Lilliefors-Test

The Lilliefors test or Kolmogorov - Smirnov - Lilliefors test is a statistical test that the frequency distribution of data from a sample deviations can be investigated by the normal distribution. It is based on a modification of the Kolmogorov -Smirnov test, in which there is a general test of fit for the specific application of normality testing. This makes it more suitable for the test of normality than the Kolmogorov -Smirnov test, but its test strength is less than that of other normality tests. It was named after Hubert Lilliefors, the first time in 1967 described him.

Test Description

To carry out the Lilliefors test, the distance is determined between the distribution of sample data and a theoretical normal distribution for which the mean and standard deviation of the sample will be accepted. The greater this distance is, the smaller the p-value. The null hypothesis of the test is the assumption that the data of the sample to be examined be normal. A p-value less than 0.05 as a test result is therefore to be interpreted as statistically significant deviation of the frequency distribution from the normal distribution of the sample, whereas a p-value greater than 0.05 does not necessarily mean the presence of normally distributed data. The decision as to whether the data from a sample are normally distributed is, among other things, important for the choice of test method for further analysis, as certain tests require normally distributed samples and deviations from the normal distribution, non-parametric tests are to be used as an alternative.

Alternative methods

Alternatives to the Lilliefors test include the Shapiro -Wilk test and the Jarque - Bera test and apply the Anderson -Darling normality test as a test. While it is more suitable for the test of normality than the Kolmogorov -Smirnov test, the Anderson -Darling apply in particular test and the Shapiro -Wilk test for their superior strength test than the Lilliefors - test.