Testing problems with nuisance parameters: Linear models under non-classical assumptions

Most testing problems involve a multidimensional parameter, only one component of which characterizes the hypotheses. In a mathematically strong sense, there are only two methods to cope with the remaining component, the so called nuisance parameter. Firstly, conditioning on a sufficient and complete statistic and, secondly, reduction by invariance. Both methods require strong assumptions on the underlying class of distributions. Therefore the local asymptotic approach is reviewed, which extends Neyman’s theory of C(α)-tests and yields, in a unified way, approximate optimal test statistics for smoothly indexed families. That device is applied to non-normal heteroscedastic linear models.