# NAG Library Routine Document

## 1Purpose

g01tcf returns a number of deviates associated with the given probabilities of the ${\chi }^{2}$-distribution with real degrees of freedom.

## 2Specification

Fortran Interface
 Subroutine g01tcf ( tail, lp, p, ldf, df, x,
 Integer, Intent (In) :: ltail, lp, ldf Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: p(lp), df(ldf) Real (Kind=nag_wp), Intent (Out) :: x(*) Character (1), Intent (In) :: tail(ltail)
#include nagmk26.h
 void g01tcf_ (const Integer *ltail, const char tail[], const Integer *lp, const double p[], const Integer *ldf, const double df[], double x[], Integer ivalid[], Integer *ifail, const Charlen length_tail)

## 3Description

The deviate, ${x}_{{p}_{i}}$, associated with the lower tail probability ${p}_{i}$ of the ${\chi }^{2}$-distribution with ${\nu }_{i}$ degrees of freedom is defined as the solution to
 $P Xi ≤ xpi :νi = pi = 1 2 νi/2 Γ νi/2 ∫ 0 xpi e -Xi/2 Xi vi / 2 - 1 dXi , 0 ≤ xpi < ∞ ; ​ νi > 0 .$
The required ${x}_{{p}_{i}}$ is found by using the relationship between a ${\chi }^{2}$-distribution and a gamma distribution, i.e., a ${\chi }^{2}$-distribution with ${\nu }_{i}$ degrees of freedom is equal to a gamma distribution with scale parameter $2$ and shape parameter ${\nu }_{i}/2$.
For very large values of ${\nu }_{i}$, greater than ${10}^{5}$, Wilson and Hilferty's Normal approximation to the ${\chi }^{2}$ is used; see Kendall and Stuart (1969).
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

## 4References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the ${\chi }^{2}$ distribution Appl. Statist. 24 385–388
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

## 5Arguments

1:     $\mathbf{ltail}$ – IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:     $\mathbf{tail}\left({\mathbf{ltail}}\right)$ – Character(1) arrayInput
On entry: indicates which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability, i.e., ${p}_{i}=P\left({X}_{i}\le {x}_{{p}_{i}}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability, i.e., ${p}_{i}=P\left({X}_{i}\ge {x}_{{p}_{i}}:{\nu }_{i}\right)$.
Constraint: ${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$ or $\text{'U'}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:     $\mathbf{lp}$ – IntegerInput
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4:     $\mathbf{p}\left({\mathbf{lp}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${p}_{i}$, the probability of the required ${\chi }^{2}$-distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left(j\right)$, .
Constraints:
• if ${\mathbf{tail}}\left(k\right)=\text{'L'}$, $0.0\le {\mathbf{p}}\left(\mathit{j}\right)<1.0$;
• otherwise $0.0<{\mathbf{p}}\left(\mathit{j}\right)\le 1.0$.
Where  and .
5:     $\mathbf{ldf}$ – IntegerInput
On entry: the length of the array df.
Constraint: ${\mathbf{ldf}}>0$.
6:     $\mathbf{df}\left({\mathbf{ldf}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\nu }_{i}$, the degrees of freedom of the ${\chi }^{2}$-distribution with ${\nu }_{i}={\mathbf{df}}\left(j\right)$, .
Constraint: ${\mathbf{df}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{ldf}}$.
7:     $\mathbf{x}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$.
On exit: ${x}_{{p}_{i}}$, the deviates for the ${\chi }^{2}$-distribution.
8:     $\mathbf{ivalid}\left(*\right)$ – Integer arrayOutput
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf}}\right)$.
On exit: ${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
 On entry, invalid value supplied in tail when calculating ${x}_{{p}_{i}}$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, invalid value for ${p}_{i}$.
${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ${\nu }_{i}\le 0.0$.
${\mathbf{ivalid}}\left(i\right)=4$
${p}_{i}$ is too close to $0.0$ or $1.0$ for the result to be calculated.
${\mathbf{ivalid}}\left(i\right)=5$
The solution has failed to converge. The result should be a reasonable approximation.
9:     $\mathbf{ifail}$ – 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, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.
${\mathbf{ifail}}=2$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lp}}>0$.
${\mathbf{ifail}}=4$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldf}}>0$.
${\mathbf{ifail}}=-99$
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 results should be accurate to five significant digits for most argument values. Some accuracy is lost for ${p}_{i}$ close to $0.0$ or $1.0$.

## 8Parallelism and Performance

g01tcf is not threaded in any implementation.

For higher accuracy the relationship described in Section 3 may be used and a direct call to g01tff made.

## 10Example

This example reads lower tail probabilities for several ${\chi }^{2}$-distributions, and calculates and prints the corresponding deviates.

### 10.1Program Text

Program Text (g01tcfe.f90)

### 10.2Program Data

Program Data (g01tcfe.d)

### 10.3Program Results

Program Results (g01tcfe.r)

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