NAG CL Interface
g01jcc (prob_​chisq_​noncentral_​lincomb)

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1 Purpose

g01jcc returns the lower tail probability of a distribution of a positive linear combination of χ2 random variables.

2 Specification

#include <nag.h>
void  g01jcc (const double a[], const Integer mult[], const double rlamda[], Integer n, double c, double *p, double *pdf, double tol, Integer maxit, NagError *fail)
The function may be called by the names: g01jcc, nag_stat_prob_chisq_noncentral_lincomb or nag_prob_lin_non_central_chi_sq.

3 Description

For a linear combination of noncentral χ2 random variables with integer degrees of freedom the lower tail probability is
P (j=1najχ2(mj,λj)c) , (1)
where aj and c are positive constants and where χ2(mj,λj) represents an independent χ2 random variable with mj degrees of freedom and noncentrality parameter λj. 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
k=0dkF (m+2k,c/β) , (2)
where F (m+2k,c/β)=P (χ2(m+2k)<c/β) , i.e., the probability that a central χ2 is less than c/β.
The value of β is set at
unless βB>1.8amin, in which case
is used, where amin=min{aj} and amax=max{aj}, for j=1,2,,n.

4 References

Farebrother R W (1984) The distribution of a positive linear combination of χ2 random variables Appl. Statist. 33(3)

5 Arguments

1: a[n] const double Input
On entry: the weights, a1,a2,,an.
Constraint: a[i]>0.0, for i=0,1,,n-1.
2: mult[n] const Integer Input
On entry: the degrees of freedom, m1,m2,,mn.
Constraint: mult[i]1, for i=0,1,,n-1.
3: rlamda[n] const double Input
On entry: the noncentrality parameters, λ1,λ2,,λn.
Constraint: rlamda[i]0.0, for i=0,1,,n-1.
4: n Integer Input
On entry: n, the number of χ2 random variables in the combination, i.e., the number of terms in equation (1).
Constraint: n1.
5: c double Input
On entry: c, the point for which the lower tail probability is to be evaluated.
Constraint: c0.0.
6: p double * Output
On exit: the lower tail probability associated with the linear combination of n χ2 random variables with mj degrees of freedom, and noncentrality parameters λj, for j=1,2,,n.
7: pdf double * Output
On exit: the value of the probability density function of the linear combination of χ2 variables.
8: tol double Input
On entry: the relative accuracy required by you in the results. If g01jcc is entered with tol greater than or equal to 1.0 or less than 10×machine precision (see X02AJC), the value of 10×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: maxit1.
10: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If on exit fail.code= NE_INT, NE_INT_ARRAY, NE_REAL or NE_REAL_ARRAY, then g01jcc returns 0.0.
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.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
The central χ2 calculation has failed to converge. This is an unlikely exit. A larger value of tol should be tried.
On entry, maxit=value.
Constraint: maxit1.
On entry, n=value.
Constraint: n1.
On entry, mult[value]=value.
Constraint: mult[i]1, for i=0,1,,n-1.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
The solution appears to be too close to 0 or 1 for accurate calculation. The value returned is 0 or 1 as appropriate.
On entry, c=value.
Constraint: c0.0.
On entry, a[value]=value.
Constraint: a[i]>0.0, for i=0,1,,n-1.
On entry, rlamda[value]=value.
Constraint: rlamda[i]0.0, for i=0,1,,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

Background information to multithreading can be found in the Multithreading documentation.
g01jcc is not threaded in any implementation.

9 Further Comments


10 Example

The number of χ2 variables is read along with their coefficients, degrees of freedom and noncentrality parameters. The lower tail probability is then computed and printed.

10.1 Program Text

Program Text (g01jcce.c)

10.2 Program Data

Program Data (g01jcce.d)

10.3 Program Results

Program Results (g01jcce.r)