The adaptive Wynn algorithm in generalized linear models with univariate response

Abstract

For a nonlinear regression model, the information matrices of designs depend on the parameter of the model. The adaptive Wynn algorithm for D-optimal design estimates the parameter at each step on the basis of the observed responses and employed design points so far, and selects the next design point as in the classical Wynn algorithm for D-optimal design. The name “Wynn algorithm” is in honor of Henry P. Wynn who established the latter “classical” algorithm in his 1970 paper (Ann. Math. Stat. 41 (1970) 1655–1664). The asymptotics of the sequences of designs and maximum likelihood estimates generated by the adaptive algorithm is studied for an important class of nonlinear regression models: generalized linear models whose (univariate) response variables follow a distribution from a one-parameter exponential family. Under the assumptions of compactness of the experimental region and of the parameter space together with some natural continuity assumptions, it is shown that the adaptive ML-estimators are strongly consistent and the design sequence is asymptotically locally D-optimal at the true parameter point. If the true parameter point is an interior point of the parameter space, then under some smoothness assumptions the asymptotic normality of the adaptive ML-estimators is obtained.