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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_nonpar_gofstat_anddar (g08ch)

## Purpose

nag_nonpar_gofstat_anddar (g08ch) calculates the Anderson–Darling goodness-of-fit test statistic.

## Syntax

[result, y, ifail] = g08ch(issort, y, 'n', n)
[result, y, ifail] = nag_nonpar_gofstat_anddar(issort, y, 'n', n)

## Description

Denote by ${A}^{2}$ the Anderson–Darling test statistic for $n$ observations ${y}_{1},{y}_{2},\dots ,{y}_{n}$ of a variable $Y$ assumed to be standard uniform and sorted in ascending order, then:
 $A2 = -n-S ;$
where:
 $S = ∑ i=1 n 2i-1 n ln⁡yi + ln 1- y n-i+1 .$
When observations of a random variable $X$ are non-uniformly distributed, the probability integral transformation (PIT):
 $Y=FX ,$
where $F$ is the cumulative distribution function of the distribution of interest, yields a uniformly distributed random variable $Y$. The PIT is true only if all parameters of a distribution are known as opposed to estimated; otherwise it is an approximation.

## 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

## 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:     $\mathrm{result}$ – double scalar
The result of the function.
2:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
If ${\mathbf{issort}}=\mathit{false}$, the data sorted in ascending order; otherwise the array is unchanged.
3:     $\mathrm{ifail}$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.

Not applicable.

None.

## Example

This example calculates the ${A}^{2}$ statistic for data assumed to arise from an exponential distribution with a sample parameter estimate and simulates its $p$-value using the NAG basic random number generator.
```function g08ch_example

fprintf('g08ch 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];
n = int64(numel(x));

% Maximum likelihood estimate of mean
beta = mean(x);
% PIT, using exponential CDF with mean beta
y = 1 - exp(-x/beta);
% Let g08ch sort the (approximately) uniform variates
issort = false;

% Calculate a-squared
[a2, y, ifail] = g08ch( ...
issort, y);

aa2 = (1+0.6/numel(y))*a2;

% Number of simulations
nsim = int64(888);
% Initialize the base generator to a repeatable sequence
seed  = [int64(206033)];
genid = int64(1);
subid = int64(-1);
[state, ifail] = g05kf( ...
genid, subid, seed);
[state, xsim, ifail] = g05sf( ...
n*nsim, beta, state);

% Simulations loop
nupper = 0;
for j=1:nsim
k = (j-1)*n;
x = xsim(k+1:k+n);
% Maximum likelihood estimate of mean
sbeta = mean(x);
% PIT
y = 1 - exp(-x/sbeta);
% Calculate a-squared
[sa2, y, ifail] = g08ch( ...
issort, y);
if sa2 > aa2
nupper = nupper + 1;
end
end

% Simulated upper tail probability value
p = nupper/(nsim+1);

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

```
```g08ch example results

H0: data from exponential distribution with mean 1.5240e+00
Test statistic, A-squared:   0.1616
Upper tail probability:      1.0000
```