g01gc returns the probability associated with the lower tail of the noncentral

χ^{2}

-distribution .

Syntax

C#
public static double g01gc( double x, double df, double rlamda, double tol, int maxit, out int ifail )

Visual Basic
Public Shared Function g01gc ( _ x As Double, _ df As Double, _ rlamda As Double, _ tol As Double, _ maxit As Integer, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01gc( double x, double df, double rlamda, double tol, int maxit, [OutAttribute] int% ifail )

F#
static member g01gc : x : float * df : float * rlamda : float * tol : float * maxit : int * ifail : int byref -> float

Parameters

x: Type: System..::..Double
On entry: the deviate from the noncentral $χ^{2}$ -distribution with $ν$ degrees of freedom and noncentrality parameter $λ$ .

Constraint: $x \geq 0.0$ .

df: Type: System..::..Double
On entry: $ν$ , the degrees of freedom of the noncentral $χ^{2}$ -distribution.

Constraint: $df \geq 0.0$ .

rlamda: Type: System..::..Double
On entry: $λ$ , the noncentrality parameter of the noncentral $χ^{2}$ -distribution.

Constraint: $rlamda \geq 0.0$ if $df > 0.0$ or $rlamda > 0.0$ if $df = 0.0$ .

tol: Type: System..::..Double
On entry: the required accuracy of the solution. If g01gc is entered with tol greater than or equal to $1.0$ or less than $10 \times machine precision$ (see x02aj), then the value of $10 \times machine precision$ is used instead.

maxit: Type: System..::..Int32
On entry: the maximum number of iterations to be performed.
Suggested value: $100$ . See [Further Comments] for further discussion.

Constraint: $maxit \geq 1$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01gc returns the probability associated with the lower tail of the noncentral

χ^{2}

-distribution .

Description

The lower tail probability of the noncentral

χ^{2}

-distribution with

ν

degrees of freedom and noncentrality parameter

λ

P (X \leq x : ν; λ)

, is defined by

P (X \leq x : ν; λ) = \sum_{j = 0}^{\infty} e^{- λ / 2} \frac{{(λ / 2)}^{j}}{j!} P (X \leq x : ν + 2 j; 0),

(1)

where

P (X \leq x : ν + 2 j; 0)

is a central

χ^{2}

-distribution with

ν + 2 j

degrees of freedom.

The value of

j

at which the Poisson weight,

e^{- λ / 2} \frac{{(λ / 2)}^{j}}{j!}

, is greatest is determined and the summation (1) is made forward and backward from that value of

j

The recursive relationship:

P (X \leq x : a + 2; 0) = P (X \leq x : a; 0) - \frac{(x^{a} / 2) e^{- x / 2}}{Γ (a + 1)}

(2)

is used during the summation in (1).

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Error Indicators and Warnings

Note: g01gc may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

If on exit

ifail = 1

2

4

5

, then g01gc returns

0.0

$ifail = 1$

On entry,	$df < 0.0$ ,
or	$rlamda < 0.0$ ,
or	$df = 0.0$ and $rlamda = 0.0$ ,
or	$x < 0.0$ ,
or	$maxit < 1$ .

$ifail = 2$: The initial value of the Poisson weight used in the summation (1) was too small to be calculated. The value of $P (x \leq x : ν; λ)$ is likely to be zero.

$ifail = 3$: The solution has failed to converge in maxit iterations.

$ifail = 4$: The value of a term required in (2) is too large to be evaluated accurately. The most likely cause of this error is both x and rlamda being very large.

$ifail = 5$: The calculations for the central $χ^{2}$ probability has failed to converge. This is an unlikely error exit. A larger value of tol should be used.

$ifail = -9000$: An error occured, see message report.

Accuracy

The summations described in [Description] are made until an upper bound on the truncation error relative to the current summation value is less than tol.

Parallelism and Performance

None.

Further Comments

The number of terms in (1) required for a given accuracy will depend on the following factors:

(i)	The rate at which the Poisson weights tend to zero. This will be slower for larger values of $λ$ .
(ii)	The rate at which the central $χ^{2}$ probabilities tend to zero. This will be slower for larger values of $ν$ and $x$ .

Example

This example reads values from various noncentral

χ^{2}

-distributions, calculates the lower tail probabilities and prints all these values until the end of data is reached.

Example program (C#): g01gce.cs

Example program data: g01gce.d

Example program results: g01gce.r