# #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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