Lack-of-fit-efficiently optimal designs to estimate the highest coefficient of a polynomial with large degree

To check regression models Bischoff and Miller (2006a. Optimal designs which are efficient for lack of fit tests. Ann. Stat., to appear.) introduced optimal designs to estimate a parameter in the class of designs which guarantee a certain efficiency with respect to the power of a lack of fit (LOF-) test. One part of such an optimal design is absolutely continuous with respect to the Lebesgue measure and the other part consists of a finite number of mass points. The optimal design to estimate the highest coefficient of a polynomial regression of fixed degree k-1k-1 (ekek-optimal design) in the class of designs with LOF-efficiency of at least r   has the same mass points as the classical ekek-optimal design if r is small enough. In this paper we investigate the set of efficiencies r with that property.