Non-central Student t distribution

The non-central t distribution, also known as the non-central Student's t distribution, is a generalization of Student's t distribution that includes a non-centrality parameter.

Definition

The non-central t distribution has degrees of freedom $ν$ and non-centrality parameter $λ$ . The probability density function (PDF) is:

f (x; ν, λ) = \frac{ν^{ν / 2} e^{- ν λ^{2} / 2 (λ^{2} + ν)}}{B (ν / 2, 1 / 2) \sqrt{π (λ^{2} + ν)}} {(1 + \frac{x^{2}}{ν})}^{- (ν + 1) / 2} M_{- (ν + 1) / 2}^{1 / 2} (\frac{λ^{2} x^{2}}{ν (λ^{2} + ν)})

where $M_{p}^{q} (z)$ is the Whittaker M function and $B (a, b)$ is the beta function.

Applications

Statistical power analysis uses it for t-tests.
Confidence interval calculations employ it when the population mean is non-zero.
Quality control uses it in process monitoring with small samples.

Properties

Statistical Properties

Property	Value
Mean	$λ \sqrt{\frac{ν}{2}} \frac{Γ (\frac{ν - 1}{2})}{Γ (\frac{ν}{2})}, ν > 1$
Variance	$\frac{ν (1 + λ^{2})}{ν - 2} - λ^{2} ν {(\frac{Γ (\frac{ν - 1}{2})}{2 Γ (\frac{ν}{2})})}^{2}, ν > 2$
Mode	No closed form

Notable properties include:

The distribution is symmetric about $λ$ when $ν \to \infty$ .
Higher moments involve complex hypergeometric functions.
The distribution has heavier tails than the normal distribution.

Relationships to Other Distributions

When $λ = 0$ , it reduces to Student's t distribution.
As $ν \to \infty$ , it approaches a normal distribution with mean $λ$ and unit variance.
It is related to the non-central F distribution through a quadratic transformation.

The NonCentralStudentTDistribution class

The non-central t distribution is implemented by the NonCentralStudentTDistribution class. It has one constructor with the degrees of freedom as its only argument.

The following constructs a non-central student t distribution with 8 degrees of freedom and non-centrality parameter 12:

var ncStudentT = new NonCentralStudentTDistribution(8, 12.0);

Visual Basic

Dim ncStudentT = New NonCentralStudentTDistribution(8, 12)

Visual Basic

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

let ncStudentT = NonCentralStudentTDistribution(8.0, 12.0)

The NonCentralStudentTDistribution 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 2. Wiley.
Abramowitz, M., & Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards.