Efficient lack of fit designs that are optimal to estimate the highest coefficient of a polynomial

Kurzfassung/Abstract

To check regression models, the authors [Optimal designs which are efficient for lack of fit tests. Ann. Stat., to appear; see also J. Stat. Plann. Inference 136, No. 12, 4239--4249 (2006; Zbl 1098.62098)] 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 $k-1$ $(\bold e_{k}$-optimal design) in the class of designs with LOF-efficiency of at least $r$ has the same mass points as the classical $\bold e_{k}$-optimal design if $r$ is small enough. We investigate the set of efficiencies $r$ with that property.