Optimal Bayesian estimation of the median effective dose

Abstract

Abstract This paper provides some properties of the Fisher information function arising in quantal response bioassay, attribute life testing and dilution assay models. It is shown that the Fisher information function (arising in Probit, Logit and extreme value models) is a totally positive function of order 2 and consequently unimodal as a function of the unknown parameter as well as a function of the design level. A method is provided for obtaining estimates of the shift parameter (median effective dose when scale parameter is known for symmetric tolerance densities) in these models from Bayesian and adaptive Bayesian points of view. We assumme that the prior distribution belongs to a class containing Polya densities of order 2 defined over the real line besides the rectangular densities. Optimality is defined in terms of maximizing the expected Fisher information function. It is shown that the optimal Bayesian estimates exist and are unique under certain conditions on the model and the prior distributions. Optimal Bayesian estimates of the median effective dose for a two stage design set up for Logit model are provided for some prior distributions. It is shown that the tables of optimal Bayesian estimates obtained for the uniform prior can be used for logistic prior as well.