Kalpana Kalpana (Editor)

Noncentral F distribution

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a 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 n2 degrees of freedom. It is also required that X and Y are statistically independent of each other.

Contents

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 λ and ν 1 degrees of freedom, and Y is a chi-squared random variable with ν 2 degrees of freedom that is statistically independent of X , then

F = X / ν 1 Y / ν 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

when f ≥ 0 and zero otherwise. The degrees of freedom ν 1 and ν 2 are positive. The noncentrality parameter λ is nonnegative. The term B ( x , y ) is the beta function, where

B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) .

The cumulative distribution function for the noncentral F-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 )

where I is the regularized incomplete beta function.

The mean and variance of the noncentral F-distribution are

E ⁡ [ F ] = { ν 2 ( ν 1 + λ ) ν 1 ( ν 2 − 2 ) ν 2 > 2 Does not exist ν 2 ≤ 2

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 ν 2 > 4 Does not exist ν 2 ≤ 4.

Differential equation

The pdf of the noncentral F-distribution is a solution of the following differential equation:

{ 4 x ( ν 2 + ν 1 x ) 2 f ″ ( x ) + f ′ ( x ) ( − 2 ν 2 2 ν 1 + 8 ν 2 2 + 16 ν 1 2 x 2 + 4 ν 2 ν 1 2 x 2 − 2 λ ν 2 ν 1 x − 2 ν 2 ν 1 2 x + 4 ν 2 2 ν 1 x + 24 ν 2 ν 1 x ) + ν 1 ( ν 2 + 2 ) f ( x ) ( − λ ν 2 − ν 2 ν 1 + 4 ν 2 + 4 ν 1 x + ν 2 ν 1 x ) = 0 , f ( 1 ) = e − λ / 2 ν 1 ν 1 2 ν 2 ν 2 2 ( ν 1 + ν 2 ) 1 2 ( − ν 1 − ν 2 ) 1 F 1 ( 1 2 ( ν 1 + ν 2 ) ; ν 1 2 ; λ ν 1 2 ( ν 1 + ν 2 ) ) B ( ν 1 2 , ν 2 2 ) , f ′ ( 1 ) = e − λ / 2 ν 1 ν 1 2 ν 2 ν 2 2 ( ν 1 + ν 2 ) 1 2 ( − ν 1 − ν 2 − 2 ) ( ν 2 ( λ 1 F 1 ( 1 2 ( ν 1 + ν 2 + 2 ) ; 1 2 ( ν 1 + 2 ) ; λ ν 1 2 ( ν 1 + ν 2 ) ) − 2 1 F 1 ( 1 2 ( ν 1 + ν 2 ) ; ν 1 2 ; λ ν 1 2 ( ν 1 + ν 2 ) ) ) − 2 ν 1 1 F 1 ( 1 2 ( ν 1 + ν 2 ) ; ν 1 2 ; λ ν 1 2 ( ν 1 + ν 2 ) ) ) 2 B ( ν 1 2 , ν 2 2 ) }

Special cases

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

Z has a noncentral chi-squared distribution if

Z = lim ν 2 → ∞ ν 1 F

where F has a noncentral F-distribution.

See also noncentral t-distribution.

Implementations

The noncentral F-distribution is implemented in the R language (e.g., pf function), in MATLAB (ncfcdf, ncfinv, ncfpdf, ncfrnd and ncfstat functions in the statistics toolbox) in Mathematica (NoncentralFRatioDistribution function), in NumPy (random.noncentral_f), and in Boost C++ Libraries.

A collaborative wiki page implements an interactive online calculator, programmed in the R language, for the noncentral t, chi-squared, and F distributions, at the Institute of Statistics and Econometrics, School of Business and Economics, Humboldt-Universität zu Berlin.

References

Noncentral F-distribution Wikipedia
(Text) CC BY-SA