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.
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.
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