Date: Feb 22, 2013 3:09 AM
Author: David Jones
Subject: Re: Trying to understand Bayes and Hypothesis
"Dave" wrote in message

news:7e66c68a-ac39-4207-a399-03d64e0277fe@googlegroups.com...

(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.

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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distribution" assumption has been replaced by a Student's t distribution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".

David Jones