#StackBounty: #confidence-interval Conservative confidence interval for linear combination of parameters

Bounty: 50

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

Take the parameter $deltaequiv f(alpha,beta)$ 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 $$Tequiv {cin mathbb{R}: exists (a,b) in C_{alpha,beta} text{ s.t. } c=f(a,b)}$$

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


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