[R] Fitting Pareto dist in a mixture

Pikounis, V. Bill v_bill_pikounis at merck.com
Wed Nov 14 20:46:52 CET 2001

Dear all:
First, apologies for cross-posting multiplicities and for a query that is
analytically related than S-language related.

The bottom-line wish is:

Could you please provide and advice, references, etc on S software
approaches for
fitting a distribution with density:

p*g(x) + (1-p)*f(x) 

where g(x) is the familiar lognormal 2-parameter density
and f(x) is Pareto as defined below?

Here is the lengthier story:
I am collaborating with a scientific colleague who has constructed a
of binding assay data, namely EC/IC50's across a wide variety of chemical
entities and experimental conditions.  Among other things, he has an
in characterizing tails of the distribution.  We have been studying various
fields in the literature to get a sense of what density forms we should

In addition to lognormal, weibull, & gamma, we found several references to
Pareto distribution.  It is of course easy enough to fit all of these with
existing S language functions (and reference to MASS and S Programming), and
even when considering lognormal mixtures as we are.

However, I have been attempting to use the Pareto
density form 

f(x) = ( a *  k^a  ) / ( x^{a + 1} ) ; k > 0, a > 0, x >= k

as part of a mixture with a lognormal.  Reparametrizing the density to 

f(x) = ( a *  k^a  ) / ( (x + k)^{a + 1} ) ; k > 0, a > 0, x > 0

seems to help but the optimization routines remain balky and / or give
convergences that do not make sense to me.  (I understand that starting
values, scales, etc. are very important.) I suspect that the support aspect
the Pareto ( i.e.  f(x) > 0 only for x >= k ) imposes an indentifiability
problem with the mixing proportion parameter.

Again, any advice is welcome.

Thanks in advance,

Bill Pikounis, Ph.D.
Biometrics Research Department
Merck Research Laboratories
PO Box 2000, MailDrop RY70-38
126 E. Lincoln Avenue
Rahway, New Jersey 07065-0900

v_bill_pikounis at merck.com

Phone: 732 594 3913
Fax: 732 594 1565

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