The transfer principle: A tool for complete case analysis

This paper gives a general method for deriving limiting distributions of complete case statistics for missing data models from corresponding results for the model where all data are observed. This provides a convenient tool for obtaining the asymptotic behavior of complete case versions of established full data methods without lengthy proofs. The methodology is illustrated by analyzing three inference procedures for partially linear regression models with responses missing at random. We first show that complete case versions of asymptotically efficient estimators of the slope parameter for the full model are efficient, thereby solving the problem of constructing efficient estimators of the slope parameter for this model. Second, we derive an asymptotically distribution free test for fitting a normal distribution to the errors. Finally, we obtain an asymptotically distribution free test for linearity, that is, for testing that the nonparametric component of these models is a constant. This test is new both when data are fully observed and when data are missing at random.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1061 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Topics in Partially Linear Single-Index Models for Longitudinal Data

The partially linear single-index model is a semiparametric model proposed to the case when some predictors are linearly associated with the response variable, while some other predictors are nonlinearly associated with the response variable. It is widely used for its flexibility in statistical modeling. Furthermore, its generalized version is a generalization of some popular models such as the generalized linear model, the partially linear model and the single-index model. However, the proper estimation in partially linear single-index models for longitudinal data, where multiple measurements are observed for each subject, is still open to discussion. Our main purpose is to establish a unified estimation method for the longitudinal partially linear single-index model and its generalized version. With this question in mind, we propose a new iterative three-stage estimation method in partially linear single-index models and generalized partially linear single-index models for longitudinal data. With the proposed method, the within-subject correlation is properly taken into consideration in the estimation of both the parameters and the nonparametric single-index function. The parameter estimators are shown to be asymptotically semiparametric efficient. The asymptotic variance of the single-index function estimator is shown to be generally less than that of existing estimators. Simulation studies are performed to demonstrate the finite sample performance. Three real data examples are also analyzed to illustrate the methodology

Structured additive regression for multicategorical space-time data: A mixed model approach

In many practical situations, simple regression models suffer from the fact that the dependence of responses on covariates can not be sufficiently described by a purely parametric predictor. For example effects of continuous covariates may be nonlinear or complex interactions between covariates may be present. A specific problem of space-time data is that observations are in general spatially and/or temporally correlated. Moreover, unobserved heterogeneity between individuals or units may be present. While, in recent years, there has been a lot of work in this area dealing with univariate response models, only limited attention has been given to models for multicategorical space-time data. We propose a general class of structured additive regression models (STAR) for multicategorical responses, allowing for a flexible semiparametric predictor. This class includes models for multinomial responses with unordered categories as well as models for ordinal responses. Non-linear effects of continuous covariates, time trends and interactions between continuous covariates are modelled through Bayesian versions of penalized splines and flexible seasonal components. Spatial effects can be estimated based on Markov random fields, stationary Gaussian random fields or two-dimensional penalized splines. We present our approach from a Bayesian perspective, allowing to treat all functions and effects within a unified general framework by assigning appropriate priors with different forms and degrees of smoothness. Inference is performed on the basis of a multicategorical linear mixed model representation. This can be viewed as posterior mode estimation and is closely related to penalized likelihood estimation in a frequentist setting. Variance components, corresponding to inverse smoothing parameters, are then estimated by using restricted maximum likelihood. Numerically efficient algorithms allow computations even for fairly large data sets. As a typical example we present results on an analysis of data from a forest health survey