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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_nonpar_gofstat_anddar_normal (g08ck)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


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.


[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)


Calculates the Anderson–Darling test statistic A2 (see nag_nonpar_gofstat_anddar (g08ch)) and its upper tail probability for the small sample correction:
Adjusted ​ A2 = A2 1+0.75/n+ 2.25/n2 ,  
for n observations.


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


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:     yn – double array
yi, for i=1,2,,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
A2, the Anderson–Darling test statistic.
4:     aa2 – double scalar
The adjusted A2.
5:     p – double scalar
p, the upper tail probability for the adjusted A2.
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:
Constraint: n>1.
issort=true and the data in y is not sorted in ascending order.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


Probabilities are calculated using piecewise polynomial approximations to values estimated by simulation.

Further Comments



This example calculates the A2 statistics for data assumed to arise from a fully-unspecified Normal distribution and the p-value.
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)
Chapter Contents
Chapter Introduction
NAG Toolbox

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