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 ​ A^{2} = A^{2} (1 + 0.75 / n + 2.25 / n^{2}),

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: $issort$ – 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.

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: $ybar$ – double scalar: The maximum likelihood estimate of mean.
2: $yvar$ – double scalar: The maximum likelihood estimate of variance.
3: $a2$ – double scalar: $A^{2}$ , the Anderson–Darling test statistic.
4: $aa2$ – double scalar: The adjusted $A^{2}$ .
5: $p$ – double scalar: $p$ , the upper tail probability for the adjusted $A^{2}$ .
6: $ifail$ – 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.

$ifail = - 99$: 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 are calculated using piecewise polynomial approximations to values estimated by simulation.

Further Comments

None.

Example

This example calculates the

A^{2}

statistics for data assumed to arise from a fully-unspecified Normal distribution and the

p

-value.

Open in the MATLAB editor: g08ck_example

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('Adjusted A-squared:        %8.4f\n', aa2);
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
Adjusted A-squared:          0.1713
Upper tail probability:      0.9312

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

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Chapter Introduction

NAG Toolbox