Optimal designs for testing the functional form of a regression via nonparametric estimation techniques

For the problem of checking linearity in a heteroscedastic nonparametric regression model under a fixed design assumption, we study maximin designs which maximize the minimum power of a nonparametric test over a broad class of alternatives from the assumed linear regression model. It is demonstrated that the optimal design depends sensitively on the used estimation technique (i.e. weighted or ordinary least-squares) and on an inner product used in the definition of the class of alternatives. Our results extend and put recent findings of Wiens (Statist. Probab. Lett. 12 (1991) 217) in a new light, who established the maximin optimality of the uniform design for lack-of-fit tests in homoscedastic multiple linear regression models.