Integer, Intent (In)	::	mult(n), n, maxit
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	a(n), rlamda(n), c, tol
Real (Kind=nag_wp), Intent (Out)	::	p, pdf, wrk(n+2*maxit)

C Header Interface

#include <nag.h>

void	g01jcf_ (const double a[], const Integer mult[], const double rlamda[], const Integer n, const double c, double p, double pdf, const double tol, const Integer maxit, double wrk[], Integer *ifail)

The routine may be called by the names g01jcf or nagf_stat_prob_chisq_noncentral_lincomb.

3 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

4 References

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

χ^{2}

random variables Appl. Statist. 33(3)

5 Arguments

1: $a (n)$ – Real (Kind=nag_wp) array Input: On entry: the weights, $a_{1}, a_{2}, \dots, a_{n}$ .

Constraint: $a (i) > 0.0$ , for $i = 1, 2, \dots, n$ .
2: $mult (n)$ – Integer array Input: On entry: the degrees of freedom, $m_{1}, m_{2}, \dots, m_{n}$ .

Constraint: $mult (i) \geq 1$ , for $i = 1, 2, \dots, n$ .
3: $rlamda (n)$ – Real (Kind=nag_wp) array Input: On entry: the noncentrality parameters, $λ_{1}, λ_{2}, \dots, λ_{n}$ .

Constraint: $rlamda (i) \geq 0.0$ , for $i = 1, 2, \dots, n$ .
4: $n$ – Integer Input: On entry: $n$ , the number of $χ^{2}$ random variables in the combination, i.e., the number of terms in equation (1).

Constraint: $n \geq 1$ .
5: $c$ – Real (Kind=nag_wp) Input: On entry: $c$ , the point for which the lower tail probability is to be evaluated.

Constraint: $c \geq 0.0$ .
6: $p$ – Real (Kind=nag_wp) Output: 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 $j = 1, 2, \dots, n$ .
7: $pdf$ – Real (Kind=nag_wp) Output: On exit: the value of the probability density function of the linear combination of $χ^{2}$ variables.
8: $tol$ – Real (Kind=nag_wp) Input: On entry: the relative accuracy required by you in the results. If g01jcf is entered with tol greater than or equal to $1.0$ or less than $10 \times machine precision$ (see x02ajf), the value of $10 \times machine precision$ is used instead.
9: $maxit$ – Integer Input: On entry: the maximum number of terms that should be used during the summation.

Suggested value: $500$ .

Constraint: $maxit \geq 1$ .
10: $wrk (n + 2 \times maxit)$ – Real (Kind=nag_wp) array Workspace
11: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $−1$ is recommended since useful values can be provided in some output arguments even when $ifail \neq 0$ on exit. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

Note: in some cases g01jcf may return useful information.

If on exit

ifail = 1

2

, then g01jcf returns

0.0

$ifail = 1$: On entry, $c = ⟨ value ⟩$ .
Constraint: $c \geq 0.0$ .

On entry, $maxit = ⟨ value ⟩$ .
Constraint: $maxit \geq 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = 2$: On entry, $a (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $a (i) > 0.0$ , for $i = 1, 2, \dots, n$ .

On entry, $mult (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $mult (i) \geq 1$ , for $i = 1, 2, \dots, n$ .

On entry, $rlamda (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $rlamda (i) \geq 0.0$ , for $i = 1, 2, \dots, n$ .

$ifail = 3$: 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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

8 Parallelism and Performance

g01jcf is not threaded in any implementation.

9 Further Comments

None.

10 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.

g01jc: FL CL CPP AD

NAG FL Interfaceg01jcf (prob_​chisq_​noncentral_​lincomb)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g01jcf (prob_chisq_noncentral_lincomb)