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Chapter Contents

Chapter Introduction

NAG Toolbox: nag_nonpar_gofstat_anddar (g08ch)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

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:

A^{2} = - n - S;

where:

S = \sum_{i = 1}^{n} \frac{2 i - 1}{n} [\ln y_{i} + \ln (1 - y_{n - i + 1})] .

When observations of a random variable

X

are non-uniformly distributed, the probability integral transformation (PIT):

Y = F (X),

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: $issort$ – 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: $result$ – double scalar: The result of the function.
2: $y (n)$ – double array: If $issort = false$ , the data sorted in ascending order; otherwise the array is unchanged.
3: $ifail$ – 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

Not applicable.

Further Comments

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.

Open in the MATLAB editor: g08ch_example

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

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Chapter Contents

Chapter Introduction

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