nag_prob_lin_non_central_chi_sq (g01jcc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_prob_lin_non_central_chi_sq (g01jcc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagg01.h>
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 χ2 random variables with integer degrees of freedom the lower tail probability is
P j=1najχ2mj,λjc , (1)
where aj and c are positive constants and where χ2mj,λj represents an independent χ2 random variable with mj degrees of freedom and noncentrality argument 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 χ2m+2k<c/β , i.e., the probability that a central χ2 is less than c/β.
The value of β is set at
β=βB=21/amin+1/amax
unless βB>1.8amin, in which case
β=βA=amin
is used, where amin=minaj and amax=maxaj, 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 doubleInput
On entry: the weights, a1,a2,,an.
Constraint: a[i]>0.0, for i=0,1,,n-1.
2:     mult[n]const IntegerInput
On entry: the degrees of freedom, m1,m2,,mn.
Constraint: mult[i]1, for i=0,1,,n-1.
3:     rlamda[n]const doubleInput
On entry: the noncentrality parameters, λ1,λ2,,λn.
Constraint: rlamda[i]0.0, for i=0,1,,n-1.
4:     nIntegerInput
On entry: n, the number of χ2 random variables in the combination, i.e., the number of terms in equation (1).
Constraint: n1.
5:     cdoubleInput
On entry: c, the point for which the lower tail probability is to be evaluated.
Constraint: c0.0.
6:     pdouble *Output
On exit: the lower tail probability associated with the linear combination of n χ2 random variables with mj degrees of freedom, and noncentrality arguments λj, for j=1,2,,n.
7:     pdfdouble *Output
On exit: the value of the probability density function of the linear combination of χ2 variables.
8:     toldoubleInput
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×machine precision (see nag_machine_precision (X02AJC)), then the value of 10×machine precision is used instead.
9:     maxitIntegerInput
On entry: the maximum number of terms that should be used during the summation.
Suggested value: 500.
Constraint: maxit1.
10:   failNagError *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: maxit1.
On entry, n=value.
Constraint: n1.
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: c0.0.
NE_REAL_ARRAY
On entry, a has an element0.0: 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  Further Comments

None.

9  Example

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

9.1  Program Text

Program Text (g01jcce.c)

9.2  Program Data

Program Data (g01jcce.d)

9.3  Program Results

Program Results (g01jcce.r)


nag_prob_lin_non_central_chi_sq (g01jcc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012