Lilliefors test

from Wikipedia, the free encyclopedia

The Lilliefors test or Kolmogorow-Smirnow-Lilliefors test is a statistical test with which the frequency distribution of the data of a sample can be examined for deviations from a normal distribution with an unknown expected value and unknown variance. It is based on a modification of the Kolmogorow-Smirnow test , which is a general adaptation test for the special application of normality testing . This makes it more suitable than the Kolmogorow-Smirnow test for the test for normality , but its test strength is less than that of other normality tests . It was named after Hubert Lilliefors, who first described it in 1967.

There is also a variant of the test for exponentially distributed random variables.

Test description

To carry out the Lilliefors test, the distance is determined between the distribution of the sample data and a theoretical normal distribution for which the expected value and the standard deviation of the sample are assumed. The larger this distance, the smaller the p-value . The test's null hypothesis is the assumption that the data of the sample to be examined are normally distributed. A p-value less than 0.05 as a test result is therefore to be interpreted as a statistically significant deviation of the frequency distribution of the sample from the normal distribution, while a p-value greater than 0.05 does not necessarily mean the availability of normally distributed data. The decision as to whether the data of a sample is normally distributed is important, among other things, for the selection of the test procedure for further analyzes, since certain tests require normally distributed samples and non-parametric tests can be used as an alternative in the event of deviations from the normal distribution .

In 1986 a corrected table of the critical values ​​of the test was published.

Alternative procedures

Alternatives to the Lilliefors test include the Shapiro-Wilk test and the Jarque-Bera test and the use of the Anderson-Darling test as a normality test. While the Lilliefors test is more suitable for testing normal distribution than the Kolmogorow-Smirnow test, the Anderson-Darling test and the Shapiro-Wilk test in particular are considered to be superior to the Lilliefors test in terms of their strength .

literature

Individual evidence

  1. ↑ Hubert W. Lilliefors: On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown . In: Journal of the American Statistical Association . tape 62 , no. 318 , 1967, ISSN  0162-1459 , pp. 399-402 , doi : 10.1080 / 01621459.1967.10482916 ( tandfonline.com [accessed November 12, 2019]).
  2. ↑ Hubert W. Lilliefors: On the Kolmogorov-Smirnov test for the exponential distribution with mean Unknown . In: Journal of the American Statistical Association . tape 64 , no. 325 , 1969, ISSN  0162-1459 , pp. 387–389 , doi : 10.1080 / 01621459.1969.10500983 ( tandfonline.com [accessed November 12, 2019]).
  3. ↑ Gerard E. Dallal, Leland Wilkinson: An Analytic Approximation to the Distribution of Lilliefors's Test Statistic for Normality . In: The American Statistician . tape 40 , no. 4 , November 1986, ISSN  0003-1305 , pp. 294–296 , doi : 10.1080 / 00031305.1986.10475419 ( tandfonline.com [accessed November 12, 2019]).