On efficiency of locally D-optimal designs under heteroscedasticity and non-Gaussianity

In the classical theory of locally optimal designs, which is developed within the framework of the center+error model, the most efficient design is the one based on MGLE, the maximum Gaussian likelihood estimator. However, practical scenarios often lack of complete information as to the governing probability model for the response measure and deviate from Gaussianity and homoscedasticity assumptions, in which, MqLE, the maximum quasi-likelihood estimator, has been advocated in the literature. In this work, we examine the locally optimal design based on the novel oracle-SLSE, the second-order least-square estimator, in the case where the underlying probability model is incompletely specified. We find that in a general setting, our oracle SLSE-based optimal design, incorporating skewness and kurtosis information, outperforms those based on MqLE or MGLE. Our numerical experiment supports this, with locally D-optimal designs based on MqLE approaching the efficiency of oracle-SLSE designs in some cases. This research guides the choice of estimators in practical scenarios departing from ideal assumptions.