Lilliefors test

The Lilliefors test is a statistical test used to assess the goodness of fit of a distribution to a dataset. Specifically, it is an adaptation of the Kolmogorov-Smirnov test designed to test the null hypothesis that data come from a normally distributed population. Unlike the standard Kolmogorov-Smirnov test, which requires the mean and variance to be specified, the Lilliefors test is used when parameters such as the mean and variance are estimated from the data.

Background

The Lilliefors test was developed by Hubert Lilliefors, an American statistician, in 1967. The test is particularly useful because it adjusts the critical values for the effect of parameter estimation. When the parameters of the normal distribution (mean and variance) are estimated from the data, the critical values from the standard Kolmogorov-Smirnov test are no longer valid, as they assume known parameters. The Lilliefors test corrects for this by using simulation or other methods to generate the appropriate critical values.

Procedure

The Lilliefors test procedure involves several steps:

1. Estimate the mean and variance from the sample data.
2. Standardize the data using the estimated mean and variance.
3. Compute the empirical cumulative distribution function (ECDF) of the standardized data.
4. Calculate the maximum difference (D) between the ECDF and the cumulative distribution function (CDF) of the standard normal distribution.
5. Compare the calculated D to a critical value from the Lilliefors distribution table. If D is greater than the critical value, the null hypothesis that the data are from a normal distribution is rejected.

Applications

The Lilliefors test is widely used in statistics to verify the assumption of normality, which is a common requirement in many parametric statistical tests and procedures. Ensuring that data are normally distributed can be crucial for the accurate application of tests such as the t-test, ANOVA, and many regression models.

Limitations

While the Lilliefors test is an effective tool for assessing normality, it has some limitations:

• It is less powerful than other tests like the Shapiro-Wilk test for small sample sizes.
• The test relies on tabulated values of critical points, which may not be available for all sample sizes and significance levels.

See also

• Normal distribution
• Statistical hypothesis testing
• Goodness of fit