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st: functional form advice?


From   Jenkins S P <stephenj@essex.ac.uk>
To   Statalist <statalist@hsphsun2.harvard.edu>
Subject   st: functional form advice?
Date   Sat, 22 May 2004 08:26:37 +0100 (BST)

I seek advice on functional forms (including potential references to
follow up).

I'm looking for classes of univariate function f(x) for a variable x
> 0, such that
f(1) = 0,
f(x) > 0 and f'(x) <= 0 for x < 1,
f(x) > 0 and f'(x) >= 0 for x > 1.

Examples of f(x) include monotonically increasing functions of:
f(x) = |x - 1| ;
f(x) = (x - 1)^2  = x(x-1) + 1 ;
f(x) = x*[ln(x)-1)] + 1

You can see what these functions look like with Stata commands like
	twoway function y = abs(x-1), range(0 2)
	twoway function y = (x-1)^2, range(0 2)
      	twoway function y = x*(ln(x) - 1) + 1, range(0 2)

I suspect that the last two cases are examples of a type of generalized
entropy class. Usually these involve functional forms like x.ln(x),
ln(1/x), and (1/a)*x^a for a!=0, but I haven't been able to figure out the
corresponding general form in this case.

Stephen
=============================================
Professor Stephen P. Jenkins <stephenj@essex.ac.uk>
Institute for Social and Economic Research (ISER)
University of Essex, Colchester CO4 3SQ, UK
Phone: +44 1206 873374.  Fax: +44 1206 873151.
http://www.iser.essex.ac.uk
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