ABSTRACT
The actual computation of the maximum likelihood estimator when is given may need successive approximations. As Kudô (1963) observed the convergence of this approximation is not fast but one can sometimes judge at a comparatively early stage of the calculation, by observing only, which of the components should be zero and which should be positive. We refer to this paper for an example concerning the computation of
Geometrically the maximum likelihood estimate is the projection of the vector along regression planes onto the positive orthant of the sample space. If one uses
the linear transformation of to a uncorrelated Y, which exists as S>0, the projection is orthogonal onto a polyhedral half cone, the affine image of the positive orthant. Thus is a vector whose components are either positive or zero. This leads to a partition of the sample space into 2p disjoint regions. Let us denote by ?k any of
the regions of the sample space with exactly k of the positive. We assume,
without any loss of generality, that the k positive are the last k components. We write
where contains the last k components of Similarly partition
Let S-1=? . Partition
with S22, ? 22 are both k×k submatrices. From Theorem 7.3.1 we get
solving we get
Using (7.81c) and (7.81d) we get
Hence
Since Theorem 7.3.1 implies
we can write
and the likelihood ratio test of H0 against H1 rejects H0 whenever where C is a constant depends on the size a of the test.