Statistical Inference For The Generalized Additive Models

Abstract

Generalized additive models (GAMs) ( Hastie and Tibshirani

1986 ) are non-parametric extensions of Generalized linear models (GLMs) . The only assumption underlyingGAMs is the functions are additive and that the components are smooth. A GAM, like a GLM, uses a link function to establish a relationship between the mean of the response variable and a 'smooth' function of the explanatory variable(s). The strength of GAMs is their ability to deal with highly non-linear and non-monotonic relationships between the response and the set of explanatory variables. Estimation in GAM is based on a combination of the local scoring algorithm (Hastie and Tibshirani (1986)) and the backfitting algorithm (Friedman , Stuetzle (1981) and Buja, Hastie and Tibshirani (1989)) . The backfitting algorithm is a general algorithm that can fit additive model using the smoothers. Our uses of GAMs rely extensively on two popular smoothers:

the spline (Schoenberg (1964) and Reinsch (1967)) and locally weighted scatterplot smoother (Cleveland (1979 and Cleveland and Devlin (1988)). A large body of statistical research establishes that these smoothers have many desirable properties. In this thesis, the problems of estimating the percentiles and constructing prediction intervals will be considered for some generalized additive models (GAMs) such as the normal, gamma, binomial, and Poisson models. Comprehensive comparisons between GAM and GLM will be carried out using real data and simulation studies .