Heteroscedasticity diagnostics in two-phase linear regression models

In two-phase linear regression models, it is a standard assumption that the random errors of two phases have constant variances. However, this assumption is not necessarily appropriate. This paper is devoted to the tests for variance heterogeneity in these models. We initially discuss the simultaneous test for variance heterogeneity of two phases. When the simultaneous test shows that significant heteroscedasticity occurs in the whole model, we construct two individual tests to investigate whether or not both phases or one of them have/has significant heteroscedasticity. Several score statistics and their adjustments based on Cox and Reid [D. R. Cox and N. Reid, Parameter orthogonality and approximate conditional inference. J. Roy. Statist. Soc. Ser. B 49 (1987), pp. 1–39] are obtained and illustrated with Australian onion data. The simulated powers of test statistics are investigated through Monte Carlo methods.     

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.     

Variance stabilizing filters#

Abstract

In this paper new filters for removing unspecified form of heteroscedasticity are proposed. The filters build on the assumption that the variance of a pre-whitened time series can be viewed as a latent stochastic process by its own. This makes the filters flexible and useful in many situations. A simulation study shows that removing heteroscedasticity before fitting a model leads to efficiency gains and bias reductions when estimating the parameters of ARMA models. A real data study shows that pre-filtering can increase the forecasting precision of quarterly US GDP growth.     

Comparing the mean vectors of two independent multivariate log-normal distributions

The multivariate log-normal distribution is a good candidate to describe data that are not only positive and skewed, but also contain many characteristic values. In this study, we apply the generalized variable method to compare the mean vectors of two independent multivariate log-normal populations that display heteroscedasticity. Two generalized pivotal quantities are derived for constructing the generalized confidence region and for testing the difference between two mean vectors. Simulation results indicate that the proposed procedures exhibit satisfactory performance regardless of the sample sizes and heteroscedasticity. The type I error rates obtained are consistent with expectations and the coverage probabilities are close to the nominal level when compared with the other method which is currently available. These features make the proposed method a worthy alternative for inferential analysis of problems involving multivariate log-normal means. The results are illustrated using three examples.     

Consistency of least-squares estimator and its jackknife variance estimator in nonlinear models

The purpose of this paper is twofold: (1) We establish the consistency of the least-squares estimator in a nonlinear modely_i = f(x_i,θ) +σ_ie_i where the range of the parameter θ is noncompact, the regression function is unbounded, and the σ_i,'s are not necessarily equal. This extends the results in Jennrich (1969) and Wu (1981). (2) Under the same model, the jackknife estimator of the asymptotic covariance matrix of the least-squares estimator is shown to be consistent, which provides a theoretical justification of the empirical results in Duncan (1978) and the use of the jackknife method in large-sample inferences.     

Designing for a Response Transformation Parameter

We investigate D -optimum designs for experiments in which a linear model holds after an unknown power transformation of the univariate response variable. This is a departure from standard D -optimal design in which an appropriate scale for the response is assumed known before data collection. The design problem that we formulate is intrinsically non-linear, requiring characterization of likely parameter values. Several applications are presented to illustrate the importance of recognizing the role of response transformations at the design stage.     

Stabilizing heteroscedasticity for butterfly-distributed residuals by the weighting absolute centered external variable

In the current study, a new method by the weighting absolute centered external variable (WCEV) was proposed to stabilize heteroscedasticity for butterfly-distributed residuals (BDRs). After giving brief information about heteroscedasticity and BDRs, WCEV was introduced. The WCEV and commonly used variance stabilizing methods are compared on a simple and a multiple regression model. The WCEV was also tested for other type of heteroscedasticity patterns. In addition to heteroscedasticity, other regression assumptions were checked for the WCEV.     

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.     

Statistical inference for heteroscedastic semi-varying coefficient EV models

This paper proposes an estimation procedure for a class of semi-varying coefficient regression models when the covariates of the linear part are subject to measurement errors. Initial estimates for the regression and varying coefficients are first constructed by the profile least-squares procedure without input from heteroscedasticity, a bias-corrected kernel estimate for the variance function then is proposed, which in turn is used to define re-weighted bias-corrected estimates of the regression and varying coefficients. Large sample properties of the proposed estimates are thoroughly investigated. The finite-sample performance of the proposed estimates is assessed by an extensive simulation study and an application to the Boston housing data set. The simulation results show that the re-weighted bias-corrected estimates outperform the initial estimates and the naive estimates.