Minimum bias designs for generalized linear models

Minimum bias (all bias) designs for the linear model were proposed by Box and Draper. In this article we extend their results to generalized linear models. We show that, in the canonical case, the minimum bias design density is: (a) proportional to the weight function reflecting the importance of each design point, and (b) inversely proportional to the observation variance at each point. We also derive minimum bias design densities in non-canonical cases. Implications for binary, Poisson and exponential data are considered in the examples. From these examples we observe that when the experimenter is mainly interested in the mean of the Binomial, Poisson or exponential distribution, rather than the canonical parameter, and if the weight function is chosen to be inversely proportional to the variance of the maximum likelihood estimator of the mean, minimum bias designs are uniform. These uniform designs automatically minimize the bias/standard error ratio and the mean square error/variance ratio.