de.lmu.ifi.dbs.elki.math.statistics.tests

Class AndersonDarlingTest

java.lang.Object

de.lmu.ifi.dbs.elki.math.statistics.tests.AndersonDarlingTest

@Reference(authors="T. W. Anderson, D. A. Darling",
           title="Asymptotic theory of certain \'goodness of fit\' criteria based on stochastic processes",
           booktitle="Annals of mathematical statistics 23(2)",
           url="https://doi.org/10.1214/aoms/1177729437",
           bibkey="doi:10.1214/aoms/1177729437")
public class AndersonDarlingTest
extends java.lang.Object

Perform Anderson-Darling test for a Gaussian distribution.

This is a test against normality / goodness of fit. I.e. you can use it to reject the hypothesis that the data is normal distributed. Such tests are sensitive to data set size: on small samples, even large deviations could be by-chance and thus not allow rejection. On the other hand, on large data sets even a slight deviation can be unlikely to happen if the data were indeed normal distributed. Thus, this test is more likely to fail to reject small data sets even when they intuitively do not appear to be normal distributed, while it will reject large data sets that originate from a distribution only slightly different from the normal distribution.

Before using, make sure you have understood statistical tests, and the difference between failure-to-reject and acceptance!

The data size should be at least 8 before the results start getting somewhat reliable. For large data sets, the chance of rejecting the normal distribution hypothesis increases a lot: no real data looks exactly like a normal distribution.

References:

T. W. Anderson, D. A. Darling
Asymptotic theory of certain 'goodness of fit' criteria based on stochastic processes
Annals of mathematical statistics 23(2)

M. A. Stephens
EDF Statistics for Goodness of Fit and Some Comparisons
Journal of the American Statistical Association 69(347)

Since:

0.7.0

Author:

Erich Schubert

Constructor Summary

Constructors

Modifier	Constructor and Description
private	AndersonDarlingTest() Private constructor.

Method Summary

Modifier and Type	Method and Description
static double	A2Noncentral(double[] sorted) Test a sorted but not standardized data set.
static double	A2StandardNormal(double[] sorted) Test a sorted data set against the standard normal distribution.
static double	removeBiasNormalDistribution(double A2, int n) Remove bias from the Anderson-Darling statistic if the mean and standard deviation were estimated from the data, and a normal distribution was assumed.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

AndersonDarlingTest

private AndersonDarlingTest()

Private constructor. Static methods only.

Method Detail

A2StandardNormal

public static double A2StandardNormal(double[] sorted)

Test a sorted data set against the standard normal distribution.

Note: the data will be compared to the standard normal distribution, i.e. with mean 0 and variance 1.

The data size should be at least 8 before the results start getting somewhat reliable. For large data sets, the chance of rejecting increases a lot: no real data looks exactly like a normal distribution.

Parameters:

sorted - Sorted input data.

Returns:

Test statistic A².

A2Noncentral

public static double A2Noncentral(double[] sorted)

Test a sorted but not standardized data set.

The data size should be at least 8!

Parameters:

sorted - Sorted input data.

Returns:

Test statistic A², after bias removal.

removeBiasNormalDistribution

@Reference(authors="M. A. Stephens",
           title="EDF Statistics for Goodness of Fit and Some Comparisons",
           booktitle="Journal of the American Statistical Association, Volume 69, Issue 347",
           url="https://doi.org/10.1080/01621459.1974.10480196",
           bibkey="doi:10.1080/01621459.1974.10480196")
public static double removeBiasNormalDistribution(double A2,
                                                                                                                                                                                                                                                                                                                                                                                          int n)

Remove bias from the Anderson-Darling statistic if the mean and standard deviation were estimated from the data, and a normal distribution was assumed.

Parameters:

A2 - A2 statistic

n - Sample size

Returns:

Unbiased test statistic