Noncentral F-distribution

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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-square distribution with n₁ degrees of freedom and the denominator Y has a central chi-square 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. One uses the noncentral F-distribution to find the power function of such a test.

1 Occurrence and Specification of the Noncentral F-distribution
2 Special cases
3 Related distributions
4 See also
5 Implementations
6 External links
7 References

[edit] Occurrence and Specification of the Noncentral F-distribution

If $X$ is a noncentral chi-square random variable with noncentrality parameter $λ$ and $ν 1$ degrees of freedom, and $Y$ is a chi-square random variable with $ν 2$ degrees of freedom that's statistically independent of $X$ , then

$F=\frac{X/\nu_1}{Y/\nu_2}$

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

$p(f) =\sum\limits_{k=0}^\infty\frac{e^{-\lambda/2}(\lambda/2)^k}{ B\left(\frac{\nu_2}{2},\frac{\nu_1}{2}+k\right) k!} \left(\frac{\nu_1}{\nu_2}\right)^{\frac{\nu_1}{2}+k} \left(\frac{\nu_2}{\nu_2+\nu_1f}\right)^{\frac{\nu_1+\nu_2}{2}+k}f^{\nu_1/2-1+k}$

when $f\ge0$ and zero otherwise. The degrees of freedom $ν 1$ and $ν 2$ are positive. The noncentrailty parameter $λ$ is nonnegative. The term $B (x, y)$ is the beta function, where

$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x,y)}.$

The mean and variance of the noncentral F-distribution are

$\mbox{E}\left[F\right]= \begin{cases} \frac{\nu_2(\nu_1+\lambda)}{\nu_1(\nu_2-2)} &\nu_2>2\\ \mbox{Does not exist} &\nu_2\le2\\ \end{cases}$

and

$\mbox{Var}\left[F\right]= \begin{cases} 2\frac{(\nu_1+\lambda)^2+(\nu_1+2\lambda)(\nu_2-2)}{(\nu_2-2)^2(\nu_2-4)}\left(\frac{\nu_2}{\nu_1}\right)^2 &\nu_2>4\\ \mbox{Does not exist} &\nu_2\le4\\ \end{cases}.$

[edit] Special cases

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

[edit] Related distributions

$Z$ has a noncentral chi-square distribution if $Z=\lim_{\nu_2\to\infty}\nu_1F$ where $F$ has a noncentral F-distribution.

[edit] See also

[edit] Implementations

The noncentral F-distribution is implemented in the R programming language (e.g., pf function), in MATLAB (ncfcdf, ncfinv, ncfpdf, ncfrnd and ncfstat functions in the statistics toolbox) and in Mathematica (NoncentralFRatioDistribution function).

[edit] External links

Free Noncentral F-distribution Cumulative Density Function (CDF) Calculator

[edit] References

^ S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, (New Jersey: Prentice Hall, 1998), p.29.

Eric W. Weisstein et al., Noncentral F-distribution, from MathWorld.

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