ABSTRACT
Corresponding to the transformation on the sufficient statistic (S, W1, W2) and S are given by
where is the induced transformation on the sufficient statistic corresponding to A maximal invariant in the parameter space is given by
Since the power function of any invariant test is constant on each orbit of the parametric space of S, there is no loss of generality in working on a class of its representatives instead of working on the original parametric space. Let
with D=(0, 0,…, 0, 1)'. The set consists of a class of representatives for the orbits of the parameter space and ?(A(d))=d2. We will show in Theorem 8.10.2 below that the locally best invariant test, of H0: d=0 against H1: d=? as ?? 0, rejects H0 whenever
is large. Note that the statistic is the product of two factors. The first factor is
equivalent to the likelihood ratio test statistic and it essentially measures the multiple correlation between X1 and X3 after removing the effect of X2, where
with Xi n×pi submatrices. The second factor is the ratio of two estimates of S11. The additional data is used to get an improved estimator in the denominator. Giri’s test (Giri, 1979) has n1=0, W1=0. The second factor provides a measure a of orthogonality with X1 and the columns of X2. The fact that this test is locally most powerful suggests that as X1 becomes more nearly orthogonal to the
columns of X2, the first factor becomes more effective in detecting near-zero
correlation. The test does not involve Y2. In this context we note that Giri’s test
uses X2 only through the projection matrix which contains no information on S22. It is not surprising that additional information on S22 is ignored.
