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

			(4)
			(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)
			(8)
			(9)
			(10)

and the first few central moments are

			(11)
			(12)

The mean and variance are therefore given by

			(13)
			(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.