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

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 , .