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erstwhile
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Joined: March 3rd, 2003, 3:18 pm

### Connection between alpha stable distributions and extreme value distributions

Alpha stable distributions are either gaussian or have power law like tails, asymptotically. Sometimes tails will be described as Pareto-like. The definition of alpha stable distributions comes from properties of sums of numbers and there is a sort of generalised central limit theorem that is central two the definition: one refers to the maximumdomain of attraction.Extreme value theory is based on the properties of maxima or minima of distributions: there is a version of the central limit theorem that refers to maxima instead of sum or average, and there is a maximum domain of attraction argument in classifying how the extreme tails of a given distribution converge to one of three extreme value distributions.Surely there must be a connection here? A given alpha stable distribution must have a fairly direct connection to a given extreme value distribution right?

erstwhile
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Joined: March 3rd, 2003, 3:18 pm

### Connection between alpha stable distributions and extreme value distributions

Intuitively infinite variance really means that the longer your data sample the higher the variance. The Cauchy (Lorentz) distribution has infinite variance but has a well defined distribution of its maxima - I think it becomes the Gumbel distribution in the deep tails though.

erstwhile
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Joined: March 3rd, 2003, 3:18 pm

### Connection between alpha stable distributions and extreme value distributions

Yes - I reckoned one of you folks here would know the answer - or maybe there isn't a straightforward connection...

Alan
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### Connection between alpha stable distributions and extreme value distributions

QuoteOriginally posted by: erstwhileAlpha stable distributions are either gaussian or have power law like tails, asymptotically. Sometimes tails will be described as Pareto-like. The definition of alpha stable distributions comes from properties of sums of numbers and there is a sort of generalised central limit theorem that is central two the definition: one refers to the maximumdomain of attraction.Extreme value theory is based on the properties of maxima or minima of distributions: there is a version of the central limit theorem that refers to maxima instead of sum or average, and there is a maximum domain of attraction argument in classifying how the extreme tails of a given distribution converge to one of three extreme value distributions.Surely there must be a connection here? A given alpha stable distribution must have a fairly direct connection to a given extreme value distribution right?Interesting question.Just thinking out loud, and looking up some definitions, you can see some *very* close connections (pre-rescaling limit). (http://en.wikipedia.org/wiki/Extreme_value_theory)Extreme value distributions are distributions of the minimum or maximum of lists{X_1,X_2,...,X_n} of n IID draws of some rv, call it X.Start with the distribution of the minimum M_n for a rv with positive support:M_n = min {X_1,X_2,...,X_n} This has complementary distribution function, if you think about it a little,(*) P(M_n >= x ) = P(X_1 >= x, X_2 >= x, ..., X_n >= x) = F(x)^n,where F(x) = P(X >= x) = complementary distribution function of X.On the other hand, consider the Laplace transform of the sum S_n of n IID draws of some other r.v. Y with positive support.S_n = Y_1 + Y_2 + ... + Y_n This sum has LT with respect to x(**) E[e^(- x S_n)] = G(x)^n where G(x) = int(0,infty) e^(-x y) p_Y(y) dy is the LT of the density of Y.Look at (*) vs (**). Can G(x) be intrepreted as a complementary distribution function?Yes! We have G(0) = 1, G(infty) = 0, G(x) >= 0, and G(x) is monotone decreasing in x.These are the properties that characterize complementary distribution functions F(x) ![Again, this is all for a r.v. supported on (0,infty)]. So (*) and (**) associate two distributions with positive support. In other words, the LT of the position of the random walk S_n, generated by draws of Y, can be interpreted as the complementary distribution function of the minimum of n draws of a corresponding rv, X. The correspondence X <==> Y is that X has a complementary distribution function equal to the LT of the density of Y. So, say given a distribution of X you could get the density of the associated Y by a Laplace inversion.This is for arbitrary distributions and arbitrary n = 1, 2, ...As for the rescaling limit n -> infty, alpha-stable, etc, I'm not sure; will stop here.
Last edited by Alan on March 29th, 2013, 11:00 pm, edited 1 time in total.

Alan
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### Connection between alpha stable distributions and extreme value distributions

Thanks.Some alpha stable <==> extreme value connections follow pretty immediately from what I wrote, as follows.Remember my stuff requires positive support.Let Y be distributed one-sided alpha-stable, which has positive support. The LT of the density (http://arxiv.org/pdf/1007.0193v3.pdf) has a simple formula: G(x) = int(0,infty) e^(-x y) p_Y(y) dy = exp(-x^alpha), for 0 < alpha < 1So, this alpha-stable Y is associated to another r.v. X, by what I wrote, namely the one with complementary distribution function P(X >= x) = exp(-x^alpha). That means the density of X is (1) p_X(x) = alpha x^(alpha-1) exp(-x^alpha), 0 < alpha < 1Now let Z be a Frechet distibuted variate.The Frechet distribution (http://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution)is one of the possible extreme value densities, in the rescaling limit, for a maximum.It has the density (2) p_Frechet(z) = beta z^(-1 - beta) exp{-z^(-beta)}, 0 < beta < infty.Notations: I am using parameter beta here instead of conventional Frechet alpha to distinguish (at least initially) from the alpha of (1). The density variable z is used for the realizations of Z. Since the Frechet is associated with maxima, we flip the IID list of maxima to minima by reciprocals:max(Z_1,Z_2, ...., Z_n) = [min (X_1,X_2,....,X_n)]^(-1), where X_i = 1/Z_i. So, if Z is Frechet distributed, then we should consider the distribution of X = 1/Z to get minima. Call the distribution of X the associated-Frechet (I just made that name up, so don't know if that is a convention, but it fits).From (2), the density of the associated-Frechet will be(3) p_AssocFrechet(x) = beta x^{beta-1} exp(-x^beta), 0 < beta < inftyCompare (1) and (3). You can see they are the same function,now identifying beta = alpha, at least in the parameter overlap range 0 < alpha < 1. So, to summarize, here's how it works for this example: The one-sided alpha-stable distribution is indeed connected to an extreme value distribution for a minimum, namely the associated-Frechet distribution, verifying erstwhile's intuition.Here's the connection, again, in words:First, an "associated-Frechet variate with parameter alpha" is definedto be the reciprocal of the Frechet variate with parameter alpha.Then, the connection is that the Laplace transform of the density of the one-sided alpha-stable variate with parameter alpha is identical to the complementary distribution function of an associated-Frechet variate with the same parameter alpha. The next step would be to try to directly relate some other cases, but not today ...
Last edited by Alan on March 29th, 2013, 11:00 pm, edited 1 time in total.

erstwhile
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Joined: March 3rd, 2003, 3:18 pm

### Connection between alpha stable distributions and extreme value distributions

Alan that was impressive, thanks - I need to let this connection sink in now...