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Noncentral F-Distribution

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The doubly noncentral F-distribution describes the distribution (X/n_1)/(Y/n_2) for two independently distributed noncentral chi-squared variables X:chi_(n_1)^2(lambda_1) and Y:chi_(n_2)^2(lambda_2) (Scheffe 1959, Bulgren 1971). If lambda_1=lambda_2=0, this becomes the usual (central) F-distribution, and if lambda_1!=0,lambda_2=0, it becomes the singly noncentral F-distribution. The case lambda_1=0,lambda_2!=0 gives a special case of the doubly noncentral distribution.

The probability density function of the doubly noncentral F-distribution is

P(n_1,n_2;lambda_1,lambda_2;x) =sum_(k=0)^inftysum_(l=0)^infty(n_1^(k+n_1/2)n_2^(l+n_2/2)x^(k+n_1/2-1)lambda_1^klambda_2^l)/(2^(k+l)k!l!e^((lambda_1+lambda_2)/2)B(k+1/2n_1,l+1/2n_2)) ×(n_2+n_1x)^(-(k+l)-(n_1+n_2)/2)
(1)

and the distribution function by

D(n_1,n_2;lambda_1,lambda_2;x)=sum_(k=0)^inftysum_(l=0)^infty((n_1x)/(n_2))^(k+n_1/2) ×(lambda_1^klambda_2^l_2F_1(k+1/2n_1,k+l+1/2(n_1+n_2);k+1+1/2n_1;-(n_1)/(n_2)x))/(2^(k+l-1)e^((lambda_1+lambda_2)/2)(2k+n_1)B(1/2n_1+k,1/2n_2+l)k!l!),
(2)

where B(p,q) is a beta function and _2F_1(a,b;c;z) is a hypergeometric function. The nth raw moment is given analytically as

mu_n^'=e^(-(lambda_1+lambda_2)/2)((n_2)/(n_1))^nGamma(1/2n_1+n)Gamma(1/2n_2-n)×_1F^~_1(1/2n_1+n;1/2n_1;1/2lambda_1)_1F^~_1(1/2n_2-n;1/2n_2;1/2lambda_2).
(3)

Here _1F^~_1(a;b;z)=_1F_1(a;b;z)/Gamma(b) is the regularized hypergeometric function, _1F_1 is the confluent hypergeometric function of the first kind, and Gamma is the gamma function. The moment of order n exists for n_2>2n.

The singly noncentral F-distribution is given by

P(x)=e^(-lambda/2+(lambdan_1x)/[2(n_2+n_1x)])n_1^(n_1/2)n_2^(n_2/2)x^(n_1/2-1)(n_2+n_1x)^(-(n_1+n_2)/2)(Gamma(1/2n_1)Gamma(1+1/2n_2)L_(n_2/2)^(n_1/2-1)(-(lambdan_1x)/(2(n_2+n_1x))))/(B(1/2n_1,1/2n_2)Gamma[1/2(n_1+n_2)])
(4)
=(e^(-lambda/2)x^(n_1/2-1)(xn_1+n_2)^(-(n_1+n_2)/2)n_1^(n_1/2)n_2^(n_2/2)_1F_1(1/2(n_1+n_2);1/2n_1;(xlambdan_1)/(2(xn_1+n_2))))/(B(1/2n_1,1/2n_2)),
(5)

where Gamma(z) is the gamma function, B(alpha,beta) is the beta function, and L_m^n(z) is a generalized Laguerre polynomial. It is implemented in the Wolfram Language as NoncentralFRatioDistribution[n1, n2, lambda].

The nth raw moment of the singly noncentral F-distribution is given analytically as

mu_n^'=((n_2)/(n_1))^n(Gamma(1/2n_1+n)Gamma(1/2n_2-n)_1F^~_1(n+1/2n_1;1/2n_1;1/2lambda))/(e^(lambda/2)Gamma(1/2n_2)).
(6)

The first few raw moments are then

mu_1^'=(n_2(lambda+n_1))/(n_1(n_2-2))
(7)
mu_2^'=(n_2^2[lambda^2+2(n_1+2)lambda+n_1(n_1+2)])/(n_1^2(n_2-2)(n_2-4))
(8)
mu_3^'=(n_2^3[lambda^3+3(n_1+4)lambda^2+3(n_1+2)(n_1+4)lambda+n_1(n_1+2)(n_1+4)])/(n_1^3(n_2-6)(n_2-4)(n_2-2))
(9)
mu_4^'=(n_2^4[lambda^4+4(n_1+6)lambda^3+6(n_1+4)(n_1+6)lambda^2+4(n_1+2)(n_1+4)(n_1+6)lambda+n_1(n_1+2)(n_1+4)(n_1+6)])/(n_1^4(n_2-8)(n_2-6)(n_2-4)(n_2-2)),
(10)

and the first few central moments are

mu_2=(2n_2^2[lambda^2+2(n_1+n_2-2)lambda+n_1(n_1+n_2-2)])/(n_1^2(n_2-2)^2(n_2-4))
(11)
mu_3=(8n_2^3[2lambda^3+6(n_1+n_2-2)lambda^2+3(n_1+n_2-2)(2n_1+n_2-2)lambda+n_1(n_1+n_2-2)(2n_1+n_2-2)])/(n_1^3(n_2-2)^3(n_2-4)(n_2-6)).
(12)

The mean and variance are therefore given by

mu=((lambda+n_1)n_2)/(n_1(n_2-2))
(13)
sigma^2=(2n_2^2[lambda^2+2(n_1+n_2-2)lambda+n_1(n_1+n_2-2)])/(n_1^2(n_2-2)^2(n_2-4)).
(14)

See also

F-Distribution

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References

Bulgren, W. G. "On Representations of the Doubly Non-Central F Distribution." J. Amer. Stat. Assoc. 66, 184, 1971.Patnaik, P. B. "The Non-Central c_2- and F-Distributions and Their Applications." Biometrika 36, 202-232, 1949.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.

Referenced on Wolfram|Alpha

Noncentral F-Distribution

Cite this as:

Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NoncentralF-Distribution.html

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