R help - Jul 2005 - Off topic -2 Ln Lambda and Chi square

Dear R :

Sorry for the off topic question, but does anyone know the reference for
the -2 Ln Lambda following a Chi Square distribution, please?

Possibly one of Bartlett's?

Thanks in advance!

Sincerely,
Laura Holt
mailto: lauraholt_983 at hotmail.com

If you meant the lambda as the likelihood ratio test statistic, the
asymptotic chi-squared distribution comes from the asymptotic normality of
the MLEs.  The proof is in a paper by Abraham Wald in 1943.  See Stuart &
Ord (Kendall's Advanced Statistics) for discussion (e.g., vol. 2, 5th
edition).

Andy
> From: Laura Holt
> 
> Dear R :
> 
> Sorry for the off topic question, but does anyone know the 
> reference for
> the -2 Ln Lambda following a Chi Square distribution, please?
> 
> Possibly one of Bartlett's?
> 
> Thanks in advance!
> 
> Sincerely,
> Laura Holt
> mailto: lauraholt_983 at hotmail.com
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide! 
> http://www.R-project.org/posting-guide.html
> 
> 
>

This is a theorem for maximum likelihood tests. See:

Theorem 12.2 (presented without proof), page 391,  in "John E. Freund's
Mathematical Statistics with Applications", Seventh Edition, by Irwin 
Miller and Marylees Miller. Upper Saddle River, N.J.: Pearson Prentice 
Hall, 2004.

    Theorem 12.2: For large n, the distribution of -2 ln Lambda 
approaches, under very general conditions, the chi-square distribution 
with 1 degree of freedom.

Theorem 6.3.1 (given with a proof), p. 335, in "Introduction to 
Mathematical Statistics", Sixth Edition, by RV Hogg, JW McKean, and AT 
Craig. 2005. Upper Saddle River, New Jersey: Pearson Prentice Hall.

Proofs are also given in:

Testing Statistical Hypotheses, Second Edition, by E. L. Lehmann. New 
York: John Wiley and Sons, Inc., 1986

and

Mathematical Statistics, by S. S. Wilkes. New York: John Wiley and Sons, 
Inc., 1962.

-- Conrad

Laura Holt wrote:
>Dear R :
>
>Sorry for the off topic question, but does anyone know the reference for
>the -2 Ln Lambda following a Chi Square distribution, please?
>
>Possibly one of Bartlett's?
>
>Thanks in advance!
>
>Sincerely,
>Laura Holt
>mailto: lauraholt_983 at hotmail.com
>
>______________________________________________
>R-help at stat.math.ethz.ch mailing list
>https://stat.ethz.ch/mailman/listinfo/r-help
>PLEASE do read the posting guide!
http://www.R-project.org/posting-guide.html
>
>  
>
-- 
Conrad Halling
conrad.halling at bifx.org