Confidence intervals that all hold at once

Question

You report confidence intervals for the mean returns of 4 strategies, and you want to be 95% confident that all four intervals simultaneously contain their true means (not just each one separately).

Using the Bonferroni idea, what confidence level must each individual interval carry?

Accepted Answer

The subtlety of "all at once." A single 95% interval misses its target 5% of the time. If you build four independent 95% intervals, the chance that at least one misses is much bigger than 5%, so the four together are far less than 95% reliable. To make the joint statement 95% confident, each interval must be individually stronger. Spread the error budget. Bonferroni splits the total 5% miss budget across the intervals. With m intervals and a joint miss budget \alpha, each interval is allowed only \alpha/m chance of missing: 	ext{per-interval miss} = \frac{\alpha}{m} = \frac{0.05}{4} = 0.0125…

Confidence intervals that all hold at once

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