# Is the Berry-Esseen theorem useful for justifying normality?

The Kolmogorov-Smirnov (KS) test tells one how confident they can be that a sample comes from a hypothesized distribution. It is my understanding that this test can be used to justify whether or not the underlying distribution of a sample is normal. If with high confidence one can say that the underlying distribution of a data source is normal then the z- or t-interval can be applied to compute confidence intervals for the mean of the data.

My question has to do with the inverse of this problem. Say we have data with a known (non-normal) distribution and want to justify under what parameters the distribution is "normal enough" to use z- and t-intervals for the mean. One way of comparing a distribution to its normal approximation that I see a lot is to compute the Kolmogorov metric between two distributions, i.e. for a random variable $X$ and it's normal approximation $X^{\ast}\sim\mathcal{N}(\Bbb E[X],\mathrm{var}[X])$ the Kolmogorov metric is $$K(X,X^{\ast})=\sup_{x\in\Bbb R}|F_{X}(x)-F_{X^{\ast}}(x)|.$$ Note that the above is not referring to the KS statistic but rather the maximum discrepancy between two theoretical cdf's. One famous result using this approach is the Berry-Esseen theorem which puts an upper bound on the Kolmogorov metric for an arbitrary distribution and it's normal approximation.

My question is this: Is there an accepted value for the Kolmogorov metric that has been used to justify sufficient normality for the purposes of confidence intervals?

Perhaps a equivalent question would be: Does the Berry-Esseen theorem have any practical application (in an analogous way to the KS-test) for justifying normality?

• This question is ambiguous: what parameter would the confidence intervals be concerned about? The answer depends on that parameter! In light of this, one part of your question definitely can be answered: there cannot possibly be any generically "accepted value" that works for all parameters.
– whuber
Jun 16, 2018 at 17:47

No. Different confidence interval procedures have difference sensitivities to normality, and various requirements of the work can influence how much deviation from the theoretical coverage is acceptable.$$^{\dagger}$$
$$^{\dagger}$$Is $$94.5\%$$ coverage good enough for a $$95\%$$ confidence interval? Under most circumstances, I would say so, but perhaps you have an application where that is inadequate.