Two Kinds of Restricted Estimators in Singular Linear Regression

In this article, the parameter estimators in singular linear model with linear equality restrictions are considered. The restricted root estimator and the generalized restricted root estimator are proposed and some properties of the estimators are also studied. Furthermore, we compare them with the restricted unified least squares estimator and show their sufficient conditions under which their superior over the restricted unified least squares estimator in terms of mean squares error, and discuss the choice of the unknown parameters of the generalized restricted root estimator.     

Estimation of group means when adjusting for covariates in generalized linear models

Generalized linear models are commonly used to analyze categorical data such as binary, count, and ordinal outcomes. Adjusting for important prognostic factors or baseline covariates in generalized linear models may improve the estimation efficiency. The model‐based mean for a treatment group produced by most software packages estimates the response at the mean covariate, not the mean response for this treatment group for the studied population. Although this is not an issue for linear models, the model‐based group mean estimates in generalized linear models could be seriously biased for the true group means. We propose a new method to estimate the group mean consistently with the corresponding variance estimation. Simulation showed the proposed method produces an unbiased estimator for the group means and provided the correct coverage probability. The proposed method was applied to analyze hypoglycemia data from clinical trials in diabetes. Copyright © 2014 John Wiley & Sons, Ltd.     

The feasible generalized restricted ridge regression estimator

The presence of autocorrelation in errors and multicollinearity among the regressors have undesirable effects on the least-squares regression. There are a wide range of methods which are proposed to overcome the usefulness of the ordinary least-squares estimator or the generalized least-squares estimator, such as the Stein-rule, restricted least-squares or ridge estimator. Therefore, we introduce a new feasible generalized restricted ridge regression (FGRR) estimator to examine multicollinearity and autocorrelation problems simultaneously for the general linear regression model. We also derive some statistical properties of the FGRR estimator and comparisons have been conducted using matrix mean-square error. Moreover, a Monte Carlo simulation experiment is performed to investigate the performance of the proposed estimator over the others.     

基于半参数方法的模型辅助抽样估计研究

在总结现有模型辅助估计方法的基础上,本文通过构造一种半参数超总体模型,同时结合广义差分估计思想提出一种新型的模型辅助估计量。该估计量比传统的非参数和半参数回归估计利用更少、更易得到的辅助信息,即只需利用和广义回归估计相同的辅助信息,但一般会比广义回归估计拥有更高的估计精度。理论证明了该估计量是渐近设计无偏和设计一致的,其渐近设计均方误差为广义差分估计量的方差。模拟结果显示：其至少与广义回归估计一样好;对于线性程度越低的超总体模型,其估计精度比广义回归估计有越明显的提高;就本文模拟而言,光滑参数在0.04～0.12间适当取值时其会取到相对较好的估计效果。     

Fractional principal components regression: a general approach to biased estimators

Several biased estimators have been proposed as alternatives to the least squares estimator when multicollinearity is present in the multiple linear regression model. The ridge estimator and the principal components estimator are two techniques that have been proposed for such problems. In this paper the class of fractional principal component estimators is developed for the multiple linear regression model. This class contains many of the biased estimators commonly used to combat multicollinearity. In the fractional principal components framework, two new estimation techniques are introduced. The theoretical performances of the new estimators are evaluated and their small sample properties are compared via simulation with the ridge, generalized ridge and principal components estimators     

Efficiency of the generalized difference-based Liu estimators in semiparametric regression models with correlated errors

In this paper, a generalized difference-based estimator is introduced for the vector parameter β in the semiparametric regression model when the errors are correlated. A generalized difference-based Liu estimator is defined for the vector parameter β in the semiparametric regression model. Under the linear nonstochastic constraint Rβ=r, the generalized restricted difference-based Liu estimator is given. The risk function for the β?_GRD(η) associated with weighted balanced loss function is presented. The performance of the proposed estimators is evaluated by a simulated data set.     

Ridge regression methodology in partial linear models with correlated errors

In this article, a generalized restricted difference-based ridge estimator is defined for the vector parameter in a partial linear model when the errors are dependent. It is suspected that some additional linear constraints may hold on to the whole parameter space. The estimator is a generalization of the well-known restricted least-squares estimator and is confined to the (affine) subspace which is generated by the restrictions. The risk functions of the proposed estimators are derived under balanced loss function. Finally, the performance of the new estimators is evaluated by a simulated data set.     

A pseudo‐empirical best linear unbiased prediction approach to small area estimation using survey weights

The authors develop a small area estimation method using a nested error linear regression model and survey weights. In particular, they propose a pseudo‐empirical best linear unbiased prediction (pseudo‐EBLUP) estimator to estimate small area means. This estimator borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. The proposed estimator also has a nice self‐benchmarking property. The authors also obtain an approximation to the model mean squared error (MSE) of the proposed estimator and a nearly unbiased estimator of MSE. Finally, they compare the proposed estimator with the EBLUP estimator and the pseudo‐EBLUP estimator proposed by Prasad & Rao (1999), using data analyzed earlier by Battese, Harter & Fuller (1988).     

Robust designs for experiments with blocks

ABSTRACT

For experiments running in field plots or over time, the observations are often correlated due to spatial or serial correlation, which leads to correlated errors in a linear model analyzing the treatment means. Without knowing the exact correlation matrix of the errors, it is not possible to compute the generalized least-squares estimator for the treatment means and use it to construct optimal designs for the experiments. In this paper, we propose to use neighborhoods to model the covariance matrix of the errors, and apply a modified generalized least-squares estimator to construct robust designs for experiments with blocks. A minimax design criterion is investigated, and a simulated annealing algorithm is developed to find robust designs. We have derived several theoretical results, and representative examples are presented.     

Liu estimator in partly linear regression models with correlated errors

This article is concerned with the parameter estimation in partly linear regression models when the errors are dependent. To overcome the multicollinearity problem, a generalized Liu estimator is proposed. The theoretical properties of the proposed estimator and its relationship with some existing methods designed for partly linear models are investigated. Finally, a hypothetical data is conducted to illustrate some of the theoretical results.     

AN EXAMINATION OF ESTIMATED RESIDUALS IN A REGRESSION WITH AN INFINITE ORDER PARAMETRIC MODEL

We consider a linear regression with the error term that obeys an autoregressive model of infinite order and estimate parameters of the models. The parameters of the autoregressive model should be estimated based on estimated residuals obtained by means of the method of ordinary least squares, because the errors are unobservable. The consistency of the coefficients, variance and spectral density of the model obeyed by the error term is shown. Further, we estimate the coefficients of the linear regression by means of the method of estimated generalized least squares. We also show the consistency of the estimator.

     

A Simple Approach to Inference in Random Coefficient Models

Random coefficient regression models have been used to analyze cross-sectional and longitudinal data in economics and growth-curve data from biological and agricultural experiments. In the literature several estimators, including the ordinary least squares and the estimated generalized least squares (EGLS), have been considered for estimating the parameters of the mean model. Based on the asymptotic properties of the EGLS estimators, test statistics have been proposed for testing linear hypotheses involving the parameters of the mean model. An alternative estimator, the simple mean of the individual regression coefficients, provides estimation and hypothesis-testing procedures that are simple to compute and teach. The large sample properties of this simple estimator are shown to be similar to that of the EGLS estimator. The performance of the proposed estimator is compared with that of the existing estimators by Monte Carlo simulation.     

A modified generalized mixed regression estimator when disturbances are nonnormal

A generalized mixed regression estimator for the estimation of the regression coefficients in the linear regression model with incomplete prior information is proposed and its properties are studied considering the risk under the general quadratic loss function when the disturbances are small and non normal.     

A generalized inverse regression estimator in multi-univariate linear calibration

A multivariate linear calibration problem, in which response variable is multivariate and explanatory variable is univariate, is considered. In this paper a class of generalized inverse regression estimators is proposed in multi-univariate linear calibration. It includes the classical estimator and the inverse regression one (or Krutchkoff estimator). For the proposed estimator we derive the expressions of bias and mean square error (MSE). Furthermore the behavior of these characteristics is investigated through an analytical method. In addition through a numerical study we confirm the existence of a generalized inverse regression estimator to improve both the classical and the inverse regression estimators on the MSE criterion.     

Statistical Inference and Applications of Mixture of Varying Coefficient Models

In this paper, we consider a new mixture of varying coefficient models, in which each mixture component follows a varying coefficient model and the mixing proportions and dispersion parameters are also allowed to be unknown smooth functions. We systematically study the identifiability, estimation and inference for the new mixture model. The proposed new mixture model is rather general, encompassing many mixture models as its special cases such as mixtures of linear regression models, mixtures of generalized linear models, mixtures of partially linear models and mixtures of generalized additive models, some of which are new mixture models by themselves and have not been investigated before. The new mixture of varying coefficient model is shown to be identifiable under mild conditions. We develop a local likelihood procedure and a modified expectation–maximization algorithm for the estimation of the unknown non‐parametric functions. Asymptotic normality is established for the proposed estimator. A generalized likelihood ratio test is further developed for testing whether some of the unknown functions are constants. We derive the asymptotic distribution of the proposed generalized likelihood ratio test statistics and prove that the Wilks phenomenon holds. The proposed methodology is illustrated by Monte Carlo simulations and an analysis of a CO₂‐GDP data set.     

Fundamental Equations of BLUE and BLUP in the Multivariate Linear Model with Applications

Ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE), and best linear unbiased predictor (BLUP) in the general linear model with new observations are generalized to the general multivariate linear model. The fundamental equations of BLUE and BLUP in the multivariate linear model are derived by two methods, including the vectorization method and projection method. By using the matrix rank method, some new results of linear BLUE-sufficiency, linear BLUP-sufficiency, and the equality of OLSE, BLUE, and BLUP are given in the multivariate linear model.     

A note on generalized ridge estimators

It is shown that a necessary and sufficient condition derived by Farebrother (1984)for a generalized ridge estimator to dominate the ordinary least-squares estimator with respect to the mean-square-error-matrix criterion in the linear regression model admits a similar interpretation as the well known criterion of Toro-Viz-carrondo and Wallace (1968)for the dominance of a restricted least-squares estimator over the ordinary least-squares estimator. Two other properties of the generalized ridge estimators, referring to the concept of admissibility, are also pointed out.     

Efficient estimation of general linear mixed effects models

An extension of the linear growth curve model (Biometrics 38 (1982) 963) was proposed by Stukel and Demidenko (Biometrics 53 (1997) 720) to study the effects of population covariates on one or more characteristics of the curve, when the characteristics are expressed as linear combinations of the growth curve parameters. In the present paper, this general growth curve model receives a comprehensive theoretical treatment. A two-stage estimator, consisting of a generalized least squares estimator under constraints for the population parameters and a moment estimator for the variance parameters, is developed for application in the non-Gaussian error situation. Two likelihood based estimators, global maximum likelihood and second-stage maximum likelihood, are also developed. It is shown that all three estimators are consistent, asymptotically normally distributed, and efficient, and are equivalent when the number of individuals tends to infinity. An expression for the bias in the estimator of the population parameters is derived under second stage model misspecification. We show that if parameters that are not of primary interest are incorrectly specified, bias may occur in parameters that are of interest using the standard growth curve model. The general growth curve model does not require specification of such nuisance parameters and is robust in terms of bias. The general linear growth curve model is used to study the effects of host sex on pancreatic tumor growth in rats.     

Monotonic change point estimation of generalized linear model-based regression profiles

In this article, a maximum likelihood estimator is derived in the generalized linear model-based regression profiles under monotonic change in Phase II. The performance of the proposed estimator is comprehensively investigated through some special cases, and compared with estimators under step change and drift. The results show that the proposed estimator has better performance in small and medium shifts under different increasing changes. Finally, the applicability of the proposed estimator is illustrated using a real case.     

Variable selection in the high-dimensional continuous generalized linear model with current status data

In survival studies, current status data are frequently encountered when some individuals in a study are not successively observed. This paper considers the problem of simultaneous variable selection and parameter estimation in the high-dimensional continuous generalized linear model with current status data. We apply the penalized likelihood procedure with the smoothly clipped absolute deviation penalty to select significant variables and estimate the corresponding regression coefficients. With a proper choice of tuning parameters, the resulting estimator is shown to be a root n/p_n-consistent estimator under some mild conditions. In addition, we show that the resulting estimator has the same asymptotic distribution as the estimator obtained when the true model is known. The finite sample behavior of the proposed estimator is evaluated through simulation studies and a real example.