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).

Usage

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:
• 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.

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.

See Also

shapiro.test for performing the Shapiro-Wilk test for normality. ad.test, cvm.test, pearson.test, sf.test for performing further tests for normality. qqnorm for producing a normal quantile-quantile plot.

Examples

lillie.test(rnorm(100, mean = 5, sd = 3))
lillie.test(runif(100, min = 2, max = 4))