LillieforsTest

LillieforsTest is a statistical procedure for assessing whether a dataset comes from a normal distribution when the mean and variance are unknown and must be estimated from the data. It was introduced by Hubert Lilliefors in 1967 as a refinement of the Kolmogorov-Smirnov test, addressing the issue that the standard KS test is not valid when distribution parameters are estimated from the sample. The null hypothesis is that the data are drawn from a normal distribution with unknown mean and variance; the alternative is that the data do not come from such a normal distribution.

To perform the test, one computes the sample mean x̄ and the sample standard deviation s, and then standardizes the data via z_i = (x_i − x̄)/s. The test statistic D_n is the maximum absolute difference between the empirical distribution function F_n of the standardized data and the CDF of the standard normal, F_0(z) = Φ(z). Because the parameters are estimated from the data, the null distribution of D_n is not the standard Kolmogorov distribution; critical values depend on the sample size. Consequently, p-values are obtained through Monte Carlo simulation under the null (or from tabulated critical values for given n).

LillieforsTest is widely used in practical data analysis and is implemented in several statistical software environments (for example, in R via the nortest package with a function commonly referred to as lillie.test). While it provides a flexible approach to normality testing when parameters are unknown, its power characteristics vary with sample size and the nature of departures from normality; in some settings alternative tests such as Shapiro-Wilk or Anderson-Darling may offer greater sensitivity.