Cdf.Noncentralchi Method

CdfNoncentralchi Method

Evaluates the noncentral chi-squared cumulative probability distribution function.

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

Syntax

public static double Noncentralchi(
	double chsq,
	double df,
	double alam
)

Public Shared Function Noncentralchi ( 
	chsq As Double,
	df As Double,
	alam As Double
) As Double

public:
static double Noncentralchi(
	double chsq, 
	double df, 
	double alam
)

static member Noncentralchi : 
        chsq : float * 
        df : float * 
        alam : float -> float

Parameters

chsq: Type: SystemDouble
A double scalar value representing the argument at which the function is to be evaluated.
df: Type: SystemDouble
A double scalar value representing the number of degrees of freedom. df must be positive.
alam: Type: SystemDouble
A double scalar value representing the noncentrality parameter. This must be nonnegative, and alam + df must be less than or equal to 200,000.

Return Value

Type: Double
A double scalar value representing the probability that a chi-squared random variable takes a value less than or equal to chsq.

Remarks

Method Cdf.Noncentralchi evaluates the distribution function, F, of a noncentral chi-squared random variable with df degrees of freedom and noncentrality parameter alam, that is, with $\nu={\rm df}$ , $\lambda={\rm alam}$ , and $\chi={\rm chsq}$ ,

$F\left(x\right)=\sum\limits_{i=0}^\infty {\frac{e^{-\lambda/2}\left(\lambda/2\right)^i}{i!}}\int_0^x{\frac{ t^{\left(\nu+2i\right)/2-1}e^{-t/2}}{2^{\left(\nu+2i\right)/2}{ \Gamma\left(\frac{\nu+2i}{2}\right)}}}dt$

where $\Gamma (\cdot)$ is the gamma function. This is a series of central chi-squared distribution functions with Poisson weights. The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.

The noncentral chi-squared random variable can be defined by the distribution function above, or alternatively and equivalently, as the sum of squares of independent normal random variables. If the have independent normal distributions with means ${\mu}_i$ and variances equal to one and

$\chi=\sum\limits_{i=1}^n{Y_i}^2$

then $\chi$ has a noncentral chi-squared distribution with

degrees of freedom and noncentrality parameter equal to

$\sum\limits_{i=1}^n{\mu_i}^2$

With a noncentrality parameter of zero, the noncentral chi-squared distribution is the same as the chi-squared distribution.

Cdf.Noncentralchi determines the point at which the Poisson weight is greatest, and then sums forward and backward from that point, terminating when the additional terms are sufficiently small or when a maximum of 1000 terms have been accumulated. The recurrence relation 26.4.8 of Abramowitz and Stegun (1964) is used to speed the evaluation of the central chi-squared distribution functions.

Noncentral Chi-Squared Distribution Function

Reference

Cdf Class

Imsl.Stat Namespace

Other Resources

Example