Lilliefors Test
- A hypothesis test for whether a sample follows a specified continuous distribution by comparing sample and theoretical cumulative distributions.
- Implements the Kolmogorov–Smirnov (KS) statistic and compares it to a critical value to decide fit.
- Commonly applied to assess normality and goodness-of-fit; limited to continuous data and may have accuracy issues for small sample sizes despite claims of usefulness in such cases.
Definition
Section titled “Definition”The Lilliefors test is a statistical test used to determine whether a given data set follows a specific distribution. The test is described as particularly useful when dealing with small sample sizes because it provides a more accurate assessment of the data’s distribution than other tests.
Explanation
Section titled “Explanation”The Lilliefors test computes the Kolmogorov–Smirnov (KS) statistic, defined in the source as the maximum difference between the cumulative distribution function of the data and the cumulative distribution function of the target distribution. The calculated KS statistic is compared to a critical value: if the KS statistic is smaller than the critical value, the data are considered to follow the specified distribution. The test is applied to assess fit of data to a chosen continuous distribution.
Examples
Section titled “Examples”Assessing normality
Section titled “Assessing normality”To test whether a data set is normally distributed, the Lilliefors test calculates the Kolmogorov–Smirnov (KS) statistic as the maximum difference between the cumulative distribution function of the data and the cumulative distribution function of the normal distribution. If the calculated KS statistic is smaller than a critical value, then the data is considered to be normally distributed.
Assessing goodness-of-fit to a specific distribution
Section titled “Assessing goodness-of-fit to a specific distribution”For goodness-of-fit, the Lilliefors test calculates the KS statistic for the given data set and compares it to the critical value for the chosen distribution. If the calculated KS statistic is smaller than the critical value, then the data is considered to fit the chosen distribution well.
Notes or pitfalls
Section titled “Notes or pitfalls”- The Lilliefors test is applicable only to continuous data sets.
- The test may not be accurate for small sample sizes because critical values are calculated based on the assumption of a large sample size.
Related terms
Section titled “Related terms”- Kolmogorov–Smirnov (KS) statistic
- Cumulative distribution function (CDF)
- Normal distribution
- Goodness-of-fit