Probability distribution generalizing the F-distribution with a noncentrality parameter
Inprobability theory andstatistics , thenoncentralF -distribution is acontinuous probability distribution that is anoncentral generalization of the (ordinary)F -distributionX /n 1 )/(Y /n 2 ), where the numeratorX has anoncentral chi-squared distribution withn 1 degrees of freedom and the denominatorY has a centralchi-squared distribution withn 2 degrees of freedom. It is also required thatX andY arestatistically independent of each other.
It is the distribution of thetest statistic inanalysis of variance problems when thenull hypothesis is false. The noncentralF -distribution is used to find thepower function of such a test.
Occurrence and specification [ edit ] IfX {\displaystyle X} noncentral chi-squared random variable with noncentrality parameterλ {\displaystyle \lambda } ν 1 {\displaystyle \nu _{1}} Y {\displaystyle Y} chi-squared random variable withν 2 {\displaystyle \nu _{2}} statistically independent ofX {\displaystyle X}
F = X / ν 1 Y / ν 2 {\displaystyle F={\frac {X/\nu _{1}}{Y/\nu _{2}}}} is a noncentralF -distributed random variable.Theprobability density function (pdf) for the noncentralF -distribution is[ 1]
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}} whenf ≥ 0 {\displaystyle f\geq 0} ν 1 {\displaystyle \nu _{1}} ν 2 {\displaystyle \nu _{2}} B ( x , y ) {\displaystyle B(x,y)} beta function , where
B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) . {\displaystyle B(x,y)={\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y)}}.} Thecumulative distribution function for the noncentralF -distribution is
F ( x ∣ d 1 , d 2 , λ ) = ∑ j = 0 ∞ ( ( 1 2 λ ) j j ! e − λ / 2 ) I ( d 1 x d 2 + d 1 x | d 1 2 + j , d 2 2 ) {\displaystyle 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)} whereI {\displaystyle I} regularized incomplete beta function .
The mean and variance of the noncentralF -distribution are
E [ F ] { = ν 2 ( ν 1 + λ ) ν 1 ( ν 2 − 2 ) if ν 2 > 2 does not exist if ν 2 ≤ 2 {\displaystyle \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
Var [ F ] { = 2 ( ν 1 + λ ) 2 + ( ν 1 + 2 λ ) ( ν 2 − 2 ) ( ν 2 − 2 ) 2 ( ν 2 − 4 ) ( ν 2 ν 1 ) 2 if ν 2 > 4 does not exist if ν 2 ≤ 4. {\displaystyle \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}}} Whenλ = 0, the noncentralF -distribution becomes theF -distribution
Related distributions [ edit ] Z has anoncentral chi-squared distribution if
Z = lim ν 2 → ∞ ν 1 F {\displaystyle Z=\lim _{\nu _{2}\to \infty }\nu _{1}F} whereF has a noncentralF -distribution.
See alsononcentral t-distribution .
The noncentralF -distribution is implemented in theR language (e.g., pf function), inMATLAB (ncfcdf, ncfinv, ncfpdf, ncfrnd and ncfstat functions in the statistics toolbox) inMathematica (NoncentralFRatioDistribution function), inNumPy (random.noncentral_f), and inBoost C++ Libraries .[ 2]
A collaborative wiki page implements an interactive online calculator, programmed in theR language , for the noncentral t,chi-squared , and F distributions, at the Institute of Statistics and Econometrics of theHumboldt University of Berlin .[ 3]
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
