On the optimality and a generalization of rao-robson's statistic

Abstract

It is shown that the statistic Q_RR (K.C.Rao and D.S. Robson 1974, Moore 1977) for testing goodness of fit is asymptotically optimal for the family of local alternatives corresponding to the asymptotic reduction of the testing problem within the class of tests based on two sets of statistics: one is the set of cell counts and the other one consists of raw data mle (or any other asymtotically equivalent estimator) of the nuisance parameter θ. This optimality is analogous to that of the usual statistic Q_PF (Pearson 1900,Fisher 1924) for a different family of local alternatives within the class of tests based only on the set of cell counts. We also propose a generalized statistic Q (asymptotically equivalent to Q_RR) which would be appropriate when the mle are not easily available and some other root n-consistent estimates are used instread. The statistic Q provides an alternative to Q_RR in the same spirit as the statistic Q_DN of Dzhaparidze and Nikulin (1974) is to Q_PF Finally, the question of a practical guideline in choosing between Q_PF (or Q_DN) and Q_RR (or Q), each being optimal with respect to its own family of alternatives, is considered. An adaptive procedure is suggested which uses a preliminary test for agreement between (grouped data mle) and (raw data mle). Some examples are presented for illustration purposes.

Keywords:

• X² tests
• raw and grouped data mle
• asymptotic optimality
• linear hypothesis
• orthogonal projector
• g-inverse