[SciPy-User] [SciPy-dev] Bug/Error with chi-squared distribution and df<1

[Bruce Southey]

On 09/22/2009 01:48 PM, Pierre GM wrote:
> On Sep 22, 2009, at 2:27 PM, josef.pktd at gmail.com wrote:
>    
>>>>> I'm not an expert on distributions, but as far as I can tell, the
>>>>> chi2
>>>>> distribution is defined for degrees of freedom>0.
>>>>>            
> Mmh, could anybody point me to a *real* case where we would have less
> than 1 degree of freedom ?
> Check the definition of the X2 on wikipedia, for example: if k
> variables are iid scaled normal, their sum is X2 w/ k degrees of
> freedom (dfs). Naturally, k, the dfs are integers, and that makes
> quite sense to prevent the use of 0<  k<  1. Or returning NaNs instead
> of raising an exception...
>
>
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>    
The easy and common one is when using a likelihood ratio test when one 
parameter is on a boundary. So when you use the LRT in a mixed effects 
model with one random term to test that variance of that term is zero 
then, you have to use a 50:50 mixture of chi-squared values at df=0 and 
df=1.

The classic paper is:
Self, S. and Liang, K-Y. (1987) Asymptotic Properties of Maximum 
Likelihood Estimators
and Likelihood Ratio Tests under Nonstandard Conditions. Journal of the 
American
Statistical Association 82, 605-610.

Bruce