Corrected p-values for tests based on estimated nuisance parameters

In some situations the asymptotic distribution of a random function T _n() that depends on a nuisance parameter is tractable when has known value. In that case it can be used as a test statistic, if suitably constructed, for some hypothesis. However, in practice, often needs to be replaced by an estimator S _n. In this paper general results are given concerning the asymptotic distribution of T _n(S _n) that include special cases previously dealt with. In particular, some situations are covered where the usual likelihood theory is nonregular and extreme values are employed to construct estimators and test statistics.