# NAG FL Interfaceg01ecf (prob_​chisq)

## 1Purpose

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

## 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 <nag.h>
 double g01ecf_ (const char *tail, const double *x, const double *df, Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01ecf or nagf_stat_prob_chisq.

## 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

NIST Digital Library of Mathematical Functions
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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value 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 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:
Note: in some cases g01ecf may return useful information.
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 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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)