Linear hypothesis

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

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)

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} $.