D-optimal design for logistic model based on more precise approximation

The approximation theory is utilized in optimal design finding by approximating the covariance matrix by the Fisher information matrix; however, the approximation accuracy depends on the sample size. Moreover, the proportion of success, p, may also be important in approximation when the underlying model is logistic. So, in the case of facing a restriction in running a large experiment in addition to facing an extreme p, the usual information matrix may not be enough accurate in approximation. In this study, a locally D-optimal design is proposed for a logistic model based on the quasi-information matrix which was obtained by applying more exact Hammersley-Chapman-Robbins (H-C-R) lower bound for variance approximation. The general equivalence theorem was deficient, so it was adapted. Under such an approach, the obtained optimal design is more efficient than the previous ones in terms of variance and mean square error of the parameter estimates. Furthermore, the probability of infinite estimate of the model parameters is reduced in experiments with extreme proportion of successes at the support points. Therefore, the proposed optimal design performs better for special practical situations.