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Consider two parameters $\alpha$ and $\beta$ and let $C_{\alpha,\beta}$ denote the joint $95\%$ confidence region of $(\alpha,\beta)$.

Take the parameter $\delta\equiv f(\alpha,\beta)$ where $f$ is a linear function of $(\alpha,\beta)$. For example, $\delta=\alpha+\beta$.

Let $C_{\delta}$ denote the joint $95\%$ confidence region of $\delta$.

Consider the interval $$T\equiv \{c\in \mathbb{R}: \exists (a,b) \in C_{\alpha,\beta} \text{ s.t. } c=f(a,b)\}$$

Question: is $T \supseteq C_{\delta}$?

TEX
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    No. But with care, something can be said. This is essentially the same question addressed at https://stats.stackexchange.com/questions/18215. Apply that one to the parameters $(\alpha, -\beta).$ I understand implicitly you assume confidence intervals are independent, for otherwise the answer *a fortiori* is no due to the dependence. – whuber Oct 06 '20 at 22:21
  • Thanks. What if instead I consider the proper joint confidence region of $(\alpha,\beta)$ (call it $C$) and I compute $T\equiv \{c\in \mathbb{R}: \exists (a,b) \in C \text{ s.t. } c=a+b\}$? – TEX Oct 07 '20 at 07:23
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    By reparameterizing your question (use, say, parameters $(\alpha,\alpha+\beta)$) it simplifies to this: is the projection of a 95% confidence region $C$ onto one of the parameters a 95% confidence interval? In this form it is easy to check, by computing any simple example, that this is not the case. – whuber Oct 07 '20 at 12:08
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    Thanks. However, I believe that the projection of the $95\%$ confidence region onto one of the parameters should contain the $95\%$ confidence interval. This is what I mean by "conservative". For example, if I look at the classical t-test versus F-test picture for OLS, the projection of the ellipse contains the individual intervals. – TEX Oct 07 '20 at 12:39
  • +1 That's a good question. – whuber Oct 07 '20 at 12:45

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