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: $issort$ – logical scalar: Set $issort = true$ if the observations are sorted in ascending order; otherwise the function will sort the observations.
2: $y (n)$ – double array: $y_{i}$ , for $i = 1, 2, \dots, n$ , the $n$ observations.

Constraint: if $issort = true$ , the values must be sorted in ascending order. Each $y_{i}$ must lie in the interval $(0, 1)$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array y.
$n$ , the number of observations.

Constraint: $n > 1$ .

Output Parameters

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

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $n > 1$ .

$ifail = 3$: $issort = true$ and the data in y is not sorted in ascending order.

$ifail = 9$: The data in y must lie in the interval $(0, 1)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: 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.

Further Comments

None.

Example

This example calculates the

A^{2}

statistic and its

p

-value for uniform data obtained by transforming exponential variates.

Open in the MATLAB editor: g08cj_example

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox