Lilliefors

Lilliefors refers to the Lilliefors test for normality, a statistical test named after Hubert Lilliefors. The test is a modification of the Kolmogorov–Smirnov test designed for situations where the parameters of the normal distribution are unknown and must be estimated from the data.

In the Lilliefors test, the null hypothesis states that the data are drawn from a normal distribution with unknown mean and variance. The alternative is that the data do not come from a normal distribution. The test statistic is the maximum absolute difference between the empirical distribution function of the observed data and the cumulative distribution function of a normal distribution with parameters estimated from the data (the sample mean and sample standard deviation).

Because the parameters are estimated from the sample, the null distribution of the test statistic is not the standard Kolmogorov distribution. Consequently, critical values and p-values are obtained through Monte Carlo simulations, producing the Lilliefors distribution. In practice, one computes the statistic from the data and compares it to simulated null distributions to determine significance.

The Lilliefors test is widely used as a practical method for assessing normality when population parameters are not known. It is implemented in various statistical software packages and is considered a standard alternative to KS-based normality tests that assume known parameters. Limitations include varying power depending on the alternative and sample size; for some small-sample situations, tests like Shapiro–Wilk may offer greater sensitivity.