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Date: 14-2-2016
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Date: 6-2-2021
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Date: 25-4-2021
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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
(4) |
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(5) |
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
(7) |
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(8) |
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(9) |
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(10) |
and the first few central moments are
(11) |
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(12) |
The mean and variance are therefore given by
(13) |
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(14) |
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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