G08 Chapter Contents
G08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG08CKF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G08CKF calculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of a fully-unspecified Normal distribution.

## 2  Specification

 SUBROUTINE G08CKF ( N, ISSORT, Y, YBAR, YVAR, A2, AA2, P, IFAIL)
 INTEGER N, IFAIL REAL (KIND=nag_wp) Y(N), YBAR, YVAR, A2, AA2, P LOGICAL ISSORT

## 3  Description

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.

## 4  References

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

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the number of observations.
Constraint: ${\mathbf{N}}>1$.
2:     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:     Y(N) – 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:     YBAR – REAL (KIND=nag_wp)Output
On exit: the maximum likelihood estimate of mean.
5:     YVAR – REAL (KIND=nag_wp)Output
On exit: the maximum likelihood estimate of variance.
6:     A2 – REAL (KIND=nag_wp)Output
On exit: ${A}^{2}$, the Anderson–Darling test statistic.
7:     AA2 – REAL (KIND=nag_wp)Output
On exit: the adjusted ${A}^{2}$.
8:     P – REAL (KIND=nag_wp)Output
On exit: $p$, the upper tail probability for the adjusted ${A}^{2}$.
9:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{N}}<2$.
${\mathbf{IFAIL}}=3$
The data in Y is not sorted in ascending order.

## 7  Accuracy

Probabilities are calculated using piecewise polynomial approximations to values estimated by simulation.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (g08ckfe.f90)

### 9.2  Program Data

Program Data (g08ckfe.d)

### 9.3  Program Results

Program Results (g08ckfe.r)