Are confidence intervals scale-invariant?

Question

Suppose I estimate a mean and construct some sort of confidence intervals (e.g. based on normal approximation or bootstrapped) around the mean. I now wish to rescale my mean from, say, the mean number of infections per hundred persons to the mean number of infections per thousand persons by multiplying the mean by 10.

Does multiplying the endpoints of the confidence intervals by 10 produce the correct confidence intervals?

If you write the corresponding probability statements, you should be able to see the answer. — Glen_b, Jul 27 '19 at 08:50
It a number is followed by a unit (scale), this number is not unit (scale) - invariant. — user158565, Jul 27 '19 at 16:24

Glen_b · Accepted Answer · 2019-07-29T05:24:42.370

3

If you have a scale family everything works:

If the probability that a random interval $[a,b]$ includes $\theta$ is some value $q \geq 1-\alpha$ then the probability that a random interval $[ka,kb]$ includes $k\theta$ is also $q$ (and so also $\geq 1-\alpha$).

Consequently, generally when you have units, like meters or kilograms, then you can rescale to other units (say, mm) and the CI generally carries through (scale equivariance being the apparent intent of the question).

It's when the thing is unit-free that you don't tend to get it (e.g. with a Poisson count, or a proportion, say).

edited Jul 29 '19 at 05:24

answered Jul 27 '19 at 08:59

Glen_b

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I'd be keen to hear the reason for the downvote; I'd happily try to improve my answer if I knew what the problem was. – Glen_b Jul 27 '19 at 14:32

Are confidence intervals scale-invariant?

1 Answers1