void	nag_prob_lin_non_central_chi_sq (const double a[], const Integer mult[], const double rlamda[], Integer n, double c, double p, double pdf, double tol, Integer maxit, NagError *fail)

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 argument

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: $a [n]$ – const doubleInput: On entry: the weights, $a_{1}, a_{2}, \dots, a_{n}$ .

Constraint: $a [i] > 0.0$ , for $i = 0, 1, \dots, n - 1$ .
2: $mult [n]$ – const IntegerInput: On entry: the degrees of freedom, $m_{1}, m_{2}, \dots, m_{n}$ .

Constraint: $mult [i] \geq 1$ , for $i = 0, 1, \dots, n - 1$ .
3: $rlamda [n]$ – const doubleInput: On entry: the noncentrality parameters, $λ_{1}, λ_{2}, \dots, λ_{n}$ .

Constraint: $rlamda [i] \geq 0.0$ , for $i = 0, 1, \dots, n - 1$ .
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: $c$ – doubleInput: On entry: $c$ , the point for which the lower tail probability is to be evaluated.

Constraint: $c \geq 0.0$ .
6: $p$ – double *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 arguments $λ_{j}$ , for $j = 1, 2, \dots, n$ .
7: $pdf$ – double *Output: On exit: the value of the probability density function of the linear combination of $χ^{2}$ variables.
8: $tol$ – doubleInput: On entry: the relative accuracy required by you in the results. If nag_prob_lin_non_central_chi_sq (g01jcc) is entered with tol greater than or equal to $1.0$ or less than $10 \times machine precision$ (see nag_machine_precision (X02AJC)), 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: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ACCURACY: 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.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONVERGENCE: The central $χ^{2}$ calculation has failed to converge. This is an unlikely exit. A larger value of tol should be tried.
NE_INT: On entry, $maxit = 〈value〉$ .
Constraint: $maxit \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
NE_INT_ARRAY: On entry, $mult [〈value〉] = 〈value〉$ .
Constraint: $mult [i] \geq 1$ , for $i = 0, 1, \dots, n - 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_PROB_BOUNDARY: The solution appears to be too close to $0$ or $1$ for accurate calculation. The value returned is $0$ or $1$ as appropriate.
NE_REAL: On entry, $c = 〈value〉$ .
Constraint: $c \geq 0.0$ .
NE_REAL_ARRAY: On entry, $a [〈value〉] = 〈value〉$ .
Constraint: $a [i] > 0.0$ , for $i = 0, 1, \dots, n - 1$ .

On entry, $rlamda [〈value〉] = 〈value〉$ .
Constraint: $rlamda [i] \geq 0.0$ , for $i = 0, 1, \dots, n - 1$ .

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

nag_prob_lin_non_central_chi_sq (g01jcc) 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 arguments. The lower tail probability is then computed and printed.

NAG Library Function Document

nag_prob_lin_non_central_chi_sq (g01jcc)

▸▿ 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