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Chapter Contents
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 A2${A}^{2}$ (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:     issort – logical scalar
Set issort = true${\mathbf{issort}}=\mathbf{true}$ if the observations are sorted in ascending order; otherwise the function will sort the observations.
2:     y(n) – double array
n, the dimension of the array, must satisfy the constraint n > 1${\mathbf{n}}>1$.
yi${y}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$, the n$n$ observations.
Constraint: if issort = true${\mathbf{issort}}=\mathbf{true}$, the values must be sorted in ascending order. Each yi${y}_{i}$ must lie in the interval (0,1)$\left(0,1\right)$.

Optional Input Parameters

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

None.

Output Parameters

1:     y(n) – double array
If issort = false${\mathbf{issort}}=\mathbf{false}$, the data sorted in ascending order; otherwise the array is unchanged.
2:     a2 – double scalar
A2${A}^{2}$, the Anderson–Darling test statistic.
3:     p – double scalar
p$p$, the upper tail probability for A2${A}^{2}$.
4:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
Constraint: n > 1${\mathbf{n}}>1$.
ifail = 3${\mathbf{ifail}}=3$
issort = true${\mathbf{issort}}=\mathbf{true}$ and the data in y is not sorted in ascending order.
ifail = 9${\mathbf{ifail}}=9$
The data in y must lie in the interval (0,1)$\left(0,1\right)$.

Accuracy

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

None.

Example

```function nag_nonpar_gofstat_anddar_unif_example
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 nag_nonpar_gofstat_anddar_unif sort the uniform variates
issort = false;

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

% Results
fprintf('\nH0: 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);
```
```

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

```
```function g08cj_example
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('\nH0: 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);
```
```

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

```