Re: Approxtn to LH Tail of NC F-Distbtn
- From: Outlier <MTBrenneman@xxxxxxxxx>
- Date: Tue, 18 Nov 2008 21:35:42 -0800 (PST)
On Nov 17, 1:03 am, Jack Tomsky <jtom...@xxxxxxxxxxxxx> wrote:
Hi,
Let F(x:n1,n2,L) denote the cdf for a non-central
F-distribution with
degrees of freedom n1 and n2 and non-centrality
parameter L. Let p be
some defined probability in the LH tail of the
distribution (either
p=0.01, 0.05, or 0.10). (In other words, p is some
missed detection
probability I wish to keep small).
Is there a decent formula for finding the approximate
value of x
satisfying:
p = F(x:n1,n2,L)?
Equivalently, this can be cast in terms of the
non-central Beta
distribution, i.e. is there a decent formula that
gives the
approximate value of y satisfying
p = G(y:a,b,L)
where G is the cdf for the non-central Beta
Distribution with a =
n1/2, b=n2/2, y = x*n1/(x*n1_n2)?
Thank you for any help you can provide as I can find
absolutely
nothing in the literature on this.
Matt
In Abramowitz and Stegun, there's an approximation of the noncentral F' by the central F.
P(F'|n1,n2,L) = P(F|n1*,n2),
where F = [n1/(n1+L)]F'
n1* = [(n1+L)^2]/[n1+2L]
You ought to be able to get the quantiles of F' from the quantiles of F by inversion. I don't know how accurate this approximation is in practice.. A&S also gives approximations in terms of the normal distribution.
Jack
There is a very nice review of the approximations for both the central
and non-central F distribution in Johnson & Klotz ('Univariate
Distributions').
The cdf for the central F-distribution when the numerator df < denom
df appears well approximated by a simple result known as Fisher's
approximation (which I believe is the normal approx to which you were
referring): the normal and central F-distbtn cdfs agree within 1/2%
for p-values less than 10%.
The cdf for the non-central F-distribution also has a similar "Fisher-
Like" distribution. There is a very beautiful paper by Nico Laubscher
("Normalizing the Non-Central t and F-distributions", from 1960 in the
Annals of Math Stats) in which he explains the derivation of the
approximation very clearly and runs a good set of representative
simulations.
FWIW,
M
.
- References:
- Approxtn to LH Tail of NC F-Distbtn
- From: Outlier
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