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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_nonpar_gofstat_anddar_unif (g08cj)

## Purpose

nag_nonpar_gofstat_anddar_unif (g08cj) calculates the Anderson–Darling goodness-of-fit test statistic and its probability for the case of standard uniformly distributed data.

## Syntax

[y, a2, p, ifail] = g08cj(issort, y, 'n', n)
[y, a2, p, ifail] = nag_nonpar_gofstat_anddar_unif(issort, y, 'n', n)

## Description

Calculates the Anderson–Darling test statistic ${A}^{2}$ (see nag_nonpar_gofstat_anddar (g08ch)) and its upper tail probability by using the approximation method of Marsaglia and Marsaglia (2004) for the case of uniformly distributed data.

## 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
Marsaglia G and Marsaglia J (2004) Evaluating the Anderson–Darling distribution J. Statist. Software 9(2)

## 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. Each ${y}_{i}$ must lie in the interval $\left(0,1\right)$.

### 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:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
If ${\mathbf{issort}}=\mathit{false}$, the data sorted in ascending order; otherwise the array is unchanged.
2:     $\mathrm{a2}$ – double scalar
${A}^{2}$, the Anderson–Darling test statistic.
3:     $\mathrm{p}$ – double scalar
$p$, the upper tail probability for ${A}^{2}$.
4:     $\mathrm{ifail}$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.
${\mathbf{ifail}}=9$
The data in y must lie in the interval $\left(0,1\right)$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

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

None.

## Example

This example calculates the ${A}^{2}$ statistic and its $p$-value for uniform data obtained by transforming exponential variates.
```function g08cj_example

fprintf('g08cj example results\n\n');

x = [0.4782745, 1.2858962, 1.1163891, 2.0410619, 2.2648109, 0.0833660, ...
1.2527554, 0.4031288, 0.7808981, 0.1977674, 3.2539440, 1.8113504, ...
1.2279834, 3.9178773, 1.4494309, 0.1358438, 1.8061778, 6.0441929, ...
0.9671624, 3.2035042, 0.8067364, 0.4179364, 3.5351774, 0.3975414, ...
0.6120960, 0.1332589];
mu = 1.65;
% PIT
y = 1 - exp(-x/mu);
% Let g08cj sort the uniform variates
issort = false;

% Calculate a-squared and probability
[y, a2, p, ifail] = g08cj( ...
issort, y);

% Results
fprintf('H0: data from exponential distribution with mean %10.4e\n', mu);
fprintf('Test statistic, A-squared: %8.4f\n', a2);
fprintf('Upper tail probability:    %8.4f\n', p);

```
```g08cj example results

H0: data from exponential distribution with mean 1.6500e+00
Test statistic, A-squared:   0.1830
Upper tail probability:      0.9945
```