Linear hypothesis
A statistical hypothesis according to which the mean of an
-dimensional normal law
(where
is the unit matrix), lying in a linear subspace
of dimension
, belongs to a linear subspace
of dimension
.
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 be a normally distributed vector with independent components and let
for
,
for
and
for
, where the quantities
are unknown. Then the hypothesis
, according to which
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is the canonical linear hypothesis.
Example. Let and
be
independent random variables, subject to normal distributions
and
, respectively, where the parameters
,
,
are unknown. Then the hypothesis
:
is the linear hypothesis, while a hypothesis
,
with
is not linear.
References
[1] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) |
Comments
However, such a linear hypothesis ,
with
does correspond to a linear hypothesis concerning the means of the transformed quantities
,
.
Linear hypothesis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_hypothesis&oldid=47657