# NAG Library Routine Document

## 1Purpose

g01ecf returns the lower or upper tail probability for the ${\chi }^{2}$-distribution with real degrees of freedom, via the routine name.

## 2Specification

Fortran Interface
 Function g01ecf ( tail, x, df,
 Real (Kind=nag_wp) :: g01ecf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x, df Character (1), Intent (In) :: tail
#include nagmk26.h
 double g01ecf_ (const char *tail, const double *x, const double *df, Integer *ifail, const Charlen length_tail)

## 3Description

The lower tail probability for the ${\chi }^{2}$-distribution with $\nu$ degrees of freedom, $P\left(X\le x:\nu \right)$ is defined by:
 $PX≤x:ν=12ν/2Γν/2 ∫0.0xXν/2-1e-X/2dX, x≥0,ν>0.$
To calculate $P\left(X\le x:\nu \right)$ a transformation of a gamma distribution is employed, i.e., a ${\chi }^{2}$-distribution with $\nu$ degrees of freedom is equal to a gamma distribution with scale parameter $2$ and shape parameter $\nu /2$.

## 4References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5Arguments

1:     $\mathbf{tail}$ – Character(1)Input
On entry: indicates whether the upper or lower tail probability is required.
${\mathbf{tail}}=\text{'L'}$
The lower tail probability is returned, i.e., $P\left(X\le x:\nu \right)$.
${\mathbf{tail}}=\text{'U'}$
The upper tail probability is returned, i.e., $P\left(X\ge x:\nu \right)$.
Constraint: ${\mathbf{tail}}=\text{'L'}$ or $\text{'U'}$.
2:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: $x$, the value of the ${\chi }^{2}$ variate with $\nu$ degrees of freedom.
Constraint: ${\mathbf{x}}\ge 0.0$.
3:     $\mathbf{df}$ – Real (Kind=nag_wp)Input
On entry: $\nu$, the degrees of freedom of the ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{df}}>0.0$.
4:     $\mathbf{ifail}$ – 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: g01ecf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If ${\mathbf{ifail}}={\mathbf{1}}$, ${\mathbf{2}}$ or ${\mathbf{3}}$ on exit, then g01ecf returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{tail}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tail}}=\text{'L'}$ or $\text{'U'}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge 0.0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}>0.0$.
${\mathbf{ifail}}=4$
The series used to calculate the gamma probabilities has failed to converge. The result returned should represent an approximation to the solution.
${\mathbf{ifail}}=-99$
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

A relative accuracy of five significant figures is obtained in most cases.

## 8Parallelism and Performance

g01ecf is not threaded in any implementation.

For higher accuracy the transformation described in Section 3 may be used with a direct call to s14baf.

## 10Example

Values from various ${\chi }^{2}$-distributions are read, the lower tail probabilities calculated, and all these values printed out, until the end of data is reached.

### 10.1Program Text

Program Text (g01ecfe.f90)

### 10.2Program Data

Program Data (g01ecfe.d)

### 10.3Program Results

Program Results (g01ecfe.r)

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