Re: Weighted sum of squared normals question

From: Ray Koopman (koopman_at_sfu.ca)
Date: 02/24/05 

Date: 24 Feb 2005 11:41:32 -0800

Herman Rubin wrote:
> [...]
> I doubt that this approximation is that good, especially in
> the tails. [...]

A scaled F, that fits the first three moments, should be better in the
tails than a scaled chi-square, that fits only the first two moments.
Let

S1 = a1 + a2 + ... + aN ,
S2 = a1^2 + a2^2 + ... + aN^2,
S3 = a1^3 + a2^3 + ... + aN^3.

Then the constant of proportionality is

    S1*(S1^2*S2 - 2*S2^2 + 4*S1*S3)
c = -------------------------------,
      S1^2*S2 - 4*S2^2 + 6*S1*S3

and the degrees of freedom are

     S1*(S1^2*S2 - 2*S2^2 + 4*S1*S3)
f1 = -------------------------------
      2*S1*S2^2 - S1^2*S3 + 2*S2*S3
and
         S2*(S1^2 + 2*S2)
f2 = 6 + ----------------.
           S1*S3 - S2^2

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