In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a noncentral generalization of the (ordinary) F-distribution. It describes the distribution of the quotient (X/n1)/(Y/n2), where the numerator X has a noncentral chi-squared distribution with n1 degrees of freedom and the denominator Y has a central chi-squared distribution with n2 degrees of freedom. It is also required that X and Y are statistically independent of each other.

It is the distribution of the test statistic in analysis of variance problems when the null hypothesis is false. The noncentral F-distribution is used to find the power function of such a test.

## Occurrence and specification

If $X$ is a noncentral chi-squared random variable with noncentrality parameter $\lambda$ and $\nu _{1)$ degrees of freedom, and $Y$ is a chi-squared random variable with $\nu _{2)$ degrees of freedom that is statistically independent of $X$ , then

$F={\frac {X/\nu _{1)){Y/\nu _{2)))$ is a noncentral F-distributed random variable. The probability density function (pdf) for the noncentral F-distribution is

$p(f)=\sum \limits _{k=0}^{\infty }{\frac {e^{-\lambda /2}(\lambda /2)^{k)){B\left({\frac {\nu _{2)){2)),{\frac {\nu _{1)){2))+k\right)k!))\left({\frac {\nu _{1)){\nu _{2))}\right)^((\frac {\nu _{1)){2))+k}\left({\frac {\nu _{2)){\nu _{2}+\nu _{1}f))\right)^((\frac {\nu _{1}+\nu _{2)){2))+k}f^{\nu _{1}/2-1+k)$ when $f\geq 0$ and zero otherwise. The degrees of freedom $\nu _{1)$ and $\nu _{2)$ are positive. The term $B(x,y)$ is the beta function, where

$B(x,y)={\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y))).$ The cumulative distribution function for the noncentral F-distribution is

$F(x\mid d_{1},d_{2},\lambda )=\sum \limits _{j=0}^{\infty }\left({\frac {\left({\frac {1}{2))\lambda \right)^{j)){j!))e^{-\lambda /2}\right)I\left({\frac {d_{1}x}{d_{2}+d_{1}x)){\bigg |}{\frac {d_{1)){2))+j,{\frac {d_{2)){2))\right)$ where $I$ is the regularized incomplete beta function.

The mean and variance of the noncentral F-distribution are

$\operatorname {E} [F]\quad {\begin{cases}={\frac {\nu _{2}(\nu _{1}+\lambda )}{\nu _{1}(\nu _{2}-2)))&{\text{if ))\nu _{2}>2\\{\text{does not exist))&{\text{if ))\nu _{2}\leq 2\\\end{cases))$ and

$\operatorname {Var} [F]\quad {\begin{cases}=2{\frac {(\nu _{1}+\lambda )^{2}+(\nu _{1}+2\lambda )(\nu _{2}-2)}{(\nu _{2}-2)^{2}(\nu _{2}-4)))\left({\frac {\nu _{2)){\nu _{1))}\right)^{2}&{\text{if ))\nu _{2}>4\\{\text{does not exist))&{\text{if ))\nu _{2}\leq 4.\\\end{cases))$ ## Special cases

When λ = 0, the noncentral F-distribution becomes the F-distribution.

## Related distributions

$Z=\lim _{\nu _{2}\to \infty }\nu _{1}F$ where F has a noncentral F-distribution.

2. ^ John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde. "Noncentral F Distribution: Boost 1.39.0". Boost.org. Retrieved 20 August 2011.((cite web)): CS1 maint: uses authors parameter (link)