Optimum Design via I-Divergence for Stable Estimation in Generalized Regression Models

Abstract

Optimum designs for parameter estimation in generalized regression models are usually based on the Fisher information matrix (cf. Atkinson et al. (J Stat Plan Inference 144:81–91, 2014) for a recent exposition). The corresponding optimality criteria are related to the asymptotic properties of maximum likelihood (ML) estimators in such models. However, in finite sample experiments there can be problems with identifiability, stability and uniqueness of the ML estimate, which are not reflected by information matrices. In Pázman and Pronzato (Ann Stat 42:1426–1451, 2014) and in Chap. 7 of Pronzato and Pázman (Design of Experiments in Nonlinear Models. Asymptotic Normality, Optimality Criteria and Small-Sample Properties. Springer, New York, 2013) is discussed how to solve some of these estimability issues at the design stage of an experiment in standard nonlinear regression. Here we want to extend this design methodology to more general models based on exponential families of distributions (binomial, Poisson, normal with parametrized variances, etc.). The main tool is the information (or Kullback-Leibler) divergence, which is closely related to ML estimation.