部分线性模型非参数部分的多项式类关系的检验

对于部分线性模型中非参数部分是否为多项式函数的检验问题,应该先确定其是否为多项式函数类。通过对部分线性模型的拟合残差进行再光滑,基于其变化的趋势性构造统计量以检验其是否为多项式函数类,给出了计算检验P-值的精确算法和三阶矩χ2逼近方法,模拟例子与实际例子充分显示了本方法的有效性。     

Inference in generalized additive mixed modelsby using smoothing splines

Generalized additive mixed models are proposed for overdispersed and correlated data, which arise frequently in studies involving clustered, hierarchical and spatial designs. This class of models allows flexible functional dependence of an outcome variable on covariates by using nonparametric regression, while accounting for correlation between observations by using random effects. We estimate nonparametric functions by using smoothing splines and jointly estimate smoothing parameters and variance components by using marginal quasi-likelihood. Because numerical integration is often required by maximizing the objective functions, double penalized quasi-likelihood is proposed to make approximate inference. Frequentist and Bayesian inferences are compared. A key feature of the method proposed is that it allows us to make systematic inference on all model components within a unified parametric mixed model framework and can be easily implemented by fitting a working generalized linear mixed model by using existing statistical software. A bias correction procedure is also proposed to improve the performance of double penalized quasi-likelihood for sparse data. We illustrate the method with an application to infectious disease data and we evaluate its performance through simulation.     

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.     

Bayesian regression with multivariate linear splines

We present a Bayesian analysis of a piecewise linear model constructed by using basis functions which generalizes the univariate linear spline to higher dimensions. Prior distributions are adopted on both the number and the locations of the splines, which leads to a model averaging approach to prediction with predictive distributions that take into account model uncertainty. Conditioning on the data produces a Bayes local linear model with distributions on both predictions and local linear parameters. The method is spatially adaptive and covariate selection is achieved by using splines of lower dimension than the data.     

两个部分线性模型的比较

内容提要:对于两个部分线性模型参数部分中模型系数是否相等的检验问题,本文基于比较原假设与备择假设下模型拟合的残差平方和的思想构造了检验统计量,并给出了计算检验_^p* 值的F分布逼近法。     

Principal components regression estimator of the parameters in partially linear models

ABSTRACT

As a compromise between parametric regression and non-parametric regression models, partially linear models are frequently used in statistical modelling. This paper is concerned with the estimation of partially linear regression model in the presence of multicollinearity. Based on the profile least-squares approach, we propose a novel principal components regression (PCR) estimator for the parametric component. When some additional linear restrictions on the parametric component are available, we construct a corresponding restricted PCR estimator. Some simulations are conducted to examine the performance of our proposed estimators and the results are satisfactory. Finally, a real data example is analysed.     

Asymptotic results for parametric estimation in inadequate two phases regression models

We study the asymptotic properties, consistency, asymptotic normality, of the least squares estimator in a non linear regression problem. The model uses a parametric class Л of functions, but we do not assume that the unknown function belongs to that class. Л is here a class of continuous functions with a discontinuity in the first derivative. The problem of making a choice between two classes of that type is also studied.     

Robust analysis of variance

The classes of R- and M-estimates contain practical robust alternatives to least squares estimation in linear models. These estimates form the basis for a robust analysis of variance. This inference procedure is described and its versatility demonstrated.     

Multivariate linear regression for heterogeneous data

The problem of multivariate regression modelling in the presence of heterogeneous data is dealt to address the relevant issue of the influence of such heterogeneity in assessing the linear relations between responses and explanatory variables. In spite of its popularity, clusterwise regression is not designed to identify the linear relationships within ‘homogeneous’ clusters exhibiting internal cohesion and external separation. A within-clusterwise regression is introduced to achieve this aim and, since the possible presence of a linear relation ‘between’ clusters should be also taken into account, a general regression model is introduced to account for both the between-cluster and the within-cluster regression variation. Some decompositions of the variance of the responses accounted for are also given, the least-squares estimation of the parameters is derived, together with an appropriate coordinate descent algorithms and the performance of the proposed methodology is evaluated in different datasets.     

M-Estimation for partially functional linear regression model based on splines

ABSTRACT

M-estimation is a widely used technique for robust statistical inference. In this paper, we study robust partially functional linear regression model in which a scale response variable is explained by a function-valued variable and a finite number of real-valued variables. For the estimation of the regression parameters, which include the infinite dimensional function as well as the slope parameters for the real-valued variables, we use polynomial splines to approximate the slop parameter. The estimation procedure is easy to implement, and it is resistant to heavy-tailederrors or outliers in the response. The asymptotic properties of the proposed estimators are established. Finally, we assess the finite sample performance of the proposed method by Monte Carlo simulation studies.     

On sufficient conditions for the strong consistency of least-squares estimates

The strong consistency of the least-squares estimates in regression models is obtained when the errors are i.i.d. with absolute moment of order r, 0<r? 2. The assumptions presented for the random error sequence will permit us to obtain improvements of the conditions on the regressors in order to obtain the strong consistency of the least-squares estimates in linear and nonlinear regression models.     

Multiple-index varying-coefficient models for longitudinal data

In haemodialysis patients, vascular access type is of paramount importance. Although recent studies have found that central venous catheter is often associated with poor outcomes and switching to arteriovenous fistula is beneficial, studies have not fully elucidated how the effect of switching of access on outcomes changes over time for patients on dialysis and whether the effect depends on switching time. In this paper, we characterise the switching access type effect on outcomes for haemodialysis patients. This is achieved by using a new class of multiple-index varying-coefficient (MIVC) models. We develop a new estimation procedure for MIVC models based on local linear, profile least-square method and Cholesky decomposition. Monte Carlo simulation studies show excellent finite sample performance. Finally, we analyse the dialysis data using our method.     

Robust Inference in Conditionally Linear Nonlinear Regression Models

Abstract.  We consider robust methods of likelihood and frequentist inference for the nonlinear parameter, say α , in conditionally linear nonlinear regression models. We derive closed-form expressions for robust conditional, marginal, profile and modified profile likelihood functions for α under elliptically contoured data distributions. Next, we develop robust exact-F confidence intervals for α and consider robust Fieller intervals for ratios of regression parameters in linear models. Several well-known examples are considered and Monte Carlo simulation results are presented.     

Estimation of the Variance Function in Heteroscedastic Linear Regression Models

Heteroscedasticity generally exists when a linear regression model is applied to analyzing some real-world problems. Therefore, how to accurately estimate the variance functions of the error term in a heteroscedastic linear regression model is of great importance for obtaining efficient estimates of the regression parameters and making valid statistical inferences. A method for estimating the variance function of heteroscedastic linear regression models is proposed in this article based on the variance-reduced local linear smoothing technique. Some simulations and comparisons with other method are conducted to assess the performance of the proposed method. The results demonstrate that the proposed method can accurately estimate the variance functions and therefore produce more efficient estimates of the regression parameters.     

Approximate regression models and splines

The literature pertaining to splines in regression analysis is reviewed. Spline regression is motivated as a simple extension of the basic polynomial regression model. Using this framework, the concepts of fixed and variable knot spline regression are developed and corresponding inferential procedures are considered. Smoothing splines are also seen to be an extension of polynomial regression and various optimality properties, as well as inferential and diagnostic methods, for these types of splines are discussed.     

Robust sampling strategies for regression estimation

For probability linear regression estimation, conditions are derived where sampling will be robust against violations of the commonly assumed heterogeneous variance model. Stratified pps (spps) and stratified random sampling (spscx) are shown to satisfy these conditions approximately and are more efficient generally than restricted simple random sampling (RSRS) for some real populations and for artificial populations with weights of k = 0, 0.5, 1.0, 1.5 and 2.0. The criteria needs some additional refinement to better predict relative efficiency of spps and spscx.     

Robust bootstrap estimates in heteroscedastic semi-varying coefficient models and applications in analyzing Australia CPI data

This article deals with the estimation of the parametric component, which is of primary interest, in the heteroscedastic semi-varying coefficient models. Based on the bootstrap technique, we present a procedure for estimating the parameters, which can provide a reliable approximation to the asymptotic distribution of the profile least-square (PLS) estimator. Furthermore, a bootstrap-type estimator of covariance matrix is developed, which is proved to be a consistent estimator of the covariance matrix. Moreover, some simulation experiments are conducted to evaluate the finite sample performance for the proposed methodology. Finally, the Australia CPI dataset is analyzed to demonstrate the application of the methods.     

Results concerning the generalized partially linear single-index model

The central topic of this article is the estimation of parameters of the generalized partially linear single-index model (GPLSIM). Two numerical optimization procedures are presented and an S-plus program based on these procedures is compared to a program by Wand in a simulation setting. The results from these simulations indicate that the estimates for the new procedures are as good, if not better, than Wand's. Also, this program is much more flexible than Wand's since it can handle more general models. Other simulations are also conducted. The first compares the effects of using linear interpolation versus spline interpolation in an optimization procedure. The results indicate that by using spline interpolation one gets more stable estimates at a cost of increased computational time. A second simulation was conducted to assess the performance of a method for estimating the variance of alpha. A third set of simulations is carried out to determine the best criterion for testing that one of the elements of alpha is equal to zero. The GPLSIM is applied to a water quality data set and the results indicate an interesting relationship between gastrointestinal illness and turbidity (cloudiness) of drinking water.     

Immaculating the inconsistent estimator of slope parameter in measurement error model with replicated data

ABSTRACT

The measurement error model with replicated data on study as well as explanatory variables is considered. The measurement error variance associated with the explanatory variable is estimated using the complete data and the grouped data which is used for the construction of the consistent estimators of regression coefficient. These estimators are further used in constructing an almost unbiased estimator of regression coefficient. The large sample properties of these estimators are derived without assuming any distributional form of the measurement errors and the random error component under the setup of an ultrastructural model.