Talk:Levy skew alpha-stable distribution

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[edit] Stable?

Nice article...

Are these the same as stable distributions? Clearly, they have the addition property, but are all "stable" distributions also "Levy skew alpha" stable d's? If so, material of stable distributions could be included - as an easy intro --J heisenberg 11:01, 21 Feb 2005 (UTC)

Hello - Yes, all stable distributions are included in what is being called here "Levy skew alpha" stable distributions. Some authors call them just stable distributions and call the case for alpha=3/2 the Levy distribution. Check out Nolan's web page. We need to get some plots of the skew distributions here too. Paul Reiser 14:22, 21 Feb 2005 (UTC)

I've included the stuff from the article above. Feel free to edit--J heisenberg 19:18, 21 Feb 2005 (UTC)

"yielding a distribution symmetric about c" <-- surely "...about mu"? -- Right. Its fixed, thanks. PAR 02:04, 4 August 2005 (UTC)

"Why if μ is the mode of the distribution, in the plot named 'Skewed centered Lévy distributions with unit scale factor' the x=0 is not the maximum?" dsalas

Good question - I fixed it. PAR 17:08, 24 September 2005 (UTC)

[edit] sign?

Can someone please explain to me what sign(t) means in the expresion for \varphi (t) Thanks in Advance.

Its just the sign of t, +1 for positive, -1 for negative, 0 for 0. I fixed it so it links to the sign function. PAR 20:19, 18 December 2005 (UTC)


[edit] References

This is a nice article to a topic where there's not too much readable information on the web. It would benefit from additional references, stating clearly which book or article the individual results are taken from. 134.155.68.246 11:04, 23 December 2005 (UTC)

I agree - I added explicit references that I am familiar with. PAR 16:09, 23 December 2005 (UTC)

[edit] Skewness

The skewing factor is indeed \beta tg({\pi \over 2}\alpha), so the skewness is limited through α, even if β is always in the range [0,1].

I am not sure if this deserves an other plot

(β=const>0, α=1->2), but perhaps it should be noted explicitely. al

[edit] Cumulative function

Is there an expression for the cumulative function when beta=0? The sidebar says that it usually isn't analytically expressible, see the text; but it isn't discussed in the text. Bubba73 (talk), 22:21, 21 June 2006 (UTC)

I think that was referring to the special cases when the distribution becomes a normal distribution (β has no effect) and the Cauchy distribution (β =0). PAR 10:36, 12 November 2006 (UTC)

[edit] Some problems of the regression of Levy distribution

I've learnt about some skills of regressing the Levy tail distribution. They are powerlaw fit (double log fit), Hill estimator(and its variations), and empirical function approximation. I wonder if there is any else method, and which one is better, why? Is there any idea about this issue? Thanks.

[edit] Confusion

 All or part of this article may be confusing or unclear.
Please help clarify the article. Suggestions may be on the talk page. (December 2006)

OK, I've taken probability classes at MIT no less, and I'm unable to figure out what a Levy distribution is after reading this article. I could probably do so after reading it a few more times, but this seems to be an indication that it is not well written for a broad technical audience. The introduction is particularly opaque. It uses α without giving a formula in which α is used, or any definition of what it might mean. "have the same distribution as the original." is a confusing phrase. What is "the original"? Is this saying that X1 and X2 are the same, and that the linear sum of the two is the same as both X1 and X2? The Y equations seem to indicate that the sum is merely of the same form. Or is that the same thing after normalizing to a zero-to-one scale? The graphics are also a little confusing, perhaps because the formula has not yet been given (reading the article top to bottom generally). At first I thought this was an example of multiple distributions being added, but I infer now it's just a bunch of sample functions that satisfy the distribution formula. It would help moving these lower so they appear after the forumla is given, but it would likely help even more showing sample X1 and X2 probability functions - one set that satisfies the additive constraint, and perhaps one that does not? -- Beland 02:52, 6 December 2006 (UTC)

Well, I took a try at fixing it. Is this an improvement? PAR 06:49, 6 December 2006 (UTC)

[edit] the range of alpha

in the text, it says [0,2], in the table it says (0,2). Is it very trivial, or some correction is needed? Jackzhp 02:04, 13 July 2007 (UTC)

I don't see where it says [0,2], but please change it if it does. It should be (0,2]. The distribution becomes a normal distribution when α=2 and in the limit of α=0 it becomes a Dirac delta function, which is undefined at t=0. PAR 03:50, 13 July 2007 (UTC)

[edit] Series representation of the pdf

As far as I can see the formula for the series representation only makes sense when \alpha\neq 1, otherwise, the Φ term contains the variable t, so the integration would not produce the formula stated. Am I missing something? —Preceding unsigned comment added by 192.193.245.16 (talk) 19:17, 17 January 2008 (UTC)

As far as I can see, you can't be missing something! The variable of integration — dt — for a definite integral can't possibly be a variable — t — in the result! Especially as it's singular at one end of the range of integration.
Adding to the confusion: it seems the dt got dropped from the integral when the Taylor series was introduced:
f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[\int_0^\infty e^{it(x-\mu)}\sum_{n=0}^\infty\frac{(-qt^\alpha)^n}{n!}\right]dt\Leftarrow (missing\ dt)
While it's not all that unclear, we should be consistent in using proper notation.
Bob Kerns (talk) 12:27, 20 January 2008 (UTC)

[edit] Improving the plots

Suggestion: It would be easier to grasp the paired plots, if the one curve in common between each pair, α = 0.5, β = 0.0, shared the same color. I'd suggest reversing the color sequence in the second plot, because the red tends to stand out best, (at least to those of us who are not color-blind). Making that one heavier would also help, and make it more accessible to those with red-green color-blindness.

Basically, the second plot takes the one curve from the first plot, and skews it. But to detect this, you have to search the curves for one of the same shape in both, and then verify your discovery by checking the key.

Ideally, the only color in common would be the curve in common. Perhaps differentiate the skew curves by decreasing saturation or brightness with increasing skew (i.e. fade to gray or black). Bob Kerns (talk) 11:46, 20 January 2008 (UTC)

[edit] Characteristic function error

Found and corrected an error in the characteristic function, the (|ct|2) term needed to be (|ct|2)/2; see history -> Nowaket. However, the symmetric probability density plots (upper right corner) were generated by the Fourier transform of the incorrect characteristic function. Clearly the apex of a zero mean, variance 1, Gaussian (α = 2) PDF should be ~0.4 not <0.3. Making this change corrects the PDF's.

Nowaket (talk) 13:57, 28 March 2008 (UTC)

No - please read the article before editing it. For c=1, the variance is σ2 = 2c2 and so the peak is at 1/\sigma\sqrt{2\pi}=0.28... Also, please leave comments at the bottom of the talk page, not the top. PAR (talk) 18:12, 28 March 2008 (UTC)