Efficient inference in general semiparametric regression models

摘要

Semiparametric regression has become very popular in the field of Statistics over theyears. While on one hand more and more sophisticated models are being developed,on the other hand the resulting theory and estimation process has become more andmore involved. The main problems that are addressed in this work are related toefficient inferential procedures in general semiparametric regression problems.We first discuss efficient estimation of population-level summaries in general semiparametricregression models. Here our focus is on estimating general population-levelquantities that combine the parametric and nonparametric parts of the model (e.g.,population mean, probabilities, etc.). We place this problem in a general context,provide a general kernel-based methodology, and derive the asymptotic distributionsof estimates of these population-level quantities, showing that in many cases the estimatesare semiparametric efficient.Next, motivated from the problem of testing for genetic effects on complex traits inthe presence of gene-environment interaction, we consider developing score test ingeneral semiparametric regression problems that involves Tukey style 1 d.f form ofinteraction between parametrically and non-parametrically modeled covariates. Wedevelop adjusted score statistics which are unbiased and asymptotically efficient andcan be performed using standard bandwidth selection methods. In addition, to over come the difficulty of solving functional equations, we give easy interpretations of thetarget functions, which in turn allow us to develop estimation procedures that can beeasily implemented using standard computational methods.Finally, we take up the important problem of estimation in a general semiparametricregression model when covariates are measured with an additive measurement errorstructure having normally distributed measurement errors. In contrast to methodsthat require solving integral equation of dimension the size of the covariate measuredwith error, we propose methodology based on Monte Carlo corrected scores to estimatethe model components and investigate the asymptotic behavior of the estimates.For each of the problems, we present simulation studies to observe the performance ofthe proposed inferential procedures. In addition, we apply our proposed methodologyto analyze nontrivial real life data sets and present the results.