NAG Library Routine Document

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 <nagmk26.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)

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) arrayInput: 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 arrayInput: 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) arrayInput: 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: $n$ – IntegerInput: 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: $maxit$ – IntegerInput: 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) arrayWorkspace
11: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if $ifail \neq 0$ on exit, the recommended value is $- 1$ . 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).

Note: g01jcf may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

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 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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.

NAG Library Routine Document

g01jcf (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

1

Purpose

2

Specification