Eight intervals, jointly 95% confident

Question

You report confidence intervals for 8 coefficients in a model and want to be 95% confident that all eight intervals simultaneously contain their true values.

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

Accepted Answer

Each must tighten. Eight separate 95% intervals miss too often as a group, so the joint "all eight hold" statement needs each interval to be stronger. Divide the budget. Bonferroni gives each interval a miss chance of \frac{\alpha}{m} = \frac{0.05}{8} = 0.00625, so each must carry confidence 1 - 0.00625 = 0.99375 = 99.375\%. Check. Each interval misses at most 0.625% of the time; across eight the worst-case total is 8 	imes 0.00625 = 0.05, so all eight hold together at least 95% of the time. Cost in width. A 99.375% interval uses a normal critical value near 2.73 (versus 1.96 for 95%), so ea…

Eight intervals, jointly 95% confident

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