3. ALTERNATIVE HYPOTHESES 
The least square adjustment is based on the null hypothesis. 
Ex /H) = 8 
An alternative hypothesis can be formulated as 
E[x /H.) - xl. uxt 
where E(.) stands for the expectation. 
The hypothesis assumes a translation of the probability distribution of 
the variates x" in the observation space. 
The translation vector vxt can also be written as 
vxls cly 
where 
ct = A specific direction vector, in the observation space on which the 
translation of the probability distribution is assumed to take 
place. 
V : À parameter, which indicates the orientation and magnitude of the 
translation -»o « V «4e, 
. 
Let us consider a system of m linearly independent vectors cl k,l = 
D]...,m, in the e;-space. These vectors may be considered to be the 
direction vectors of m alternative hypotheses H, : 
k 
The linear manifold formed with the system of the m vectors , can be 
Seen as a composite alternative hypothesis H, which assumes a tranla- 
tion of the probability distribution of the variates x" in the subspace 
spanned by the vectors a Hence we can write 
E{x /H} = & + c: vH 
-— « Vl « qo kl = 10V... , m° (3,1) 
In the sequel a decision rule will be found for testing H, against a. 
3.1 The test quantity 
i . 
The vectors e expressed in the e,-space, are 
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