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

`LillieTest(x)`

A list of class `htest`

, containing the following components:

- statistic
the value of the Lilliefors (Kolomogorv-Smirnov) statistic.

- p.value
the p-value for the test.

- method
the character string “Lilliefors (Kolmogorov-Smirnov) normality test”.

- data.name
a character string giving the name(s) of the data.

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

Juergen Gross <gross@statistik.uni-dortmund.de>

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.

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.

`shapiro.test`

for performing the Shapiro-Wilk test for normality.
`AndersonDarlingTest`

, `CramerVonMisesTest`

,
`PearsonTest`

, `ShapiroFranciaTest`

for performing further tests for normality.
`qqnorm`

for producing a normal quantile-quantile plot.

```
LillieTest(rnorm(100, mean = 5, sd = 3))
LillieTest(runif(100, min = 2, max = 4))
```

Run the code above in your browser using DataLab