[R] Tail area of sum of Chi-square variables

[Klaus Nordhausen]

Hi,

thanks everyone! 

pchisqsum() in the "survey" package does exactly what I was looking for!

Best wishes,

Klaus


-------- Original-Nachricht --------
Datum: Thu, 29 Mar 2007 07:45:15 -0700 (PDT)
Von: Thomas Lumley <tlumley at u.washington.edu>
An: S Ellison <S.Ellison at lgc.co.uk>
CC: klausch at gmx.de, Achim.Zeileis at wu-wien.ac.at, r-help at stat.math.ethz.ch, Christian.Kleiber at unibas.ch
Betreff: Re: [R] Tail area of sum of  Chi-square variables

> 
> The Satterthwaite approximation is surprisingly good, especially in the 
> most interesting range in the right tail (say 0.9 to 0.999). There is also
> another, better, approximation with a power of a chi-squared distribution 
> that has been used in the survey literature.
> 
> However, since it is easy to write down the characteristic function and 
> perfectly feasible to invert it by numerical integration, we might as well
> use the right answer.
> 
>  	-thomas
> 
> On Thu, 29 Mar 2007, S Ellison wrote:
> >> I was wondering if there are any R functions that give the tail area
> >> of a sum of chisquare distributions of the type:
> >>         a_1 X_1 + a_2 X_2
> >> where a_1 and a_2 are constants and X_1 and X_2 are independent
> >> chi-square variables with different degrees of freedom.
> >
> > You might also check out Welch and Satterthwaite's (separate) papers on
> effective degrees of freedom for compound estimates of variance, which led
> to a thing called the welch-satterthwaite equation by one (more or less
> notorious, but widely used) document called the ISO Guide to Expression of
> Uncertainty in Measurement (ISO, 1995). The original papers are
> > B. L. Welch, J. Royal Stat. Soc. Suppl.(1936)  3 29-48
> > B. L. Welch, Biometrika, (1938) 29 350-362
> > B. L. Welch, Biometrika, (1947) 34 28-35
> >
> > F. E. Satterthwaite, Psychometrika (1941) 6 309-316
> > F. E. Satterthwaite, Biometrics Bulletin, (1946) 2 part 6 110-114
> >
> > The W-S equation - which I believe is a special case of Welch's somewhat
> more general treatment - says that if you have multiple independent
> estimated variances v[i] (could be more or less equivalent to your a_i X_i?) with
> degrees of freedom nu[i], the distribution of their sum is approximately a
> scaled chi-squared distribution with effective degrees of freedom
> nu.effective given by
> >
> > nu.effective =  sum(v[i])^2 / sum(    (v[i]^2)/nu[i]     )
> >
> > If I recall correctly, with an observed variance s^2 (corresponding to
> the sum(v[i] above if those are observed varianes), nu*(s^2 /sigma^2) is
> distributed as chi-squared with degrees of freedom nu, so the scaling factor
> for quantiles would come out of there (depending whether you're after the
> tail areas for s^2 given sigma^2 or for a confidence interval for sigma^2
> given s^2)
> >
> > However, I will be most interested to see what a more exact calculation
> provides!
> >
> > Steve Ellison
> >
> >
> > *******************************************************************
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> >
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> >
> 
> Thomas Lumley			Assoc. Professor, Biostatistics
> tlumley at u.washington.edu	University of Washington, Seattle

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