One-Step Bootstrapping in Generalized Linear Models

Abstract

Some very commonly used statistical models are members of the class of generalized linear models introduced by Neider & Wedderbum(1972). Included are probit and logit models for binomially distributed data, log-linear models for Poisson and also classical linear models for normally distributed response data. In all these examples the underlying distribution is an element of an one-parametric exponential family. The ML-estimator for the mean vector has an interpretation as projection of the observation vector onto the space of all possible mean values. The computation is in most situations an iteratively weighted least-square estimation. However, existence and uniqueness of this estimator is not always guaranteed (see Wedderburn(1976) for a set of sufficient conditions). The following example presents logistic regression with explanatory variables x_n and only 10 0–1-observations γ_n. Here the probability of the existence of the ML-estimator β(γ₁,…, γ₁₀) is only around 80%.