F-test

F-test for whether the true predicted value $\theta = \beta_1 x_1 + ... + \beta_k x_k $ is in a sub-vector space.

There is a null-hypothesis $H_0$: $\theta = \beta_1 x_1 + ... + \beta_k x_k $

There is atest for this null hypothesis, which involves an $F$-distribution.

Recall: $F$-distribution are ration of two $\chi^2$'s. For instance, when we divide two sample variances by each other.

Usual linear model notation:

• $Y = \text{responcse variable: }n \text{ data points}$
• $x=(x_1,...,x_k) = k\text{ observed explanatory variables each with }n\text{ data points}$

We write $\theta$ as above. Note that $k=\text{dimension of the linear space spanned by }x_1, ..., x_k$

Then, consider a smaller space $V_0$ with dimension $k_0 < k$.