Non-central F Distribution

The non-central F distribution, also known as the doubly non-central F distribution, is a generalization of the F distribution. It is used to model the quotient of two non-central chi-square variables.

Definition

The non-central F distribution has degrees of freedom parameters $ν_{1}$ , $ν_{2}$ , and non-centrality parameter $λ$ . The probability density function (PDF) is:

f (x) = e^{- λ / 2} \sum_{r = 0}^{\infty} \frac{(λ / 2)^{r}}{r!} \frac{Γ (\frac{ν_{1} + ν_{2} + 2 r}{2})}{Γ (\frac{ν_{2}}{2}) Γ (\frac{ν_{1} + 2 r}{2})} {(\frac{ν_{1} x}{ν_{2}})}^{(ν_{1} + 2 r - 2) / 2} (1 + \frac{ν_{1} x}{ν_{2}})^{- (ν_{1} + ν_{2} + 2 r) / 2}

where $ν_{1}, ν_{2} > 0$ and $λ \geq 0$ .

Applications

Analysis of variance (ANOVA) uses it for power calculations.
Regression analysis employs it in testing linear hypotheses.
Quality control uses it in process capability studies.

Properties

Statistical Properties

Property	Value ( $ν_{2} > k$ where k is the moment order)
Mean	$\frac{ν_{2} (ν_{1} + λ)}{ν_{1} (ν_{2} - 2)}$
Variance	$\frac{2 ν_{2}^{2} [(ν_{1} + λ)^{2} + (ν_{1} + 2 λ) (ν_{2} - 2)]}{ν_{1}^{2} (ν_{2} - 2)^{2} (ν_{2} - 4)}$
Mode	No closed form

Notable properties include:

The distribution is defined for positive real numbers.
Higher moments exist only when $ν_{2}$ is sufficiently large.
The distribution is always right-skewed.

Relationships to Other Distributions

When $λ = 0$ , it reduces to the central F-distribution.
It is the distribution of the ratio of a non-central chi-square to a central chi-square variable, each divided by their degrees of freedom.
For large $ν_{1}$ and $ν_{2}$ , it approaches a normal distribution.

The NonCentralFDistribution class

The non-central F distribution is implemented by the NonCentralFDistribution class. It has one constructor which takes the three parameters mentioned above as arguments. The following constructs a non-central F distribution with 4 degrees of freedom for the numerator, and 25 degrees of freedom for the denominator, and non-centrality parameter 15:

var ncf = new NonCentralFDistribution(4, 25, 15.0);

Visual Basic

Dim ncf = New NonCentralFDistribution(4, 25, 15.0)

Visual Basic

No code example is currently available or this language may not be supported.

let ncf = NonCentralFDistribution(4.0, 25.0, 15.0)

The NonCentralFDistribution class has three specific properties. NumeratorDegreesOfFreedom and DenominatorDegreesOfFreedom return the degrees of freedom of the numerator and the denominator, respectively. The NonCentralityParameter property returns the non-centrality parameter.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2. Wiley.
Hogg, R. V., McKean, J. W., & Craig, A. T. (2019). Introduction to Mathematical Statistics. Pearson.