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## Lilliefors |

This test, due to Lilliefors (1967, [LIL1]) is essentially a variant of the Kolmogorov-Smirnov (K-S) test. It tests to see if a sample comes from a distribution in the Normal family with unknown population mean and variance (these are estimated from the sample), against the alternative that it does not come from a Normal distribution. With the K-S test it is assumed that the distribution parameters are known, which is often not the case. Following Lilliefors, the test procedure is as follows:

Given a sample of n observations, one determines D, where:

D=sup[F(x)-G(x)]

where sup means supremum, or largest value of a set, with G(x) being the sample cumulative distribution function and F(x) is the cumulative Normal distribution function with mean μ=the sample mean, and variance σ2=the sample variance, defined with denominator n-1. If the value of D exceeds the critical value one rejects the hypothesis that the observations are from a Normal population. Critical values were obtained by Monte Carlo simulation for the sample size range 3-30 and are provided automatically in modern software. On the basis of relatively modest simulation experiments Lilliefors argued that the test is an improvement on the chi-square test when sample sizes are small (<30) and has greater power than the K-S test.

References

[LIL1] Lilliefors H W (1967) On the Kolmogorov-Smirnov test for normality with mean and variance unknown. J of the American Statistical Association, 62, 399–402