3

To make things concrete, take a simple linear model:

$$E[y \vert x_1, x_2] = \alpha + \beta_1 x_1 +\beta_2 x_2$$

In a one-sided hypothesis test, like $\beta_1 \ge k$ vs. $\beta_1 \lt k$, the confidence interval for $\beta_1$ is unbounded on one side: $[a,+\infty)$ rather than $[a,b]$ (or bounded by the limit of the sample space on one side, like $[a,1]$ with a binary outcome).

If you are interested in the compound null like $\beta_1 \ne 0$ and $\beta_2 \ne 0$, the confidence region now looks like an ellipse.

What does that region look like if you have a one-sided confidence interval for the nulls $\beta_1 \ge k$ and $\beta_2 \le m$?

How does that change if only one of the two hypotheses is one-sided?

dimitriy
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  • I upvoted (earlier) and reopened (now). To see there's an issue here, consider what this "ellipse" looks like when you ignore (say) the condition on $\beta_2$ in $H_0:$ the confidence region is a half-plane. Although that's not dispositive -- a half-plane could be seen as a limit of ellipses -- it provides a little intuition concerning why this confidence region is unlikely to be a true ellipse or even a portion of one. (That's a hint to its solution, too.) – whuber Mar 26 '24 at 18:04
  • @whuber I am not sure I understand what it means to ignore a condition. In my linked example, the one-sided 95% CI for $\beta_1 = \beta_{triceps} \ge 0$, I get $(-\infty, 9.598815]$. If I ignore the other coefficient, is the half-plane boundary the vertical line through the upper bound of that interval? – dimitriy Mar 26 '24 at 19:12

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