Lambert series
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 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 for the noncentral F-distribution is [1]
when and zero otherwise. The degrees of freedom and are positive. The noncentrality parameter is nonnegative. 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.
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
- ↑ S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, (New Jersey: Prentice Hall, 1998), p. 29.
- ↑ Template:Cite web
- ↑ Template:Cite web
References
External links
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