Lambert series

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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.

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 ν1 degrees of freedom, and Y is a chi-squared random variable with ν2 degrees of freedom that is statistically independent of X, then

F=X/ν1Y/ν2

is a noncentral F-distributed random variable. The probability density function for the noncentral F-distribution is [1]

p(f)=k=0eλ/2(λ/2)kB(ν22,ν12+k)k!(ν1ν2)ν12+k(ν2ν2+ν1f)ν1+ν22+kfν1/21+k

when f0 and zero otherwise. The degrees of freedom ν1 and ν2 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|d1,d2,λ)=j=0((12λ)jj!eλ2)I(d1xd2+d1x|d12+j,d22)

where I is the regularized incomplete beta function.


The mean and variance of the noncentral F-distribution are

E[F]={ν2(ν1+λ)ν1(ν22)ν2>2Does not existν22

and

Var[F]={2(ν1+λ)2+(ν1+2λ)(ν22)(ν22)2(ν24)(ν2ν1)2ν2>4Does not existν24.

Special cases

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

Related distributions

Z has a noncentral chi-squared distribution if

Z=limν2ν1F

where F has a noncentral F-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.[2]

A collaborative wiki page implements an interactive online calculator, programmed in R language, for noncentral t, chisquare, and F, at the Institute of Statistics and Econometrics, School of Business and Economics, Humboldt-Universität zu Berlin.[3]

Notes

  1. S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, (New Jersey: Prentice Hall, 1998), p. 29.
  2. Template:Cite web
  3. Template:Cite web

References

External links

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