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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_nonpar_gofstat_anddar_normal (g08ck)

## Purpose

nag_nonpar_gofstat_anddar_normal (g08ck) calculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of a fully-unspecified Normal distribution.

## Syntax

[ybar, yvar, a2, aa2, p, ifail] = g08ck(issort, y, 'n', n)
[ybar, yvar, a2, aa2, p, ifail] = nag_nonpar_gofstat_anddar_normal(issort, y, 'n', n)

## Description

Calculates the Anderson–Darling test statistic ${A}^{2}$ (see nag_nonpar_gofstat_anddar (g08ch)) and its upper tail probability for the small sample correction:
 $Adjusted ​ A2 = A2 1+0.75/n+ 2.25/n2 ,$
for $n$ observations.

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{issort}$ – logical scalar
Set ${\mathbf{issort}}=\mathit{true}$ if the observations are sorted in ascending order; otherwise the function will sort the observations.
2:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the $n$ observations.
Constraint: if ${\mathbf{issort}}=\mathit{true}$, the values must be sorted in ascending order.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array y.
$n$, the number of observations.
Constraint: ${\mathbf{n}}>1$.

### Output Parameters

1:     $\mathrm{ybar}$ – double scalar
The maximum likelihood estimate of mean.
2:     $\mathrm{yvar}$ – double scalar
The maximum likelihood estimate of variance.
3:     $\mathrm{a2}$ – double scalar
${A}^{2}$, the Anderson–Darling test statistic.
4:     $\mathrm{aa2}$ – double scalar
The adjusted ${A}^{2}$.
5:     $\mathrm{p}$ – double scalar
$p$, the upper tail probability for the adjusted ${A}^{2}$.
6:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{n}}>1$.
${\mathbf{ifail}}=3$
${\mathbf{issort}}=\mathit{true}$ and the data in y is not sorted in ascending order.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

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

None.

## Example

This example calculates the ${A}^{2}$ statistics for data assumed to arise from a fully-unspecified Normal distribution and the $p$-value.
```function g08ck_example

fprintf('g08ck example results\n\n');

y = [0.3131132, 0.2520412, 1.5788841, 1.4416712,-0.8246043,-1.6466685, ...
0.7943184, 1.2874915,-0.8347250, 0.3352505, 0.9434467, 2.1099520, ...
-0.2801654,-0.7843009, 0.6218187, 2.0963809, 1.7170403,-0.1350142, ...
0.7982763,-0.2980977, 1.2283043, 1.5576090,-0.4828757, 2.6070754, ...
0.1213996, 0.1431621];
% Let g08ck sort the data
issort = false;

% Calculate a-squared and probability
[ybar, yvar, a2, aa2, p, ifail] = ...
g08ck(issort, y);

% Results
fprintf('H0: data from normal distribution\n');
fprintf('                with mean  %8.4f\n', ybar);
fprintf('             and variance  %8.4f\n', yvar);
fprintf('Test statistic, A-squared: %8.4f\n', a2);
fprintf('Upper tail probability:    %8.4f\n', p);

```
```g08ck example results

H0: data from normal distribution
with mean    0.5639
and variance    1.1386
Test statistic, A-squared:   0.1660