The Gamma interpretation is very likely correct given 26.6.3:
http://hcohl.shell42.com/as/page_946.htm
Anyway Gamma of a 1/2-integer only involves a sqrt(pi). See:
http://hcohl.shell42.com/as/page_255.htm
Keith
Dr. Keith M. Briggs
Senior Mathematician, Complexity Research, BT Exact
http://research.btexact.com/teralab/keithbriggs.html (new page, BT intranet only at the moment)
http://more.btexact.com/people/briggsk2/
http://members.lycos.co.uk/keithmbriggs/keithbriggs.html
phone: +44(0)1473 work: 641 911 home: 610 517 fax: 642 161
mail: Keith Briggs, Polaris 134, Adastral Park, Martlesham, Suffolk IP5 3RE, UK
-----Original Message-----
From: john leong [mailto:[log in to unmask]]
Sent: 15 October 2003 16:17
To: [log in to unmask]
Subject: F distribution
You really have to have the book: Handbook of
Mathematical Functions (9th Print)
to look at this problem, if you dun have this book the
following might not make too much sense.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables
~Milton Abramowitz, Irene A. Stegun
Dover Publications
Paperback - 1 June, 1974
I programmed different distributions functions more or
less a year ago. I have a problem
with F distribution that been bugging me for all this
time. Like they say it is "a splinter
in the mind". I have just started my Part time MSc in
Applied Statistics & Operational Research
and my lecturer gave me this mail group to join, so I
think may be I should ask you people for help.
I have decided to follow the method: "series
expansions" set out in Handbook of Mathematical
Functions (9th print) for the distributions as I have
run into accuracy problem with approximating
any integration function. Following the Series
Expansions so far I have got T, chisqrt, and normal
distribution working and to all digit accuracy (15 in
Visual Basic).
For F distribution the Series Expansion function are
working fine as it is, the formula can be found
in the book's page 946, as illustrated in 26.6.4,
26.6.5, 26.6.6 and 26.6.7. with each of them working
to very high accuracy. However, the function on 26.6.8
(when v1, v2 both odd for the F distribution),
it starts to have problem.
Basically the function can be divided into 2 parts
A:[A(t/v2)] and B:[B(v1,v2)] as how it been named
in the book.
I am quite happy that part A is okay. As when v1 = 0
part B will be reduced to 0. Cross checking
with the table in the book it appeared to be correct.
Lets Part B =b1*b2*b3
Where
b1 = 2/ (pi^(1/2))
b2= (((v2-1)/2)! / ((v2-2)/2)! )
b3 Sin(theta)Cos(v2,theta)(1+... etc etc
My problem is with section b2
if v1, v2 are both odd so v2-2 will also be odd.
Hence (v2-2)/2 will be a N.5 and can not work in the
factorial function.
I believe that it is a Printing error with missing
element such as v1 such that (((v1+v2)-2)/2)!
or something similar. I tried many combinations of b2
it did not return the right answer.
I have also asked the others opinion, a few lecturer
from different universities, they suggested
me to look into the Gamma Function or look through the
book to look for special treatment for factorial
function when the number is not integer.
Whereas I do not buy the "gamma" case, I cannot find
any special mention for the special treatment
for factorial-ing a fraction.
So people:
Do you know anything about the printing mistake on
26.6.8? or
Do you have another method to get the right F pdf? *
Any suggestion / discussion will be welcomed.
Many thanks for reading it all the way here, I know it
is long email.
JL
* I have also seen another method using beta function,
I do not go through that route as it involve
Gamma function what I have problem to get it right via
integration and / or approximation.
________________________________________________________________________
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