Noncentral F Distribution

Definition

Similar to the noncentral χ² distribution, the toolbox calculates noncentral F distribution probabilities as a weighted sum of incomplete beta functions using Poisson probabilities as the weights.

$F (x | ν_{1}, ν_{2}, δ) = \sum_{j = 0}^{\infty} (\frac{{(\frac{1}{2} δ)}^{j}}{j!} e^{\frac{- δ}{2}}) I (\frac{ν_{1} \cdot x}{ν_{2} + ν_{1} \cdot x} | \frac{ν_{1}}{2} + j, \frac{ν_{2}}{2})$

I(x|a,b) is the incomplete beta function with parameters a and b, and δ is the noncentrality parameter.

Background

As with the χ² distribution, the F distribution is a special case of the noncentral F distribution. The F distribution is the result of taking the ratio of χ² random variables each divided by its degrees of freedom.

If the numerator of the ratio is a noncentral chi-square random variable divided by its degrees of freedom, the resulting distribution is the noncentral F distribution.

The main application of the noncentral F distribution is to calculate the power of a hypothesis test relative to a particular alternative.

Examples

Compute Noncentral F Distribution pdf

Open Live Script

Compute the pdf of a noncentral F distribution with degrees of freedom NU1 = 5 and NU2 = 20, and noncentrality parameter DELTA = 10. For comparison, also compute the pdf of an F distribution with the same degrees of freedom.

x = (0.01:0.1:10.01)';
p1 = ncfpdf(x,5,20,10);
p = fpdf(x,5,20);

Plot the pdf of the noncentral F distribution and the pdf of the F distribution on the same figure.

figure;
plot(x,p1,'b-','LineWidth',2)
hold on
plot(x,p,'g--','LineWidth',2)
legend('Noncentral F','F distribution')

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Noncentral F, F distribution.