The “or any other measure of spreadout-ness” can be dropped here
What I meant is that, if you restrict attention to normal distributions with a fixed mean, then any reasonable measure of how spread out it is (including any of the E[|x-mean|^p]) will be a sufficient statistic, because any such measure, in order to be reasonable, must increase as variance increases (for normal distributions), so this function can be inverted to recover the variance. In other words, any other such measure will indeed be isomorphic to variance when restricted to normal distributions.
The value of m minimizing E[|X-m|] should change if I decrease the minimum X-value a lot, while leaving everything else constant
This does not change the minimizer of E[|X-m|] because it increases E[|X-m|] by the same amount for every m>min(X).
In general, you can’t decrease E[|X-m|] by moving m from median to median-d for d>0 because, for x≥median (half the distribution), you increase |X-m| by d, and for the other half, you decrease |X-m| by at most d.
“Any other such measure will indeed be isomorphic to variance when restricted to normal distributions.”
It’s true, but you should not restrict to normal distributions in this context. It is possible to find some distributions X1 and X2 with different variances but same value E(|x-mean|^p) for p≠2. Then X1 and X2 looks the same to this p-variance, but their normalized sample average will converge to different normal distributions. Hence variance is indeed the right and only measure of spreadout-ness to consider when applying the central limit theorem.
That’s exactly what I was trying to say, not a disagreement with it. The only step where I claimed all reasonable ways of measuring spreadout-ness agree was on the result you get after summing up a large number of iid random variables, not the random variables that were being summed up.
What I meant is that, if you restrict attention to normal distributions with a fixed mean, then any reasonable measure of how spread out it is (including any of the E[|x-mean|^p]) will be a sufficient statistic, because any such measure, in order to be reasonable, must increase as variance increases (for normal distributions), so this function can be inverted to recover the variance. In other words, any other such measure will indeed be isomorphic to variance when restricted to normal distributions.
This does not change the minimizer of E[|X-m|] because it increases E[|X-m|] by the same amount for every m>min(X).
In general, you can’t decrease E[|X-m|] by moving m from median to median-d for d>0 because, for x≥median (half the distribution), you increase |X-m| by d, and for the other half, you decrease |X-m| by at most d.
I don’t agree with the argument on the variance :
“Any other such measure will indeed be isomorphic to variance when restricted to normal distributions.”
It’s true, but you should not restrict to normal distributions in this context. It is possible to find some distributions X1 and X2 with different variances but same value E(|x-mean|^p) for p≠2. Then X1 and X2 looks the same to this p-variance, but their normalized sample average will converge to different normal distributions. Hence variance is indeed the right and only measure of spreadout-ness to consider when applying the central limit theorem.
That’s exactly what I was trying to say, not a disagreement with it. The only step where I claimed all reasonable ways of measuring spreadout-ness agree was on the result you get after summing up a large number of iid random variables, not the random variables that were being summed up.
Ah, these make sense. Thanks.