity is accompanied with the presence of high leverage data. In this article, the presence of high leverage data is taken into account to evaluate the performance of the adaptive estimator in terms of efficiency. Furthermore, our numerical work also evaluates the performance of the robust standard errors based on this efficient estimator in terms of interval estimation and null rejection rate (NRR).     

MODELING AND ESTIMATING VARIANCES IN REGRESSION

ABSTRACT

It is well known that ignoring heteroscedasticity in regression analysis adversely affects the efficiency of estimation and renders the usual procedure for constructing prediction intervals inappropriate. In some applications, such as off-line quality control, knowledge of the variance function is also of considerable interest in its own right. Thus the modeling of variance constitutes an important part of regression analysis. A common practice in modeling variance is to assume that a certain function of the variance can be closely approximated by a function of a known parametric form. The logarithm link function is often used even if it does not fit the observed variation satisfactorily, as other alternatives may yield negative estimated variances. In this paper we propose a rich class of link functions for more flexible variance modeling which alleviates the major difficulty of negative variances. We suggest also an alternative analysis for heteroscedastic regression models that exploits the principle of “separation” discussed in Box (Signal-to-Noise Ratios, Performance Criteria and Transformation. Technometrics 1988, 30, 1–31). The proposed method does not require any distributional assumptions once an appropriate link function for modeling variance has been chosen. Unlike the analysis in Box (Signal-to-Noise Ratios, Performance Criteria and Transformation. Technometrics 1988, 30, 1–31), the estimated variances and their associated asymptotic variances are found in the original metric (although a transformation has been applied to achieve separation in a different scale), making interpretation of results considerably easier.     

Inference under Heteroscedasticity of Unknown Form Using an Adaptive Estimator

The heteroscedasticity consistent covariance matrix estimators are commonly used for the testing of regression coefficients when error terms of regression model are heteroscedastic. These estimators are based on the residuals obtained from the method of ordinary least squares and this method yields inefficient estimators in the presence of heteroscedasticity. It is usual practice to use estimated weighted least squares method or some adaptive methods to find efficient estimates of the regression parameters when the form of heteroscedasticity is unknown. But HCCM estimators are seldom derived from such efficient estimators for testing purposes in the available literature. The current article addresses the same concern and presents the weighted versions of HCCM estimators. Our numerical work uncovers the performance of these estimators and their finite sample properties in terms of interval estimation and null rejection rate.