All,
I'm involved in research where we're using maximum likelihood and
AICc methods to evaluate the fit of our observed data to various
model distributions. The methods are rather straightforward, but we
have hit a snag for one model, and I was hoping a member of this
listserv could recommend where to find a solution.
Simplifying things here to make it easier, let's say we're comparing
our continuous (i.e., non-discrete) data to normal and uniform
distributions. (These are reasonable distributions to use for our
data.) For the normal distribution, it's a simple task to compare
our observations to a normal distribution with maximum likelihood
estimators (MLEs) estimated by the mean and variance of our observed
data. For the uniform distribution, the relevant MLEs are the
observed minimum and the maximum. The problem is that when we do
this, we are finding a consistent bias in favor of the uniform model
(even when we use test data that are clearly non-uniformly
distributed.) We suspect the cause of the bias is two-fold. First,
minima and maxima are biased estimates of population parameters (in
contrast to those for the normal distribution). In other words,
sampled extremes will always underestimate population extremes in
practice. Yet, these observed extremes *are* the appropriate MLEs
for the uniform distribution. Second, the uniform model is
theoretically bounded by the minimum and the maximum whereas the
normal is theoretically unbounded (meaning values can range from
negative to positive infinity, even if the likelihood of such
observations are essentially negligible.) The result of both
problems seems to result in a biased fit for the uniform model (which
is bounded by observed values), but not of the normal distribution
(which are not so bounded). We suspect that the second problem is
the more problematical one in our case (in large part because we're
using the appropriate MLEs for each model). We (including a
statistician) have been unable to find discussion of this issue--bias
caused by comparing an internally bounded model to an unbounded
one--in the statistical or ecological literature and we're hoping
that you might be able to offer advice.
Feel free to reply directly to me off-line.
Sincerely,
Phil
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Phil
Novack-Gottshall pnova...@westga.edu
Assistant Professor
Department of Geosciences
University of West Georgia
Carrollton, GA 30118-3100
Phone: 678-839-4061
Fax: 678-839-4071
http://www.westga.edu/~pnovackg
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