Noncentral F-Distribution
المؤلف:
Patnaik, P. B.
المصدر:
"The Non-Central c_2- and F-Distributions and Their Applications." Biometrika 36
الجزء والصفحة:
...
10-4-2021
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Noncentral F-Distribution
The doubly noncentral 
-distribution describes the distribution 
for two independently distributed noncentral chi-squared variables 
and 
(Scheffe 1959, Bulgren 1971). If 
, this becomes the usual (central) F-distribution, and if 
, it becomes the singly noncentral 
-distribution. The case 
gives a special case of the doubly noncentral distribution.
The probability density function of the doubly noncentral 
-distribution is
 |
(1)
|
and the distribution function by
 |
(2)
|
where 
is a beta function and 
is a hypergeometric function. The 
th raw moment is given analytically as
 |
(3)
|
The singly noncentral 
-distribution is given by
where 
is the gamma function, 
is the beta function, and 
is a generalized Laguerre polynomial. It is implemented in the Wolfram Language as NoncentralFRatioDistribution[n1, n2, lambda].
The 
th raw moment of the singly noncentral 
-distribution is given analytically as
 |
(6)
|
The first few raw moments are then
and the first few central moments are
The mean and variance are therefore given by
REFERENCES:
Patnaik, P. B. "The Non-Central 
- and 
-Distributions and Their Applications." Biometrika 36, 202-232, 1949.
Bulgren, W. G. "On Representations of the Doubly Non-Central 
Distribution." J. Amer. Stat. Assoc. 66, 184, 1971.
Scheffé, H. The Analysis of Variance. New York: Wiley, pp. 135 and 415, 1959.
Stuart, A.; and Ord, J. K. Kendall's Advanced Theory of Statistics, Vol. 2A: Classical Inference & the Linear Model, 6th ed. New York: Oxford University Press, p. 893, 1999.
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