Non-central Chi Square Distribution

The non-central chi square (χ²) distribution, also known as the generalized chi square distribution, is a generalization of the chi square distribution. It is used in the power analysis of statistical tests, including likelihood ratio tests.

Definition

The non-central chi square distribution with $n$ degrees of freedom and non-centrality parameter $λ$ is defined by its probability density function (PDF):

f (x; n, λ) = \frac{1}{2} e^{- (x + λ) / 2} {(\frac{x}{λ})}^{(n - 2) / 4} I_{(n - 2) / 2} (\sqrt{λ x})

where $I_{ν} (z)$ is the modified Bessel function of the first kind.

The parameters must satisfy $n > 0$ and $λ \geq 0$ .

Applications

Statistical power analysis uses it for chi-square and F-tests.
Signal processing employs it in radar detection theory.
Quality control uses it in process monitoring.

Properties

Statistical Properties

Property	Value
Mean	$n + λ$
Variance	$2 (2 λ + n)$
Skewness	$2 \sqrt{2} (3 λ + n) / (2 λ + n)^{3 / 2}$
Excess Kurtosis	$12 (4 λ + n) / (2 λ + n)^{2}$
Mode	$λ + n - 2$ (for $n \geq 2$ )

Notable properties include:

The characteristic function is $ϕ (t) = (1 - 2 i t)^{- n / 2} e^{λ i t / (1 - 2 i t)}$ .
The distribution is always right-skewed.
The moment generating function is $M (t) = (1 - 2 t)^{- n / 2} e^{λ t / (1 - 2 t)}$ .

Relationships to Other Distributions

When $λ = 0$ , it reduces to the central chi-square distribution.
It is the distribution of the sum of squares of independent normal variables with non-zero means.
For large $n$ , it approaches a normal distribution.

The NonCentralChiSquareDistribution class

The non-central chi square distribution is implemented by the NonCentralChiSquareDistribution class. It has one constructor which takes the degrees of freedom and the non-centrality parameter as arguments. The following constructs a non-central chi square distribution with 10 degrees of freedom and non-centrality parameter 15:

var ncChiSquare = new NonCentralChiSquareDistribution(10, 15);

Visual Basic

Dim ncChi = New NonCentralChiSquareDistribution(10, 15)

Visual Basic

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

let ncChiSquare = NonCentralChiSquareDistribution(10.0, 15.0)

The NonCentralChiSquareDistribution class has two specific properties. DegreesOfFreedom returns the degrees of freedom of the distribution. NonCentralityParameter returns the non-centrality parameter.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 1 (2nd ed.). Wiley.
Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to the Theory of Statistics (3rd ed.). McGraw-Hill.