View source for Linear hypothesis

<!--
l0592801.png
$#A+1 = 36 n = 0
$#C+1 = 36 : ~/encyclopedia/old_files/data/L059/L.0509280 Linear hypothesis
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A statistical hypothesis according to which the mean  $  a $
of an  $  n $-
dimensional normal law  $  N _ {n} ( a , \sigma  ^ {2} I ) $(
where  $  I $
is the unit matrix), lying in a linear subspace  $  \Pi  ^ {s} \subset  \mathbf R  ^ {n} $
of dimension  $  s < n $, 
belongs to a linear subspace  $  \Pi  ^ {r} \subset  \Pi  ^ {s} $
of dimension  $  r < s $.

Many problems of mathematical statistics can be reduced to the problem of testing a linear hypothesis, which is often stated in the following so-called canonical form. Let  $  X = ( X _ {1} \dots X _ {n} ) $
be a normally distributed vector with independent components and let  $  {\mathsf E} X _ {i} = a _ {i} $
for  $  i = 1 \dots s $, 
$  {\mathsf E} X _ {i} = 0 $
for  $  i = s + 1 \dots n $
and  $  {\mathsf D} X _ {i} = \sigma  ^ {2} $
for  $  i = 1 \dots n $, 
where the quantities  $  a _ {1} \dots a _ {s} $
are unknown. Then the hypothesis  $  H _ {0} $, 
according to which

$$ 
a _ {1}  = \dots =  a _ {r}  =  0 ,\ \ 
r < s < n ,
$$

is the canonical linear hypothesis.

Example. Let  $  Y _ {1} \dots Y _ {n} $
and  $  Z _ {1} \dots Z _ {m} $
be  $  n + m $
independent random variables, subject to normal distributions  $  N _ {1} ( a , \sigma  ^ {2} ) $
and  $  N _ {1} ( b , \sigma  ^ {2} ) $, 
respectively, where the parameters  $  a $, 
$  b $, 
$  \sigma  ^ {2} $
are unknown. Then the hypothesis  $  H _ {0} $: 
$  a = b = 0 $
is the linear hypothesis, while a hypothesis  $  a = a _ {0} $, 
$  b = b _ {0} $
with  $  a _ {0} \neq b _ {0} $
is not linear.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>

====Comments====
However, such a linear hypothesis  $  a = a _ {0} $, 
$  b = b _ {0} $
with  $  a _ {0} \neq b _ {0} $
does correspond to a linear hypothesis concerning the means of the transformed quantities  $  Y _ {i}  ^  \prime  = Y _ {i} - a _ {0} $, 
$  Z _ {i}  ^  \prime  = Z _ {i} - b _ {0} $.