The Asymptotic Powers of Certain Tests Based on Multiple Correlations

Abstract

Summary Pitman (1948) has proposed a criterion of the relative power of two tests of an hypothesis specified by a single parameter, based on the estimating efficiencies of the two test statistics, in case these test statistics are asymptotically normal. This criterion can be immediately generalized to many cases where the two statistics have limiting distributions of the same analytical form and, in particular, enables two tests based on statistics which are asymptotically distributed as chi-square to be compared, provided the numbers of degrees of freedom are the same. This was already recognized by Andrews (1954) who showed that the efficiencies of certain rank tests for analysis of variance could be compared in this way. Bradley (1955) also used the concept of asymptotic relative efficiency in relation to rank tests based on statistics which are asymptotically distributed as chi-square. The first section of this paper is therefore merely a slight generalization of Andrew’s and Bradley’s work. The asymptotic powers of multiple correlations may then be compared since these statistics are asymptotically equivalent to statistics distributed as chi-square. In some cases which have arisen the two multiple correlations to be compared have not been based on the same numbers of regressors. In such cases the comparison requires the use of tables of non central chi-square and Pitman’s criterion is no longer directly applicable. However the distances, on which the asymptotic powers of the tests depend, can be simply expressed in terms of the covariance matrix of the regressors on the null hypothesis and in some instances clear-cut results may still be simply derived. The applications in this paper relate to previous work by the author (Hannan (1955a), (1955b), (1956)).