This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 BUILT-IN MATHEMATICA SYMBOL

# NoncentralFRatioDistribution

 NoncentralFRatioDistribution represents a noncentral F-ratio distribution with n numerator degrees of freedom, m denominator degrees of freedom, and numerator noncentrality parameter . NoncentralFRatioDistributionrepresents a doubly noncentral F-ratio distribution with numerator noncentrality parameter and denominator noncentrality parameter .
• The noncentral F-ratio distribution is the distribution of the ratio of a noncentral random variable and a random variable divided by their respective degrees of freedom.
• The doubly noncentral F-ratio distribution is the distribution of a ratio of two noncentral -distributed random variables divided by their respective degrees of freedom.
Probability density function:
Probability density function for a doubly noncentral F-ratio distribution:
Cumulative distribution function:
Cumulative distribution function for a doubly noncentral F-ratio distribution:
Mean and variance:
Mean and variance for doubly noncentral:
Probability density function:
 Out[1]=
 Out[2]=
 Out[3]=
 Out[4]=

Probability density function for a doubly noncentral F-ratio distribution:
 Out[1]=

Cumulative distribution function:
 Out[1]=
 Out[2]=
 Out[3]=

Cumulative distribution function for a doubly noncentral F-ratio distribution:
 Out[1]=

Mean and variance:
 Out[1]=
 Out[2]=
Mean and variance for doubly noncentral:
 Out[3]=
 Out[4]=
 Scope   (8)
Generate a set of pseudorandom numbers that have a noncentral F-ratio distribution:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare the density histogram of the sample with the PDF of the estimated distribution:
Skewness depends on degrees of freedom m, n as well as noncentrality :
Kurtosis depends on degrees of freedom m, n as well as noncentrality :
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Hazard function:
Hazard function of a doubly noncentral F-ratio distribution:
Quantile function:
 Applications   (1)
NoncentralFRatioDistribution appears in computation of the power function of a hypothesis test about coefficients of linear model fit. The following 21 sample points were measured in an experiment:
Construct the linear model for the data in the form :
Hypothesis testing about coefficients and having simultaneously particular values is done using the -statistic that follows FRatioDistribution with 2 and 19 degrees of freedom, respectively:
Compute the value of the -statistic under the null hypothesis that and :
The critical value at the 5% significance level:
Hence the alternative hypothesis can not be rejected:
Assuming that the true values are actually 1.37 and 2.88, -statistic will follow a NoncentralFRatioDistribution with noncentrality parameter :
The power of the test, assuming true values of and :
Plot the power as a function of the noncentrality parameter:
Parameter influence on the CDF for each :
Relationships to other distributions:
Noncentral F-ratio distribution simplifies to FRatioDistribution:
Doubly noncentral F-ratio distribution simplifies to FRatioDistribution:
Doubly noncentral F-ratio distribution simplifies to noncentral F-ratio distribution:
The ratio of two NoncentralChiSquareDistribution follows a noncentral F-ratio distribution:
NoncentralFRatioDistribution is not defined when n or m is not a positive real number:
NoncentralFRatioDistribution is not defined when is not a positive real number:
The characteristic function of a noncentral F-ratio distribution has no closed-form representation:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: