# NAG Library Routine Document

## 1Purpose

g08cjf calculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of standard uniformly distributed data.

## 2Specification

Fortran Interface
 Subroutine g08cjf ( n, y, a2, p,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: y(n) Real (Kind=nag_wp), Intent (Out) :: a2, p Logical, Intent (In) :: issort
#include <nagmk26.h>
 void g08cjf_ (const Integer *n, const logical *issort, double y[], double *a2, double *p, Integer *ifail)

## 3Description

Calculates the Anderson–Darling test statistic ${A}^{2}$ (see g08chf) and its upper tail probability by using the approximation method of Marsaglia and Marsaglia (2004) for the case of uniformly distributed data.

## 4References

Anderson T W and Darling D A (1952) Asymptotic theory of certain ‘goodness-of-fit’ criteria based on stochastic processes Annals of Mathematical Statistics 23 193–212
Marsaglia G and Marsaglia J (2004) Evaluating the Anderson–Darling distribution J. Statist. Software 9(2)

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}>1$.
2:     $\mathbf{issort}$ – LogicalInput
On entry: set ${\mathbf{issort}}=\mathrm{.TRUE.}$ if the observations are sorted in ascending order; otherwise the routine will sort the observations.
3:     $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the $n$ observations.
On exit: if ${\mathbf{issort}}=\mathrm{.FALSE.}$, the data sorted in ascending order; otherwise the array is unchanged.
Constraint: if ${\mathbf{issort}}=\mathrm{.TRUE.}$, the values must be sorted in ascending order. Each ${y}_{i}$ must lie in the interval $\left(0,1\right)$.
4:     $\mathbf{a2}$ – Real (Kind=nag_wp)Output
On exit: ${A}^{2}$, the Anderson–Darling test statistic.
5:     $\mathbf{p}$ – Real (Kind=nag_wp)Output
On exit: $p$, the upper tail probability for ${A}^{2}$.
6:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
${\mathbf{ifail}}=3$
${\mathbf{issort}}=\mathrm{.TRUE.}$ and the data in y is not sorted in ascending order.
${\mathbf{ifail}}=9$
The data in y must lie in the interval $\left(0,1\right)$.
${\mathbf{ifail}}=-99$
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

Probabilities greater than approximately $0.09$ are accurate to five decimal places; lower value probabilities are accurate to six decimal places.

## 8Parallelism and Performance

g08cjf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

This example calculates the ${A}^{2}$ statistic and its $p$-value for uniform data obtained by transforming exponential variates.

### 10.1Program Text

Program Text (g08cjfe.f90)

### 10.2Program Data

Program Data (g08cjfe.d)

### 10.3Program Results

Program Results (g08cjfe.r)