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 {\displaystyle X} is a noncentral chi-squared random variable with noncentrality parameter λ {\displaystyle \lambda } and ν 1 {\displaystyle \nu _{1}} degrees of freedom, and Y {\displaystyle Y} is a chi-squared random variable with ν 2 {\displaystyle \nu _{2}} degrees of freedom that is statistically independent of X {\displaystyle X} , then
F = X / ν 1 Y / ν 2 {\displaystyle 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 ) = ∑ k = 0 ∞ e − λ / 2 ( λ / 2 ) k B ( ν 2 2 , ν 1 2 + k ) k ! ( ν 1 ν 2 ) ν 1 2 + k ( ν 2 ν 2 + ν 1 f ) ν 1 + ν 2 2 + k f ν 1 / 2 − 1 + k {\displaystyle 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 ≥ 0 {\displaystyle f\geq 0} and zero otherwise. The degrees of freedom ν 1 {\displaystyle \nu _{1}} and ν 2 {\displaystyle \nu _{2}} are positive. The term B ( x , y ) {\displaystyle B(x,y)} is the beta function, where
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