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https://issues.apache.org/jira/browse/MATH-460?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13263984#comment-13263984
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Andrew Waterman commented on MATH-460:
--------------------------------------
Hi Sebastian,
I've been working with a local mathematician, and we've been able to resolve a
few of the tests. It looks like the mean and standard deviation for Levy should
return +∞. From my friend (in response to a few questions):
(1) According to the summary of properties of the Lévy distrib given in the
right-hand margin of the Wikipedia article
(https://en.wikipedia.org/wiki/L%C3%A9vy_distribution), the mean for the Lévy
distrib is indeed +∞.
(2) If you search the article https://en.wikipedia.org/wiki/Standard_deviation
for the word "variance", you'll find several places that say that the standard
deviation is the square root of the variance. Since the summary mentioned in
(1) says that the variance of the Lévy distrib is +∞, its standard deviation
should be +∞ as well. (just as Mathematica says). From what I recall of
statistics and probability, that conclusion is consistent with the statement in
the article mentioned in (1), that the formula for the moments of a Levy
distrib "diverges for all n > 0 so that the moments of the Lévy distribution do
not exist." (The variance is the second moment [n=2].)
The GNU Scientific Library has an implementation of Levy for "alpha-stable
distributions" and for "skew alpha-stable Distribution" which might provide
some insight and provides something to test against and compare to:
[http://bzr.savannah.gnu.org/lh/gsl/trunk/annotate/head:/randist/levy.c]
I'll continue to discuss with my friend, but I'm still not quite sure how to
implement the failed tests for:
testQuantiles
testCumulativeProbabilities
testInverseCumulativeProbabilities
testConsistency
testSampling
Any feedback and insight/suggestions you might want to pass on would be very
helpful.
> Levy Distribution
> -----------------
>
> Key: MATH-460
> URL: https://issues.apache.org/jira/browse/MATH-460
> Project: Commons Math
> Issue Type: New Feature
> Reporter: Pavel Ryzhov
> Priority: Minor
> Fix For: 3.1
>
> Attachments: levy_math_460.patch
>
>
> Pretty straightforward implementation of Levy Distribution (not Levy
> alpha-stable) according to http://en.wikipedia.org/wiki/Lévy_distribution.
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