Re: Least Squares solution for fitting beta distribution to empirical distrbution
From: Herman Rubin (hrubin_at_odds.stat.purdue.edu)
Date: 02/23/05
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Date: 23 Feb 2005 15:10:20 -0500
In article <1109155518.005943.271990@l41g2000cwc.googlegroups.com>,
<clemenr@wmin.ac.uk> wrote:
>Hi.
>I would like to look at the following case. I have a set of data that I
>can use as a probability distribution using kernel density estimation.
>I would like to find the best possible (I'm thinking at the moment,
>least squares) beta distribution to approximate this empirical
>distribution.
Why should you use least squares? Maximum likelihood is
tractable for this problem, although if you have unknown
endpoints, there may be reasons for making some modifications.
Also, I suggest you do this on the original data, not the
results of kernel smoothing, which loses information, and
which will in general increase the range, causing errors.
If the density is
(x-A)^{a-1}*(B-x)^{b-1}*\Gamma(a+b)/(\Gamma(a)*\Gamma(b)*(B-A)^(a+b-1)),
the mle of a and b given A and B correspond to setting the
average value of ln((x-A)/(B-A)) to its expected value
\Psi(a) - \Psi(a+b), and correspondingly for ln((B-x)/(B-A)).
Then one can plug this in to the likelihood function, but
observe that if a <= 1, the mle of A is the smallest order
statistic, and similarly for the case b <=1 for B. If this
is an appropriate model, kernel estimators are likely to be
quite poor near any such endpoint.
-- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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