The equality of comparable extended families of classical-type and Hausman-type statistics

An extended family of statistics is defined as a set of asymptotically-equivalent quadratic forms in a given vector where each component of a weighting matrix is any consistent estimator of an appropriate matrix. Given a type of information matrix (IM) equality, two statistics are said to be comparable if both statistics are constructed with the IM-type equality imposed or both constructed with the IM-type equality not imposed. Similarly, two families are said to be comparable if all members of both families are comparable statistics. It is well known that comparable classical-type and Hausman-type statistics are asymptotically equivalent under appropriate conditions. In contrast, given the convergence in probability of appropriate quantities, this paper shows the equality of comparable extended families of classical-type and Hausman-type statistics, a result that implies (but is not implied by) the well-known asymptotic equivalences.