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Linear hypothesis

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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 , .

How to Cite This Entry:
Linear hypothesis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_hypothesis&oldid=47657
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article