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 $Y _ {i} ^ \prime = Y _ {i} - a _ {0}$, $Z _ {i} ^ \prime = Z _ {i} - b _ {0}$.