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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