# Do all these estimates of kurtosis and skewness have the same (asymptotic) distribution under normal sample distribution?

I have seen five types of estimates of kurtosis and skewness:

For an iid normal sample, do all those types of estimates of kurtosis and skewness have the same distribution for each sample size, and have the same asymptotic distribution?

In tests for normality using skewness and kurtosis (such as D'Agostino's K-squared test, and Jarque–Bera test),

• does it matter which type of estimates for skewness and kurtosis are used in building the test statistics?
• Will the null distributions of the test statistics based on those types of estimates for skewness and kurtosis be the same?

Thanks!

For an iid normal sample, do all those types of estimates of kurtosis and skewness have the same distribution for each sample size,

Obviously not; some are clearly larger or smaller (you know that $n-2+1/n < n$, right?), so obviously their means differ for a start.

In the cases where you have a pair of constants out the front of the numerator and denominator that are such that you can't immediately see 'larger' or 'smaller', pull their ratio out the front of both sets of terms, cancel out any terms in common (either in the ratio, or across the two constants) and evaluate what's left at some particular small and medium values of $n$; if they differ, you'll quickly find some values of $n$ where they do.

and have the same asymptotic distribution?

Yes, asymptotically, they all go to the same thing. Slutsky takes care of denominators, CLT takes care of numerators, the only things that might be left over would be constants of the form $(1\pm k/n+o(1/n))$ out the front, and they'll all go to 1.

does it matter which type of estimates for skewness and kurtosis are used in building the test statistics?

Obviously it makes some difference since the distributions differ at finite $n$, and the tests are conducted at finite sample sizes.

But since the J-B is only justified by an asymptotic argument, you shouldn't use it for small (or even moderate) sample sizes if you want it to have close to its nominal properties. I'd suggest $n=300$ may not be enough for some people's purposes, for example - the joint distribution goes to bivariate normal more slowly even than the marginals, which aren't that quick.

See here and the four plots near the bottom here

Will the null distributions of the test statistics based on those types of estimates for skewness and kurtosis be the same?

Not at finite sample sizes, though asymptotically they all go to the same chi-square and they soon look very very similar. They'll be very alike a good deal quicker than the J-B attains something similarly alike its asymptotic distribution so by the time $n$ is big enough that you use the J-B, you don't care.

If you want to use the small-sample joint distribution of the skewness and kurtosis (as Bowman and Shenton attempted, for example, but it's possible to do better than they did), then you just use whichever definition of skewness and kurtosis was used in deriving the small sample distribution.

• Sample estimates for skewness and kurtosis have upper bounds. In some cases, this can lead to samples denying their own parentage. A tutorial review is freely available at stata-journal.com/sjpdf.html?articlenum=st0204 – Nick Cox Feb 3 '14 at 9:35