g01jc returns the lower tail probability of a distribution of a positive linear combination of

χ^{2}

random variables.

Syntax

C#
public static void g01jc( double[] a, int[] mult, double[] rlamda, int n, double c, out double p, out double pdf, double tol, int maxit, out int ifail )

Visual Basic
Public Shared Sub g01jc ( _ a As Double(), _ mult As Integer(), _ rlamda As Double(), _ n As Integer, _ c As Double, _ <OutAttribute> ByRef p As Double, _ <OutAttribute> ByRef pdf As Double, _ tol As Double, _ maxit As Integer, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g01jc ( _
	a As Double(), _
	mult As Integer(), _
	rlamda As Double(), _
	n As Integer, _
	c As Double, _
	<OutAttribute> ByRef p As Double, _
	<OutAttribute> ByRef pdf As Double, _
	tol As Double, _
	maxit As Integer, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g01jc( array<double>^ a, array<int>^ mult, array<double>^ rlamda, int n, double c, [OutAttribute] double% p, [OutAttribute] double% pdf, double tol, int maxit, [OutAttribute] int% ifail )

Visual C++

public:
static void g01jc(
	array<double>^ a, 
	array<int>^ mult, 
	array<double>^ rlamda, 
	int n, 
	double c, 
	[OutAttribute] double% p, 
	[OutAttribute] double% pdf, 
	double tol, 
	int maxit, 
	[OutAttribute] int% ifail
)

F#
static member g01jc : a : float[] * mult : int[] * rlamda : float[] * n : int * c : float * p : float byref * pdf : float byref * tol : float * maxit : int * ifail : int byref -> unit

Parameters

a: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: the weights, $a_{1}, a_{2}, \dots, a_{n}$ .

Constraint: $a [i] > 0.0$ , for $i = 0, 1, \dots, n - 1$ .

mult: Type: array<System..::..Int32>[]()[][]
An array of size [n]
On entry: the degrees of freedom, $m_{1}, m_{2}, \dots, m_{n}$ .

Constraint: $mult [i] \geq 1$ , for $i = 0, 1, \dots, n - 1$ .

rlamda: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: the noncentrality parameters, $λ_{1}, λ_{2}, \dots, λ_{n}$ .

Constraint: $rlamda [i] \geq 0.0$ , for $i = 0, 1, \dots, n - 1$ .

n: Type: System..::..Int32
On entry: $n$ , the number of $χ^{2}$ random variables in the combination, i.e., the number of terms in equation (1).

Constraint: $n \geq 1$ .

c: Type: System..::..Double
On entry: $c$ , the point for which the lower tail probability is to be evaluated.

Constraint: $c \geq 0.0$ .

p: Type: System..::..Double%
On exit: the lower tail probability associated with the linear combination of $n$ $χ^{2}$ random variables with $m_{j}$ degrees of freedom, and noncentrality parameters $λ_{j}$ , for $j = 1, 2, \dots, n$ .

pdf: Type: System..::..Double%
On exit: the value of the probability density function of the linear combination of $χ^{2}$ variables.

tol: Type: System..::..Double
On entry: the relative accuracy required by you in the results. If g01jc 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 terms that should be used during the summation.
Suggested value: $500$ .

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]).

Description

For a linear combination of noncentral

χ^{2}

random variables with integer degrees of freedom the lower tail probability is

P (\sum_{j = 1}^{n} a_{j} χ^{2} (m_{j}, λ_{j}) \leq c),

(1)

where

a_{j}

and

c

are positive constants and where

χ^{2} (m_{j}, λ_{j})

represents an independent

χ^{2}

random variable with

m_{j}

degrees of freedom and noncentrality parameter

λ_{j}

. The linear combination may arise from considering a quadratic form in Normal variables.

Ruben's method as described in Farebrother (1984) is used. Ruben has shown that (1) may be expanded as an infinite series of the form

\sum_{k = 0}^{\infty} d_{k} F (m + 2 k, c / β),

(2)

where

F (m + 2 k, c / β) = P (χ^{2} (m + 2 k) < c / β)

, i.e., the probability that a central

χ^{2}

is less than

c / β

The value of

β

is set at

β = β_{B} = \frac{2}{(1 / a_{\min} + 1 / a_{\max})}

unless

β_{B} > 1.8 a_{\min}

, in which case

β = β_{A} = a_{\min}

is used, where

a_{\min} = \min \{a_{j}\}

and

a_{\max} = \max \{a_{j}\}

, for

j = 1, 2, \dots, n

References

Farebrother R W (1984) The distribution of a positive linear combination of

χ^{2}

random variables Appl. Statist. 33(3)

Error Indicators and Warnings

Note: g01jc 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

, then g01jc returns

0.0

$ifail = 1$

On entry,	$n < 1$ ,
or	$maxit < 1$ ,
or	$c < 0.0$ .

$ifail = 2$

On entry,	a has an element $\leq 0.0$ ,
or	mult has an element $< 1$ ,
or	rlamda has an element $< 0.0$ .

$ifail = 3$: The central $χ^{2}$ calculation has failed to converge. This is an unlikely exit. A larger value of tol should be tried.

$ifail = 4$: The solution has failed to converge within maxit iterations. A larger value of maxit or tol should be used. The returned value should be a reasonable approximation to the correct value.

$ifail = 5$: The solution appears to be too close to $0$ or $1$ for accurate calculation. The value returned is $0$ or $1$ as appropriate.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

The series (2) is summed until a bound on the truncation error is less than tol. See Farebrother (1984) for further discussion.

Parallelism and Performance

None.

Further Comments

None.

Example

The number of

χ^{2}

variables is read along with their coefficients, degrees of freedom and noncentrality parameters. The lower tail probability is then computed and printed.

Example program (C#): g01jce.cs

Example program data: g01jce.d

Example program results: g01jce.r