How to derive test statistic under composite null hypothesis?

Suppose I have a multiple linear regression of the form:

$$y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} +\epsilon$$

Then suppose I am interested in testing this set of hypotheses:

$$H_0: \beta_1 = 0 \text{ OR } \beta_2 = 0$$ $$H_A: \beta_1 \neq 0 \text{ AND } \beta_2 \neq 0$$

I am interested in deriving a test statistic for this hypothesis. I understand that I need to derive this under the null hypothesis. However, it is not as straightforward as just plugging in $\beta_1 = 0 \text{ and } \beta_2 = 0$, since our null is that either/or is zero, not both.

Any input would be much appreciated, thank you!

Solution 1:

Your null hypothesis is equivalent to $$ H_0: \min \{ |\beta_1|, |\beta_2| \}= 0 $$

In general, you are interested in the following problem: $$ H_0 : \theta \in \bigcup_{i=1}^k \Theta_i, \qquad \text{vs.} \qquad H_1 : \theta \in \bigcap_{i=1}^k \Theta_i^C $$

If we let $R_i, i=1,\dots,k$ be the rejection regions for the local tests $H_{i0}: \theta \in \Theta_i$ versus $H_{i1}=\theta \in \Theta_i^C$, then (under some conditions) the optimal test to use is the IUT (Intersection Union Test), which states that the rejection region of the global test is the one with $R = \cup_{i=1}^k R_i$.

This is covered in the following paper, see section 1.3 for a description of a very similar problem to yours.

So to perform the test in your question, the approach is to consider the individual tests:

$$ T_1: \quad H_{10}: \beta_1 = 0 \qquad \text{vs.}\qquad H_{11}: \beta_1 \neq 0, $$ $$ T_2: \quad H_{20}: \beta_2 = 0 \qquad \text{vs.}\qquad H_{21}: \beta_2 \neq 0, $$

Find a rejection region for each test using the standard theory, then take an intersection to find a rejection region for the global test.