nortest (version 1.0-4)

# lillie.test: Lilliefors (Kolmogorov-Smirnov) test for normality

## Description

Performs the Lilliefors (Kolmogorov-Smirnov) test for the composite hypothesis of normality, see e.g. Thode (2002, Sec. 5.1.1).

lillie.test(x)

## Arguments

x
a numeric vector of data values, the number of which must be greater than 4. Missing values are allowed.

## Value

• A list with class htest containing the following components:
• statisticthe value of the Lilliefors (Kolomogorv-Smirnov) statistic.
• p.valuethe p-value for the test.
• methodthe character string Lilliefors (Kolmogorov-Smirnov) normality test.
• data.namea character string giving the name(s) of the data.

## Details

The Lilliefors (Kolmogorov-Smirnov) test is an EDF omnibus test for the composite hypothesis of normality. The test statistic is the maximal absolute difference between empirical and hypothetical cumulative distribution function. It may be computed as $D=\max{D^{+}, D^{-}}$ with $$D^{+} = \max_{i=1,\ldots, n}{i/n - p_{(i)}}, D^{-} = \max_{i=1,\ldots, n}{p_{(i)} - (i-1)/n},$$ where $p_{(i)} = \Phi([x_{(i)} - \overline{x}]/s)$. Here, $\Phi$ is the cumulative distribution function of the standard normal distribution, and $\overline{x}$ and $s$ are mean and standard deviation of the data values. The p-value is computed from the Dallal-Wilkinson (1986) formula, which is claimed to be only reliable when the p-value is smaller than 0.1. If the Dallal-Wilkinson p-value turns out to be greater than 0.1, then the p-value is computed from the distribution of the modified statistic $Z=D (\sqrt{n}-0.01+0.85/\sqrt{n})$, see Stephens (1974), the actual p-value formula being obtained by a simulation and approximation process.

## References

Dallal, G.E. and Wilkinson, L. (1986): An analytic approximation to the distribution of Lilliefors' test for normality. The American Statistician, 40, 294--296. Stephens, M.A. (1974): EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69, 730--737. Thode Jr., H.C. (2002): Testing for Normality. Marcel Dekker, New York.