Re: st: Testing for differences in skewness and kurtosis?

[Nick Cox <njcoxstata@gmail.com>]

This question provokes various comments from me. They are all
variations on a single theme, that this is much more problematic than
you imply.

1. Large-sample standard errors for skewness and kurtosis _if_ the
parent is normal are given in many mathematical statistics texts and
could be the basis for a test. But if the normality of the
distributions is even slightly in doubt I have the impression that
they aren't worth much. Even if it is not in doubt, the large-sample
results do not kick in quickly. (There is at least one command in
official Stata that is wildly cavalier about this point! Some
economists use the so-called Jarque-Bera test based on asymptotic
standard errors as a test for normality.)

2. This is linked to the fact that skewness and kurtosis, as dependent
on third and fourth powers of deviations from the mean, can be very
sensitive to departures of any kind. I don't know in consequence what
a robust test of skewness and kurtosis would look like; that sounds a
contradictory request to make. If you are interested in robust
comparisons, skewness and kurtosis are not where you start.

3. You sound very confident that your distributions are normal but
with real data that is at best an approximation. It is an
approximation that you can get away frequently with tests on means,
less frequently with tests on variances, and even less frequently with
higher moments. Box in Biometrika 1953 remains pertinent.

4. Let's imagine that you applied such a test. As parent normals both
have skewness 0 and kurtosis 3, a difference in skewness or kurtosis
between two samples would be likely to arise if at least one
distribution was really not normal. This is just a restatement of the
fact that if two quantities are different, they can't be equal, and so
not equal to any particular constant. So, you would need to consider
that possibility of non-normality directly. If you concluded that that
was so, you could end up undermining the results of your previous
tests.  So, I don't think you can ignore the question of whether the
samples come from the same distribution. It's an assumption behind
what you are doing, even if it is not independently interesting.

5. In some ways the most direct way to compare skewness and kurtosis
would be to bootstrap, but that would ignore the question of whether
the distributions are normal, which is a crucial assumption so far as
you are concerned. Perhaps better is to simulate normal samples with
the same sizes, means and SDs.

When you have two distributions, a -qqplot- is a restatement of _all_
the information on those distributions. It would throw light on
whether apparent non-normal skewness or kurtosis arises from
individual outliers or something more systematic.

Nick

Box, G.E.P. 1953. Non-normality and tests on variances. Biometrika 40: 318-335.

On Sat, May 5, 2012 at 3:34 PM, George Murray <george.murray16@gmail.com> wrote:

> I am currently working with a very simple dataset, with two variables,
> V0 and V1 (around 150 obs each), each normally distributed, and the
> difference of the means of the distribution of the variables are
> (statistically) different, but the standard deviations are equal. I
> would like to test whether there exists any significant difference in
> the skewness of these two variables. Can this be done through
> hypothesis testing, or is this only possible through some simulation
> technique (bootstrapping?) Is there a test that is robust to the
> aforementioned conditions? Is there an equivalent test for kurtosis?
> Is anyone aware of how this can be calculated with Stata? (And no, I
> am not trying to test whether they come from the same distribution)
>
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