The Berry-Essen theorem uses Kolmogorov-Smirnov distance to measure similarity to Gaussian—what’s the maximum difference between the CDF of the two distributions across all values of x?
As this measure is on absolute difference rather than fractional difference it doesn’t really care about the tails and so skew is the main thing stopping this measure approaching Gaussian. In this case the theorem says error reduces with root n.
From other comments it seems skew isn’t the best measure for getting kurtosis similar to a Gaussian, rather kurtosis (and variance) of the initial function(s) is a better predictor and skew only effects it inasmuch as skew and kurtosis/variance are correlated.
The Berry-Essen theorem uses Kolmogorov-Smirnov distance to measure similarity to Gaussian—what’s the maximum difference between the CDF of the two distributions across all values of x?
As this measure is on absolute difference rather than fractional difference it doesn’t really care about the tails and so skew is the main thing stopping this measure approaching Gaussian. In this case the theorem says error reduces with root n.
From other comments it seems skew isn’t the best measure for getting kurtosis similar to a Gaussian, rather kurtosis (and variance) of the initial function(s) is a better predictor and skew only effects it inasmuch as skew and kurtosis/variance are correlated.
Great theorem! Altho note that it’s “Esseen” not “Essen”.
Ha, I don’t know how many times I have read that in the last couple of days and completely failed to notice!
I think this is a very useful measure for practical applications.