# NAG Library Routine Document

## 1Purpose

g01jcf returns the lower tail probability of a distribution of a positive linear combination of ${\chi }^{2}$ random variables.

## 2Specification

Fortran Interface
 Subroutine g01jcf ( a, mult, n, c, p, pdf, tol, wrk,
 Integer, Intent (In) :: mult(n), n, maxit Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a(n), rlamda(n), c, tol Real (Kind=nag_wp), Intent (Out) :: p, pdf, wrk(n+2*maxit)
#include nagmk26.h
 void g01jcf_ (const double a[], const Integer mult[], const double rlamda[], const Integer *n, const double *c, double *p, double *pdf, const double *tol, const Integer *maxit, double wrk[], Integer *ifail)

## 3Description

For a linear combination of noncentral ${\chi }^{2}$ random variables with integer degrees of freedom the lower tail probability is
 $P ∑j=1najχ2mj,λj≤c ,$ (1)
where ${a}_{j}$ and $c$ are positive constants and where ${\chi }^{2}\left({m}_{j},{\lambda }_{j}\right)$ represents an independent ${\chi }^{2}$ random variable with ${m}_{j}$ degrees of freedom and noncentrality argument ${\lambda }_{j}$. 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
 $∑k=0∞dkF m+2k,c/β ,$ (2)
where $F\left(m+2k,c/\beta \right)=P\left({\chi }^{2}\left(m+2k\right), i.e., the probability that a central ${\chi }^{2}$ is less than $c/\beta$.
The value of $\beta$ is set at
 $β=βB=21/amin+1/amax$
unless ${\beta }_{B}>1.8{a}_{\mathrm{min}}$, in which case
 $β=βA=amin$
is used, where ${a}_{\mathrm{min}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left\{{a}_{j}\right\}$ and ${a}_{\mathrm{max}}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{a}_{j}\right\}$, for $\mathit{j}=1,2,\dots ,n$.

## 4References

Farebrother R W (1984) The distribution of a positive linear combination of ${\chi }^{2}$ random variables Appl. Statist. 33(3)

## 5Arguments

1:     $\mathbf{a}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the weights, ${a}_{1},{a}_{2},\dots ,{a}_{n}$.
Constraint: ${\mathbf{a}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
2:     $\mathbf{mult}\left({\mathbf{n}}\right)$ – Integer arrayInput
On entry: the degrees of freedom, ${m}_{1},{m}_{2},\dots ,{m}_{n}$.
Constraint: ${\mathbf{mult}}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:     $\mathbf{rlamda}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the noncentrality parameters, ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}$.
Constraint: ${\mathbf{rlamda}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of ${\chi }^{2}$ random variables in the combination, i.e., the number of terms in equation (1).
Constraint: ${\mathbf{n}}\ge 1$.
5:     $\mathbf{c}$ – Real (Kind=nag_wp)Input
On entry: $c$, the point for which the lower tail probability is to be evaluated.
Constraint: ${\mathbf{c}}\ge 0.0$.
6:     $\mathbf{p}$ – Real (Kind=nag_wp)Output
On exit: the lower tail probability associated with the linear combination of $n$ ${\chi }^{2}$ random variables with ${m}_{\mathit{j}}$ degrees of freedom, and noncentrality arguments ${\lambda }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$.
7:     $\mathbf{pdf}$ – Real (Kind=nag_wp)Output
On exit: the value of the probability density function of the linear combination of ${\chi }^{2}$ variables.
8:     $\mathbf{tol}$ – Real (Kind=nag_wp)Input
On entry: the relative accuracy required by you in the results. If g01jcf is entered with tol greater than or equal to $1.0$ or less than  (see x02ajf), the value of  is used instead.
9:     $\mathbf{maxit}$ – IntegerInput
On entry: the maximum number of terms that should be used during the summation.
Suggested value: $500$.
Constraint: ${\mathbf{maxit}}\ge 1$.
10:   $\mathbf{wrk}\left({\mathbf{n}}+2×{\mathbf{maxit}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Note: g01jcf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If on exit ${\mathbf{ifail}}={\mathbf{1}}$ or ${\mathbf{2}}$, then g01jcf returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{c}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{c}}\ge 0.0$.
On entry, ${\mathbf{maxit}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{maxit}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{a}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
On entry, ${\mathbf{mult}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mult}}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
On entry, ${\mathbf{rlamda}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rlamda}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ifail}}=3$
The central ${\chi }^{2}$ calculation has failed to converge. This is an unlikely exit. A larger value of tol should be tried.
${\mathbf{ifail}}=4$
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.
${\mathbf{ifail}}=5$
The solution appears to be too close to $0$ or $1$ for accurate calculation. The value returned is $0$ or $1$ as appropriate.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The series (2) is summed until a bound on the truncation error is less than tol. See Farebrother (1984) for further discussion.

## 8Parallelism and Performance

g01jcf is not threaded in any implementation.

None.

## 10Example

The number of ${\chi }^{2}$ variables is read along with their coefficients, degrees of freedom and noncentrality arguments. The lower tail probability is then computed and printed.

### 10.1Program Text

Program Text (g01jcfe.f90)

### 10.2Program Data

Program Data (g01jcfe.d)

### 10.3Program Results

Program Results (g01jcfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017