Re: Goodness of fit measures for a distribution


Unknown wrote:
> Hi everyone,
> I have a basic question regarding what are some common quantitative
measures
> for the goodness of (parametric) fit of a distribution.

But this is a different question from what you're trying to do:
>
> Here is what I am trying to do. I have some sample data which lets
say it
> has a lognormal distribution. I can get some hints of how it is
distributed
> from the histogram.

A histogram is the WORST you can possibly do.

Here's my unpublished "Theorem" :-) :

You recognize (or best distinguish) a PERSON by his/her face
and BODY; you recognize a distribution by its TAIL.


> What I want to do, is fit a few distributions (e.g.
> Lognromal, Beta, Gamma, inverse Gaussian etc) and find out which fits
the
> best.

You get yourself immediately into the unnecssary complication of
"fitting" and what metric to use to judge "best" or departure from
the fit.

The good-ole PROBABILITY PAPER plot is the idea you should use.
in all probability papers, the accent is on the departure of the
TAIL of the empirical distribution from the cdf of the theoretical
distribution.


> I have carried out some fits uing maximum likelihood and I can plot
the
> pdfs, or cdfs over my data to see which fits the best.

Do a PP plot or QQ plot. Just LOOK.


> However, I need some
> quantitative results (i.e. numbers).

Why? Like the drunk who uses a lampost for support rather than light?
:-)


> Just to point out that I am not really
> interested at the parameters of the distributions, but only which
fits the
> best.


> I could do Kolmogorov-Smirnov and chi-square tests but that's as far
as I
> know.

Chi-square is based on histograms -- it's worthless.

Kolmogorov uses on ONE POINT in the difference between the empirical
and theoretical cdfs, the point of maximum departure.

Your EYEBALLS can do an infinitely better job than that, looking at
the plot of the entire cdfs.

>
> Would someone be able to tell me what sort of metrics I can use for
my
> problem?
>
> Regards,
> V.Z.

You're hung up on the traditional "confirmatory data analysis", which
sheds little or no light on your REAL problem. I am suggesting
something along the line of John Tukey's "exploratory data analysis"
without any of Tukey's cryptic acronyms, as the most suitable way
of addressing your problem.

-- Bob.

.

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