Noncentral F-distribution
In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a noncentral 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 with 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 is a noncentral chi-squared random variable with noncentrality parameter and degrees of freedom, and is a chi-squared random variable with degrees of freedom that is statistically independent of , then
is a noncentral F-distributed random variable. The probability density function (pdf) for the noncentral F-distribution is[1]
when and zero otherwise. The degrees of freedom and are positive. The term is the beta function, where
The cumulative distribution function for the noncentral F-distribution is
where is the regularized incomplete beta function.
The mean and variance of the noncentral F-distribution are
and
Special cases
When λ = 0, the noncentral F-distribution becomes the F-distribution.
Related distributions
Z has a noncentral chi-squared distribution if
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.[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
- ^ S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, (New Jersey: Prentice Hall, 1998), p. 29.
- ^ 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.
- ^ Sigbert Klinke (10 December 2008). "Comparison of noncentral and central distributions". Humboldt-Universität zu Berlin.
References
- Weisstein, Eric W.; et al. "Noncentral F-distribution". MathWorld. Wolfram Research, Inc. Retrieved 20 August 2011.
- v
- t
- e
univariate
with finite support | |
---|---|
with infinite support |
univariate
univariate
continuous- discrete |
---|
(joint)
- Discrete:
- Ewens
- Multinomial
- Continuous:
- Dirichlet
- Multivariate Laplace
- Multivariate normal
- Multivariate stable
- Multivariate t
- Normal-gamma
- Matrix-valued:
- LKJ
- Matrix normal
- Matrix t
- Matrix gamma
- Wishart
- Univariate (circular) directional
- Circular uniform
- Univariate von Mises
- Wrapped normal
- Wrapped Cauchy
- Wrapped exponential
- Wrapped asymmetric Laplace
- Wrapped Lévy
- Bivariate (spherical)
- Kent
- Bivariate (toroidal)
- Bivariate von Mises
- Multivariate
- von Mises–Fisher
- Bingham
and singular
- Degenerate
- Dirac delta function
- Singular
- Cantor
Category
Commons