The Lilliefors Test for Composite Hypotheses
A corrected version of the Kolmogorov-Smirnov normality test that accounts for the fact that its critical values change when the distribution's mean and standard deviation are estimated from the same sample being tested.
Prerequisites: The Kolmogorov-Smirnov Test
The Kolmogorov-Smirnov (KS) test checks whether a sample plausibly came from a fully specified distribution — a specific normal with a stated mean and standard deviation given in advance, not estimated from the data. Its textbook critical values are only valid under that condition. In practice, though, almost nobody testing "are these daily returns normal?" already knows the population mean and standard deviation; they estimate both from the very sample being tested, which is a subtly different, "composite" hypothesis-testing problem.
Estimating the parameters from the sample makes the fitted distribution track the data more closely than a distribution specified independently would, which mechanically shrinks the KS test statistic and would make the ordinary KS critical values too conservative — the test would fail to reject non-normal data more often than its stated significance level suggests. Lilliefors addressed this by deriving, via simulation, a separate table of critical values specifically for the case where the mean and standard deviation are both estimated from the sample, applied to the same KS statistic (the largest gap between the sample's empirical distribution function and the fitted normal's CDF).
The practical rule is simple: whenever you standardize a sample using its own sample mean and standard deviation before running a KS-style normality check — which is the standard workflow for checking whether returns "look normal" — use the Lilliefors critical values, not the plain KS table, or the test will be systematically miscalibrated.
The Lilliefors test applies the Kolmogorov-Smirnov statistic to check normality when the mean and standard deviation are estimated from the sample itself, using specially derived critical values because plugging in sample estimates shrinks the KS statistic and makes the ordinary KS table too conservative.
The common mistake is standardizing a sample by its own mean and standard deviation, then looking up the result against the standard Kolmogorov-Smirnov table rather than the Lilliefors table. Doing so understates the true test statistic's distribution and can make clearly non-normal data appear to pass a normality check.
Practice in interviews
Further reading
- Lilliefors, On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown (1967)