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Science Forum Index » Statistics - Math Forum » testing skewness or symmetry
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| Brett |
 Posted: Sun Mar 25, 2007 3:42 pm |
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Hi everyone
Is there a way to test whether a given sample follows a symmetric distribution? Please give me some tips if you know .
Bret |
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| Jack Tomsky |
| Posted: Sun Mar 25, 2007 5:30 pm |
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Quote: Hi everyone
Is there a way to test whether a given sample follows
a symmetric distribution? Please give me some tips if
you know .
Bret
Here's a simple nonparametric test for symmetry.
Calculate the first order differences: D(i) = x(i+1)-x(i). Then compare if D(i) > D(n-i) for i = 1, ..., n/2, and do a sign test on the count.
Jack |
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| Richard Ulrich |
| Posted: Sun Mar 25, 2007 9:49 pm |
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On Sun, 25 Mar 2007 21:42:11 EDT, Brett <bret2007@gmail.com> wrote:
Quote: Hi everyone
Is there a way to test whether a given sample follows a symmetric distribution? Please give me some tips if you know .
If it is symmetrical, then half the numbers will be above
the mean, and half of them below. So you can test for
a binomial at 50%. Or, if you were interested in tails,
you could compare counts beyond some stated distance
from the mean.
The most common test is probably on the third central
moment - skewness. I think that the usual standard
error assumes normality.
--
Rich Ulrich, wpilib@pitt.edu
http://www.pitt.edu/~wpilib/index.html |
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| Brett |
| Posted: Mon Mar 26, 2007 3:00 am |
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Thanks to Richard and Jack.
I figured that I could probably do this using wilcoxon sign rank test. So if the wilcoxon test fails in rejecting the null hypothesis of the symmetric distribution of median around zero, that means data is symmetric. Please correct me if there is anything wrong in my understanding.
I am applying a permutation testin my experimental analysis, and a reviewer has questioned the validity of the symmetry assumption that is needed in permutation test. I hope if I can demonstrate that the data is symmtrically distributed around zero median (which is equal to mean if the null hypotheis is true), I can justify the condition of symmetry in my permutation test. Do you agree with me?
Bret |
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| Jack Tomsky |
| Posted: Mon Mar 26, 2007 5:02 pm |
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Quote: On Mar 25, 8:30 pm, Jack Tomsky
jtom...@ix.netcom.com> wrote:
Here's a simple nonparametric test for symmetry.
Calculate the first order differences: D(i) =
x(i+1)-x(i).
Then compare if D(i) > D(n-i) for i = 1, ..., n/2,
and do a sign test on the count.
Jack
I've Monte Carlo'ed this for a few distributions. For
"reasonable"
distributions (e.g., uniform, normal, logistic), the
actual variance
of the count seems to be a little less than the
nominal variance.
For distributions with very long tails (e.g., Cauchy)
or with no tails
(e.g., X = U^b/(U^b + (1-U)^b), where U ~
Uniform[0,1] and b > 1),
the actual variance is greater than the nominal
variance, sometimes
much greater.
Thanks, Ray. It's interesting that sgn(D(i)-D(n-i+1)) for i=1, ..., n/2, are not independent for all symmetric distributions. What looked like a good idea at first glance doesn't seem to work too well in practice.
Jack |
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| Ray Koopman |
| Posted: Mon Mar 26, 2007 7:16 pm |
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On Mar 25, 8:30 pm, Jack Tomsky <jtom...@ix.netcom.com> wrote:
Quote: Here's a simple nonparametric test for symmetry.
Calculate the first order differences: D(i) = x(i+1)-x(i).
Then compare if D(i) > D(n-i) for i = 1, ..., n/2,
and do a sign test on the count.
Jack
I've Monte Carlo'ed this for a few distributions. For "reasonable"
distributions (e.g., uniform, normal, logistic), the actual variance
of the count seems to be a little less than the nominal variance.
For distributions with very long tails (e.g., Cauchy) or with no tails
(e.g., X = U^b/(U^b + (1-U)^b), where U ~ Uniform[0,1] and b > 1),
the actual variance is greater than the nominal variance, sometimes
much greater. |
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| Luis A. Afonso |
| Posted: Tue Mar 27, 2007 3:28 am |
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Brett
Since Mar 6, 2007, 7:33 AM the critical values of the Pearson’s Skewness Coefficient I found were posted, concerning 10(5)100 sample sizes, and the problem is solved. Further discussions on the matter are absolutely worthless.
The reason that the 3rd central moment is inadequate (Richard Ulrich) to test Skewness is simple: it depends on variance (or sd, standard deviation). The division by var^(3/2) (or by sd^3) leads it to be invariant whatever the dispersion.
_______licas (Luis A. Afonso) |
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| Michel PETITJEAN |
| Posted: Tue Mar 27, 2007 12:20 pm |
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Newsgroups: sci.stat.math
Subject: Re: testing skewness or symmetry
On Sun, 25 Mar 2007 21:42:11 EDT, Brett <bret2007@gmail.com> wrote:
Quote: Hi everyone
Is there a way to test whether a given sample follows a symmetric distribution? Please give me some tips if you know .
The chiral index may be used to test symmetry
(see my web page with references).
It is such that \chi = 0 <=> distribution is symmetric
(not true for the skewness).
It assumes the the variance exists, not the 3-rd order moment.
For a sample, it is at least as simple to compute than the skewness.
Its distribution has to be simulated for various parent laws
(easy).
Michel Petitjean, Email: petitjean@itodys.jussieu.fr
ITODYS (CNRS, UMR 7086) ptitjean@ccr.jussieu.fr
1 rue Guy de la Brosse Phone: +33 (0)1 44 27 48 57
75005 Paris, France. FAX : +33 (0)1 44 27 68 14
http://petitjeanmichel.free.fr/itoweb.petitjean.skewness.html |
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