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# Re: st: Skewness estimates with svyset data

 From "Stas Kolenikov" To statalist@hsphsun2.harvard.edu Subject Re: st: Skewness estimates with svyset data Date Mon, 3 Nov 2008 16:01:57 -0500

```On 11/3/08, Richard Palmer-Jones <richard.palmerjones@gmail.com> wrote:
>  Could you please clarify the nlcom formula, please?
>
>  nlcom (_b[ht3] - 3*_b[ht2] * _b[ht] + 2*_b[ht])/(_b[ht2] - _b[ht] *  _b[ht])^3/2
>
>  (ht being height, ht = ht^2 and ht3 = ht^3)
>
>  I suspect my translation is not correct as the coefficients I get are
>  not close to skewness (and is it the formula for skewness?).

Uhm... I was doing this all in my head, so let me think aloud here for
another couple minutes.

So, skewness = E[(x-m)/sigma]^3 = E(x-m)^3/( E(x-m)^2 )^(3/2) = E[ x^3
- 3m x^2 + 3 x m^2 - m^3]/(E[ x^2 - 2m x + m^2])^1.5 = E[ x^3 - 3m x^2
+ 2 m^3 ] / (E[ x^2 -m^2])^1.5.

The last expression in the first set of parentheses should be
_b[ht]^3. See if that is producing sensible skewness numbers.

Note that those are asymptotic expressions. An unbiased estimator for
variance, for instance, would be 1/(n-1) (sum x^2 - mean(x)^2), rather
than 1/n times the same thing. The latter however will have smaller
MSE.

If you are checking the skewness for some 20 groups, you would need to
have some multiple testing corrections in mind. Bonferroni says,
divide your usual alpha by the number of tests -- so you would be
rejecting for alpha = 0.05/20 = 0.0025.

To Nick: yes, I've used skewness and kurtosis to test for normality a
bunch of times (and there's a famous Mardia's multivariate
generalization that I programmed up :)). But frankly I personally
don't remember seeing confidence intervals on skewness anywhere at
all. Estimation and testing are two related ways of looking at data
with statistics, but with skewness and kurtosis you really estimate
something to see that it is close enough to zero... and sometimes you
don't even estimate a thing and go straight to the test statistic.

--
Stas Kolenikov, also found at http://stas.kolenikov.name
Small print: I use this email account for mailing lists only.
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