ABSTRACT
To compute we need the minimum of the quadratic form when µ=0 and S is known. In general this can be done by quadratic programming (for example see Nüesch (1966)).
A geometric interpretation of the statistic which will give us an actual picture useful for its computation as well as for the derivation of its distribution, is as follows. Since S>0, there exists a p×p nonsingular matrix A such that ASA'=I. Let
where Hence is proportional to the difference between the square length of the vector Y in the p-dimensional Euclidean space of m and the distance between a point Y and a closed convex polyhedral cone C defined by the inequalities
If then the second term in (7.81b) vanishes and we have
Complication arises if In any case there exists a vector such that i=1,…, p and
The point is the maximum likelihood estimate of µ under H1. The following
theorem gives some insight about Theorem 7.3.1. The point is the maximum likelihood estimate of µ under H1 if
and only if one of the ith components of the two vectors and is zero and the other is non-negative.
