Lilliefors test for normality with the Dallal-Wilkinson p-value

The Kolmogorov-Smirnov distance to the normal distribution function with the sample mean and standard deviation plugged in, \(D = \max(D^+, D^-)\). Because the two parameters are estimated, the null distribution of \(D\) is not Kolmogorov's; the p-value is the Dallal and Wilkinson (1986) analytic approximation below 0.1 and Stephens' (1974) modified statistic \((\sqrt n - 0.01 + 0.85/\sqrt n) D\) above it, as in nortest::lillie.test. Identical to the Python arm morie.fn.lilef.lilef.

Usage

lilef(x, axis = 0, cdf = NULL)

Arguments

x

Numeric sample (at least four values). For a matrix the first slice along axis is tested.

axis

0 tests the first row of a matrix, 1 the first column.

cdf

Ignored; kept for signature parity with the Python arm.

Value

A list with statistic, p_value, critical_value (Lilliefors' large-sample 5 percent point 0.886/sqrt(n)), interpretation ("reject" when the p-value is below 0.05), mean and std.

References

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

Dallal, G. E. and Wilkinson, L. (1986). An analytic approximation to the distribution of Lilliefors's test statistic 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.

Examples

lilef(c(2.1, 3.4, 1.9, 5.6, 2.8, 3.1, 9.9, 2.5, 3.3, 2.7))$p_value
#> [1] 0.0008043292