com.numericalmethod.suanshu.stats.test.distribution.normality
Class Lilliefors

java.lang.Object
  com.numericalmethod.suanshu.stats.test.HypothesisTest
      com.numericalmethod.suanshu.stats.test.distribution.normality.Lilliefors

public class Lilliefors
extends HypothesisTest

Lilliefors test tests the null hypothesis that data come from a normally distributed population with an estimated sample mean and variance. The test statistic is the maximal absolute difference between empirical and the hypothetical NormalDistribution distribution function.

Since the hypothesized CDF has been moved closer to the data by estimation based on the data, the "null distribution" of the test statistic, i.e. its probability distribution assuming the null hypothesis is true, is stochastically smaller than the Kolmogorov–Smirnov distribution, hence the Lilliefors distribution.

We first compute the p-value from the Dallal-Wilkinson (1986) formula, which is claimed to be only reliable when the p-value is smaller than 0.1. When this p-value is > 0.1, we compute again the p-value from the distribution of the modified statistic (Stephen 1974).

The R equivalent function is lillie.test in package nortest.

See Also:

• Wikipedia: Lilliefors test
• Dallal, G.E. and Wilkinson, L. "An analytic approximation to the distribution of Lilliefors' test for normality," The American Statistician, 40, 294–296. 1986.
• Stephens, M.A. "EDF statistics for goodness of fit and some comparisons," Journal of the American Statistical Association, 69, 730–737. 1974.

Field Summary

Fields inherited from class com.numericalmethod.suanshu.stats.test.HypothesisTest

k, N, pValue, testStatistics

Constructor Summary

Lilliefors(double[] sample)
Perform the Lilliefors test to test for the null hypothesis that data come from a normally distributed population with an estimated sample mean and variance.

Method Summary

java.lang.String	alternativeHypothesis() Get a description of the alternative hypothesis.
java.lang.String	nullHypothesis() Get a description of the null hypothesis.

Methods inherited from class com.numericalmethod.suanshu.stats.test.HypothesisTest

oneSidedPvalue, pValue, rejectNull, testStatistics

Methods inherited from class java.lang.Object

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

Constructor Detail

Lilliefors

public Lilliefors(double[] sample)

Perform the Lilliefors test to test for the null hypothesis that data come from a normally distributed population with an estimated sample mean and variance.

Parameters:

sample - a sample

Method Detail

nullHypothesis

public java.lang.String nullHypothesis()

Description copied from class: HypothesisTest

Get a description of the null hypothesis.

Specified by:

nullHypothesis in class HypothesisTest

Returns:

the null hypothesis description

See Also:

Wikipedia: Null hypothesis

alternativeHypothesis

public java.lang.String alternativeHypothesis()

Description copied from class: HypothesisTest

Get a description of the alternative hypothesis.

Specified by:

alternativeHypothesis in class HypothesisTest

Returns:

the alternative hypothesis description

See Also:

Wikipedia: Alternative hypothesis