Noncentral F-distribution

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

If X is a noncentral chi-squared random variable with noncentrality parameter \lambda and \nu_1 degrees of freedom, and Y is a chi-squared random variable with \nu_2 degrees of freedom that is statistically independent of X, then


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[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 \nu_1 and \nu_2 are positive. The noncentrality parameter \lambda is nonnegative. The term B(x,y) is the beta function, where


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

The cumulative distribution function for the noncentral F-distribution is


F(x|d_1,d_2,\lambda)=\sum\limits_{j=0}^\infty\left(\frac{\left(\frac{1}{2}\lambda\right)^j}{j!}e^{-\frac{\lambda}{2}}\right)I\left(\frac{d_1x}{d_2 + d_1x}\bigg|\frac{d_1}{2}+j,\frac{d_2}{2}\right)

where I is the regularized incomplete beta function.

The mean and variance of the noncentral F-distribution are


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

and


\operatorname{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\\
\text{Does not exist}
&\nu_2\le4.\\
\end{cases}

Differential equation[edit]

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

\left\{\begin{array}{l}
4 x \left(\nu _2+\nu_1 x\right){}^2 f''(x)+f'(x) \left(-2 \nu _2^2 \nu _1+8 \nu _2^2+
  16 \nu _1^2 x^2+4 \nu_2 \nu_1^2 x^2-2 \lambda \nu_2 \nu _1 x-2 \nu_2 \nu_1^2 x+4 \nu_2^2
   \nu_1 x+24 \nu_2 \nu_1 x\right)+\nu_1 \left(\nu_2+2\right) f(x) \left(-\lambda
  \nu_2-\nu_2 \nu_1+4 \nu_2+4 \nu_1 x+\nu_2 \nu_1 x\right)=0, \\[12pt]
f(1)=\frac{e^{-\lambda /2} \nu_1^{\frac{\nu_1}{2}} \nu_2^{\frac{\nu _2}{2}}
  \left(\nu _1+\nu _2\right){}^{\frac{1}{2} \left(-\nu _1-\nu _2\right)} \,
  _1F_1\left(\frac{1}{2} \left(\nu _1+\nu _2\right);\frac{\nu_1}{2};
  \frac{\lambda  \nu _1}{2 \left(\nu _1+\nu _2\right)}\right)}{B\left(\frac{\nu_1}{2},
  \frac{\nu_2}{2}\right)}, \\[12pt]
f'(1)=\frac{e^{-\lambda/2} \nu_1^{\frac{\nu_1}{2}} \nu_2^{\frac{\nu_2}{2}}
   \left(\nu _1+\nu _2\right){}^{\frac{1}{2} \left(-\nu_1-\nu_2-2\right)}
   \left(\nu_2 \left(\lambda \, _1F_1\left(\frac{1}{2} \left(\nu_1+\nu_2+2\right);
   \frac{1}{2} \left(\nu_1+2\right);\frac{\lambda \nu_1}{2 \left(\nu_1+\nu_2\right)}\right)-2 \,
   _1F_1\left(\frac{1}{2} \left(\nu_1+\nu_2\right);\frac{\nu_1}{2};
   \frac{\lambda\nu_1}{2 \left(\nu_1+\nu _2\right)}\right)\right)-2 \nu _1 \,
   _1F_1\left(\frac{1}{2} \left(\nu_1+\nu_2\right);\frac{\nu_1}{2};
   \frac{\lambda \nu_1}{2 \left(\nu_1+\nu_2\right)}\right)\right)}
  {2 B\left(\frac{\nu_1}{2},\frac{\nu_2}{2}\right)}
\end{array}\right\}

Special cases[edit]

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

Related distributions[edit]

Z has a noncentral chi-squared distribution if

 Z=\lim_{\nu_2\to\infty}\nu_1 F

where F has a noncentral F-distribution.

Implementations[edit]

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 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.[3]

Notes[edit]

  1. ^ S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, (New Jersey: Prentice Hall, 1998), p. 29.
  2. ^ John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde. "Noncentral F Distribution: Boost 1.39.0". Boost.org. Retrieved 20 August 2011. 
  3. ^ Sigbert Klinke (10 December 2008). "Comparison of noncentral and central distributions". Humboldt-Universität zu Berlin. 

References[edit]

External links[edit]