NAG Library Function Document

nag_prob_lin_non_central_chi_sq (g01jcc)

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)), then 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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ACCURACY: The required accuracy could not be met in $〈value〉$ iterations.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONVERGENCE: The central Chi square has failed to converge.
NE_INT: On entry, $maxit = 〈value〉$ .
Constraint: $maxit \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
NE_INT_ARRAY: On entry, mult has an $element < 1$ : $mult [〈value〉] = 〈value〉$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_PROB_BOUNDARY: Calculated probability at boundary.
NE_REAL: On entry, $c = 〈value〉$ .
Constraint: $c \geq 0.0$ .
NE_REAL_ARRAY: On entry, a has an $element \leq 0.0$ : $a [〈value〉] = 〈value〉$ .

On entry, rlamda has an $element < 0.0$ : $rlamda [〈value〉] = 〈value〉$ .

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

Not applicable.

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 Documentnag_prob_lin_non_central_chi_sq (g01jcc)

▸▿ Contents

1 Purpose

2 Specification