Date: Feb 22, 2013 3:09 AM
Author: David Jones
Subject: Re: Trying to understand Bayes and Hypothesis
"Dave" wrote in message
(1) Theory says the "errors" should be normally distributed and no one
argues that a variety of goodness of fit measures reject it at p<.001 or
wherever the table stops.
(2) Theory says I should be able to minimize variance choosing an
expectation or maximize an expectation choosing a variance. Of course you
cannot do that with a Cauchy distribution.
Step (2) is incorrect, given the results of step(1). Given step (1),
"theory" says either:
(a) Chose an appropriate likelihood function, based on an acceptable
distribution. Use a large sample argument to justify a chi-squared test
based on a likelihood ratio test.
(b) Choose an appropriate objective function (goodness-of-fit measure), such
as a mean absolute difference. (Although this might need to be modified if
you are fitting both location and scale.) Construct a test statistic based
on this objective function, such as the improvement in the objective
function on moving to the wider model. Construct critical values for the
test statistic by undertaking a simulation study based on what you think are
acceptable null distributions.
If you were happy enough to do a Bayesian analysis, you might note that
several recent works have been implemented with structures where the "normal
distribution" assumption has been replaced by a Student's t distribution
with fixed but low degrees of freedom, which includes the Cauchy
distribution. Hence there is a good chance that you could find a Bayesian
analysis package that includes this facility, and this might prove a viable
route for you. Of course, you might find a"frequentist" package to do
something similar, if you need to look for pre-existing code ... you might
look under "general linear model".