# NAG Library Routine Document

## 1Purpose

g08ckf calculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of a fully-unspecified Normal distribution.

## 2Specification

Fortran Interface
 Subroutine g08ckf ( n, y, ybar, yvar, a2, aa2, p,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: y(n) Real (Kind=nag_wp), Intent (Out) :: ybar, yvar, a2, aa2, p Logical, Intent (In) :: issort
#include nagmk26.h
 void g08ckf_ (const Integer *n, const logical *issort, const double y[], double *ybar, double *yvar, double *a2, double *aa2, double *p, Integer *ifail)

## 3Description

Calculates the Anderson–Darling test statistic ${A}^{2}$ (see g08chf) and its upper tail probability for the small sample correction:
 $Adjusted ​ A2 = A2 1+0.75/n+ 2.25/n2 ,$
for $n$ observations.

## 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
Stephens M A and D'Agostino R B (1986) Goodness-of-Fit Techniques Marcel Dekker, New York

## 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
On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the $n$ observations.
Constraint: if ${\mathbf{issort}}=\mathrm{.TRUE.}$, the values must be sorted in ascending order.
4:     $\mathbf{ybar}$ – Real (Kind=nag_wp)Output
On exit: the maximum likelihood estimate of mean.
5:     $\mathbf{yvar}$ – Real (Kind=nag_wp)Output
On exit: the maximum likelihood estimate of variance.
6:     $\mathbf{a2}$ – Real (Kind=nag_wp)Output
On exit: ${A}^{2}$, the Anderson–Darling test statistic.
7:     $\mathbf{aa2}$ – Real (Kind=nag_wp)Output
On exit: the adjusted ${A}^{2}$.
8:     $\mathbf{p}$ – Real (Kind=nag_wp)Output
On exit: $p$, the upper tail probability for the adjusted ${A}^{2}$.
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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
${\mathbf{ifail}}=3$
${\mathbf{issort}}=\mathrm{.TRUE.}$ and the data in y is not sorted in ascending order.
${\mathbf{ifail}}=-99$
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 are calculated using piecewise polynomial approximations to values estimated by simulation.

## 8Parallelism and Performance

g08ckf 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 ${A}^{2}$ statistics for data assumed to arise from a fully-unspecified Normal distribution and the $p$-value.

### 10.1Program Text

Program Text (g08ckfe.f90)

### 10.2Program Data

Program Data (g08ckfe.d)

### 10.3Program Results

Program Results (g08ckfe.r)

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