Conservative confidence interval for linear combination of parameters

Cross Validated Asked on December 7, 2020

Consider two parameters $$alpha$$ and $$beta$$ and let $$C_alpha$$, $$C_beta$$ be their $$95%$$ confidence intervals, respectively. Now, take the parameter $$deltaequiv f(alpha,beta)$$ where $$f$$ is a linear function of $$(alpha,beta)$$. Consider the interval $$Tequiv {cin mathbb{R}: exists (a,b) in C_alpha times C_beta text{ s.t. } c=f(a,b)}$$

1) Is the $$95%$$ confidence interval of $$delta$$ contained in $$T$$?

Instead, let $$C_{alpha,beta}$$ denote the joint $$95%$$ confidence region of $$(alpha,beta)$$. Consider the interval $$T_{joint}equiv {cin mathbb{R}: exists (a,b) in C_{alpha,beta} text{ s.t. } c=f(a,b)}$$

2) Is the $$95%$$ confidence interval of $$delta$$ contained in $$T_{joint}$$?

Remark: here I’m not assuming that the $$C_alpha$$, $$C_beta$$ come from two independent samples. They could be obtained using the same sample, or two dependent samples.

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