Quasi-minimax estimation in the general linear regression model

This paper is mainly concerned with minimax estimation in the general linear regression model y=Xβ+ε$y=X\beta +\epsilon$ under ellipsoidal restrictions on the parameter space and quadratic loss function. We confine ourselves to estimators that are linear in the response vector y  . The minimax estimators of the regression coefficient β$\beta$ are derived under homogeneous condition and heterogeneous condition, respectively. Furthermore, these obtained estimators are the ridge-type estimators and mean dispersion error (MDE) superior to the best linear unbiased estimator b=(X^′W^-1X)^-1X^′W^-1y$b=(X^{\prime }{W}^{-1}X{)}^{-1}{X}^{\prime }{W}^{-1}y$ under some conditions.     

Transformed generalized linear models

The estimation of data transformation is very useful to yield response variables satisfying closely a normal linear model. Generalized linear models enable the fitting of models to a wide range of data types. These models are based on exponential dispersion models. We propose a new class of transformed generalized linear models to extend the Box and Cox models and the generalized linear models. We use the generalized linear model framework to fit these models and discuss maximum likelihood estimation and inference. We give a simple formula to estimate the parameter that index the transformation of the response variable for a subclass of models. We also give a simple formula to estimate the r$r$th moment of the original dependent variable. We explore the possibility of using these models to time series data to extend the generalized autoregressive moving average models discussed by Benjamin et al. [Generalized autoregressive moving average models. J. Amer. Statist. Assoc. 98, 214–223]. The usefulness of these models is illustrated in a simulation study and in applications to three real data sets.     

Confidence interval estimation of the difference between paired AUCs based on combined biomarkers

In many diagnostic studies, multiple diagnostic tests are performed on each subject or multiple disease markers are available. Commonly, the information should be combined to improve the diagnostic accuracy. We consider the problem of comparing the discriminatory abilities between two groups of biomarkers. Specifically, this article focuses on confidence interval estimation of the difference between paired AUCs based on optimally combined markers under the assumption of multivariate normality. Simulation studies demonstrate that the proposed generalized variable approach provides confidence intervals with satisfying coverage probabilities at finite sample sizes. The proposed method can also easily provide P-values for hypothesis testing. Application to analysis of a subset of data from a study on coronary heart disease illustrates the utility of the method in practice.     

A note on the equality of the OLSE and the BLUE of the parametric function in the general Gauss–Markov model

In this note we consider the equality of the ordinary least squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of the estimable parametric function in the general Gauss–Markov model. Especially we consider the structures of the covariance matrix V for which the OLSE equals the BLUE. Our results are based on the properties of a particular reparametrized version of the original Gauss–Markov model.      

ASSOCIATIONS BETWEEN AIR POLLUTION AND HOSPITAL VISITS FOR CARDIOVASCULAR DISEASES IN THE ELDERLY IN SYDNEY USING BAYESIAN STATISTICAL METHODS

Using generalized linear models (GLMs), Jalaludin  et al. (2006;  J. Exposure Analysis and Epidemiology   16 , 225–237) studied the association between the daily number of visits to emergency departments for cardiovascular disease by the elderly (65+) and five measures of ambient air pollution. Bayesian methods provide an alternative approach to classical time series modelling and are starting to be more widely used. This paper considers Bayesian methods using the dataset used by Jalaludin  et al.  (2006) , and compares the results from Bayesian methods with those obtained by Jalaludin  et al.  (2006) using GLM methods.     

Trajectory of problem behaviours of Korean adopted children: using piecewise hierarchical linear growth modelling

This study examined the trajectory of problem behaviours in domestically adopted children in South Korea as they aged. This study used the Panel Study on Korean Adopted Children's longitudinal, three‐wave data (2006, 2008 and 2010). Although data were collected at three time points, our data consisted of six time points, which covered the 5–10 years age range of the adopted children at the time of survey. One hundred sixty‐four children were included in the analysis, 75 of whom contributed to one time point, 74 to two time points and 15 to three time points. The trajectory of the problem behaviours of adopted children was examined using a piecewise hierarchical linear growth model. Because the initial exploration of the data suggested non‐linear changes in behaviour problems over time, we split the growth trajectory into two time periods: Time 1 (5–7 years) and Time 2 (7–10 years). A two‐rate model was used to estimate separate slopes for the two time periods. Results showed that the externalizing and internalizing problems of adopted children have different trajectories. Internalizing problems did not show significant changes after 5 years of age, while externalizing problems increased until 7 years of age and decreased significantly thereafter.     

Who is leading the campaign charts? Comparing individual popularity on old and new media

Attention in the mass media is seen as crucial for electoral success. However, most ordinary candidates hardly get any attention in the news. With social media outlets becoming ever more popular, the question is whether the overall asymmetry in attention for candidates still holds today. Do candidates who dominate the traditional media during the campaign also dominate the social media? Or can candidates make up for a lack of mass media coverage by attracting attention on these new media platforms? This paper aims to answer these questions by pairing Twitter activity and Twitter popularity with newspaper attention for a large number of individual candidates in the 2014 Belgian election campaign. We expand the normalization versus equalization debate by not only looking at how much a new medium is used, but also at its success in terms of popularity and audience reach. Our findings show that the two platforms are indeed related, mainly because a small political elite dominates both old and new media. Twitter popularity and Twitter activity (albeit to a lesser extent) are higher among powerful politicians. We elaborate on why these findings are so much in line with the normalization hypothesis.     

Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

In this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.     

On the asymptotic normality of the R-estimators of the slope parameters of simple linear regression models with associated errors

ABSTRACT

The purpose of this paper is to prove, under mild conditions, the asymptotic normality of the rank estimator of the slope parameter of a simple linear regression model with stationary associated errors. This result follows from a uniform linearity property for linear rank statistics that we establish under general conditions on the dependence of the errors. We prove also a tightness criterion for weighted empirical process constructed from associated triangular arrays. This criterion is needed for the proofs which are based on that of Koul [Behavior of robust estimators in the regression model with dependent errors. Ann Stat. 1977;5(4):681–699] and of Louhichi [Louhichi S. Weak convergence for empirical processes of associated sequences. Ann Inst Henri Poincaré Probabilités Statist. 2000;36(5):547–567].