NoncentralFRatioDistribution

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:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Hazard function of a doubly noncentral F-ratio distribution:

Quantile function:

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:

NoncentralBetaDistribution is a transformation of NoncentralFRatioDistribution:

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: