Cdf.NoncentralstudentsT Method

CdfNoncentralstudentsT Method

Evaluates the noncentral Student's t cumulative probability distribution function.

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

public static double NoncentralstudentsT(
	double t,
	int idf,
	double delta
)

Public Shared Function NoncentralstudentsT ( 
	t As Double,
	idf As Integer,
	delta As Double
) As Double

public:
static double NoncentralstudentsT(
	double t, 
	int idf, 
	double delta
)

static member NoncentralstudentsT : 
        t : float * 
        idf : int * 
        delta : float -> float

Parameters

t: Type: SystemDouble
A double scalar value representing the argument at which the function is to be evaluated.
idf: Type: SystemInt32
An int scalar value representing the number of degrees of freedom. This must be positive.
delta: Type: SystemDouble
A double scalar value representing the noncentrality parameter.

Return Value

Type: Double
A double scalar value representing the probability that a noncentral Student's t random variable takes a value less than or equal to t.

Remarks

Method Cdf.NoncentralstudentsT evaluates the distribution function F of a noncentral t random variable with idf degrees of freedom and noncentrality parameter delta; that is, with $\nu=\mbox{idf}$ , $\delta=\mbox{delta}$ , and $t_{0}=\mbox{t}$ ,

$F{\left({t_0}\right)}=\int_{-{\infty}}^{ t_{0}}{\frac{\nu^{\nu/2}e^{{-\delta^2}/2}}{{\sqrt{\pi}\Gamma\left( \nu/2\right)\left(\nu+x^2\right)}^{\left(\nu+1\right)/2}}}\sum \limits_{i=0}^\infty{\Gamma\left(\left(\nu+i+1\right)/2\right) \left(\frac{\delta^i}{i!}\right)\left(\frac{2x^2}{\nu+x^2}\right)^{ i/2}dx}$

where $\Gamma(\cdot)$ is the gamma function. The value of the distribution function at the point $t_{0}$ is the probability that the random variable takes a value less than or equal to $t_{0}$ .

The noncentral t random variable can be defined by the distribution function above, or alternatively and equivalently, as the ratio of a normal random variable and an independent chi-squared random variable. If w has a normal distribution with mean $\delta$ and variance equal to one, has an independent chi-squared distribution with $\nu$ degrees of freedom, and

$x = w/\sqrt{u/\nu}$

then

has a noncentral

distribution with $\nu$ degrees of freedom and noncentrality parameter $\delta$ .

The distribution function of the noncentral can also be expressed as a double integral involving a normal density function (see, for example, Owen 1962, page 108). The method Cdf.NoncentralstudentsT uses the method of Owen (1962, 1965), which uses repeated integration by parts on that alternate expression for the distribution function.

Noncentral Student's t Distribution Function

Reference

Cdf Class

Imsl.Stat Namespace

Other Resources

Example