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MathGroup Archive 1994

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Re: distributions

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: Re: distributions
  • From: bobk at decide.com
  • Date: Thu, 30 Dec 93 10:40:28 PST

Dear Dave, You have two separate problems.

>
>Dear MathGroup
>
>The Levy family of probability distributions is defined as
>
>Integrate[Exp[ -t^a] Cos[t x], {t, 0, Infinity}]/Pi
>
>where a is parameter and x is the random variable.
>
>In general, there is no closed-form for this integral. For the case
>a = 1, the integral becomes the Cauchy Distribution. For the case
>a = 2, the integral becomes a Gaussian distribution.
>
>A colleague and I are interested in cases for 1 2 a 2 2 as a possible
>model for distributions with long tails to the right, i.e., large
>positive skew.
>

There is a fair amount that is known for this distribution. The
distribution is related to the Pareto distribution, in the sense that the
tails eventually become indistinguishable from a Pareto if you are far
enough out. The log of the characteristic function of this 4 parameter
family is

log phi(t) = I d t - g Abs[t]^a -I b (t/Abs[t]) Tan[a Pi/2]

(parameters are a, b, d and g)

b controls the skew. Most work has been carried out for b==0 (no skew).
There are series for the density, cumulative and asymptotic results for the
cumulative when x is large.

see the following articles:
(Same Mandelbrot as that of fractals and Mandelbrot set fame)

Mandelbrot, B. (1960) The Pareto-Levy law and the distribution of income,
International Economic Review, 1, 79-106

Mandelbrot, B. (1963) The variation of certain speculative prices, Journal
of Business, University of Chicago, 36, 394-419.

Mandelbrot, B. (1967) The variation of some other speculative prices. Journal
of Business, University of Chicago, 40, 393-413.

(Fama and Roll of the efficient markets fame)

Fama, E. F. (1963) Mandelbrot and the stable Paretian hypothesis, Journal
of Business, University of Chicago, 36, 420-429.

Fama, E. F. and Roll, R. (1968) Some properties of symmetric stable 
distributions, JASA, 63, 420-429.

Mandelbrot suggests that if you are careful, you can use the characteristic
function and the FFT to numerically invert the Fourier Transform with good
success.

>For the case a = (3/2), Mma gives an analytical expression for the 
>integral that includes many Gamma functions and HypergeometricPFQ 
>functions. 

You have a different problem here. The Hypergeometrics are a truly awesome
set of special functions which provide tremendous power. They are also
notoriously unstable for calculation purposes. However, there are also
literally thousands of transformations from one set of Hypergeometrics to
another set. An expression which is "impossible" to compute in one form
will be trivial in another form once you have applied the right
tranformation.

The basic question comes down to:

"Do you have to do these computations once (essentially) or many times
(like the calculation is imbedded in a controller or software)?"

If the answer is once, you simply use Mma's extended precision for the
calculations. Try the following sequence of precision: 50 digits, 100
digits, 200 digits, ... Eventually, you will get convergence (look at the
cumulative near 0). Computer time is essentially free and if it has to be
done once who cares how inefficient the method of arriving at the result
was.

>So, now for the question. We want to Plot the PDF for x from x = 0 to
>x = 8 or so. At x = 0, the PDF is undefined (it should be 0???) but 
>the real problem is apparent numeric instability for 0 < x < 0.04 or so.
>
>The problem is that the graph produced by Plot in Mma appears to 
>oscillate from -10^30 to +10^30 -- clearly a problem.
>
>Question: Can any of you propose a method to contain these wild (and
>incorrect) oscillations near the origin???

If the answer is many times, then the problem is one of numerical analysis.
Finding the right transformation, using the FFT, asymptotics and many other
things will all come into play. This is a fairly major undertaking and I
wish you good luck.


>
>Many thanks for you help
>
>Dave
>Dave
>David E. Burmaster, Ph.D.
>Alceon Corporation
>Cambridge, MA 02238


Peace/Bob

"Predictions can only be probable. However solidly  | Bob Korsan
founded a prediction may appear to us, we are never | Decisions, Decisions!
absolutely sure that experience will not refute it."| 2136 Lyon Avenue
                      -- Henri Poincare             | Belmont, CA 
INTERNET:Bobk at Decide.COM    ...!decwrl!decide!bobk  |    94002-1639






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