Semiparametric Methods for the Generalized Linear Model

The generalized linear model (GLM) is a popular model in many research areas. In theGLM, each outcome of the dependent variable is assumed to be generated from a particulardistribution function in the exponential family. The mean of the distribution depends onthe independent variables. The link function provides the relationship between the linearpredictor and the mean of the distribution function. In this dissertation, two semiparametric extensions of the GLM will be developed. In the first part of this dissertation, wehave proposed a new model, called a semiparametric generalized linear model with a log-concave random component (SGLM-L). In this model, the estimate of the distribution ofthe random component has a nonparametric form while the estimate of the systematic parthas a parametric form. In the second part of this dissertation, we have proposed a model,called a generalized semiparametric single-index mixed model (GSSIMM). A nonparametriccomponent with a single index is incorporated into the mean function in the generalized linear mixed model (GLMM) assuming that the random component is following a parametricdistribution.In the first part of this dissertation, since most of the literature on the GLM deals withthe parametric random component, we relax the parametric distribution assumption for therandom component of the GLM and impose a log-concave constraint on the distribution. Aniterative numerical algorithm for computing the estimators in the SGLM-L is developed. Weconstruct a log-likelihood ratio test for inference. In the second part of this dissertation, weuse a single index model to generalize the GLMM to have a linear combination of covariatesenter the model via a nonparametric mean function, because the linear model in the GLMMis not complex enough to capture the underlying relationship between the response and itsassociated covariates. The marginal likelihood is approximated using the Laplace method.A penalized quasi-likelihood approach is proposed to estimate the nonparametric functionand parameters including single-index coe±cients in the GSSIMM. We estimate variancecomponents using marginal quasi-likelihood. Asymptotic properties of the estimators aredeveloped using a similar idea by Yu (2008). A simulation example is carried out to comparethe performance of the GSSIMM with that of the GLMM. We demonstrate the advantage ofmy approach using a study of the association between daily air pollutants and daily mortalityadjusted for temperature and wind speed in various counties of North Carolina.