Re: Distribution Fitting - Generalised Gaussian based Distribution?

From: Robert Israel (israel_at_math.ubc.ca)
Date: 10/29/04 

Date: 29 Oct 2004 18:10:09 GMT

In article <kwfgd.40310$Z14.14908@news.indigo.ie>,
Ellarco <nobrow@eircom.net> wrote:
>I wish to fit some distribution to my data ... I cant.

>My data fits well to a modified form of the Generalised Gaussian
>distribution (GG). The GG is;

>f(x) = A e ^ (-|bx| ^ c)

>where,

>b = (1/sigma) * (Gamma(3/c) / Gamma(1/c)) ^ (1/2)
>A = b c / (2Gamma(1/c))

>GG does not fit well.

>However my data can be fitted nicely to;

>f1(x) = P(m) e ^ (-|bx| ^ c)

>where,

>P(m) = probability of the modal value.

Shouldn't be a probability: it's a density. Maybe that's the problem.

>Example params are;
>b = 9.39
>c = 0.424
>P(m) = 0.2667

>Unfortunatley f1 is not a pdf!

What are you using to fit your data to f1, and why do you say it's
a good fit?

It's not a pdf because int_{-infty}^infty f1(x) dx is not 1 (with your
numbers it's about 0.162). I question how you could say your
data fit this well when the integral is so far off: e.g. if you calculate
the empirical cdf F(x) = fraction of data points that are <= x,
F(x) = 1 for x >= the maximum of your data, but int_{-infty}^x f1(t) dt
will be less than 0.162. At the median of your data, F will be
approximately 1/2 and the integral will be even less. It certainly
doesn't seem like a good fit to me.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
 

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