Math Forum / Mathematics / Statistics / July 2009

Hi,

I have an empirical cummulative distribution function (ECDF) I want to
"best-fit" to an extreme value distribution (EVD). One source I came
across suggested that rather than trying to estimate the params of the
EVD from the moments of the ECDF, instead use the params computed from
the moments as an initial estimate and then refine the params by best
fitting the ECDF to the CDF for the EVD.

Visual inspection suggests that this approach produces a significant
improvement in the fit. I started playing around and using different
criteria for the fitting: the total abs devtn is not bad, but using
the slope of the ECDF to weight the abs devtn produces a better fit
(since you're weighting the errors where the "meat" of the cdf is).

I noticed that using the slope-weighted approach, the ends and center
of the ECDF and EVD CDF fit well, but the "sigmoidal" portions of the
function in between these regions was a bit off. I tried using a fit
with the ECDF second derivative (but its too noisy), and I tried using
curvature (but this weights the regions right where the sigmoid region
*starts*). Any suggestions on a metric that will hit this
preferentially weight this portion of the curve?

(BTW, the rationale for wanting to weight the sigmoid-like part of the
curve, is that if I can get this portion of the curve to fit well,
perhaps the rest of the curve will follow (since these regions require
more of a linear fit in nature). I'm going to be out for a few days
after I post this, so if I don't respond to anyone's post for a couple
of days, it isn't because I've lost interest in this thread!

TIA,

Matt