COc..saNI- INSTITUTE -^-.
so that for the sample values the mean vector of x-. is
equal to zero. Then
M = I
where M is the moment matrix of x. , or the matrix of the
second order moment of x. around the mean.
For simplicity of notation^ we shall omit the tilde of
(2.9) and it immediately follows that the ABC solution is
determined so as to minimize x-Im x-, under the condition
that Pq + P-iX-, = c. I.e.,
(2.10) X = —^—K .
where Y = p^^ is the mean of Y values in the sample. If the
mean vector of x-, is not equal to 0, then x-, in (2.10) should
be replaced for x-, - x-. , where x-, is the mean vector.
Similarly, for the AB function, x-, is determined as
(2.11) X = X +:^^^ ^
3. Exact Formula for the First Two Moments of the Error
So far the justification for particular fnnctions is
given in asymptotic terms. In this section we shall consider
the finite sample property of the procedures of the previous
We shall consider the following quantity
p'Mp + a^
for an ABC function a = in (3.1) and for an AB function
a = a . More generally, we shall consider the case where
a - ka and k is a positive constant.
Without loss of generality, we can assume that c = 1,
and we consider the first two moments of the error, i.e.,
E(p'x-l) and E(p'x-l)^ .
For the moment we shall assume that a is a constant,
which corresponds to the case when a is known. Under the
normality assumption, it is well known that p'Mp is distributed
2 A /^ , 2
normally with mean p ' Mp and variance p'Mpa . Also p'Mp/a xs
distributed according to a non-central chi-square distribution
with p degrees of freedom and with the noncentrality parameter
equal to t = p'Mp/a .
Let us define