Kolmogorov-Smirnov Test for Normality

Tests whether a one-dimensional signal is normally distributed by comparing its empirical cumulative distribution function (ECDF) to that of a normal distribution fitted to the sample (mean and SD). The statistic is the Kolmogorov-Smirnov distance \(D = \sup_x |F_n(x) - \Phi((x-\mu)/\sigma)|\), the largest vertical gap between the two CDFs; the two-sided asymptotic (Kolmogorov) p-value follows. The \(D\) statistic reproduces stats::ks.test and scipy.stats.kstest bit-for-bit.

Usage

ksNormalityTest(x)

Arguments

x: A numeric vector (the time series).

Value

A list with statistic (the KS distance \(D\)), p_value (two-sided asymptotic), and n (sample size).

Details

A distribution-shape (empirical-CDF) normality test, complementary to the moment-based jarqueBeraTest: KS responds to the overall shape of the distribution, Jarque-Bera to its skewness and kurtosis (the tails). The two can disagree – on a long, mildly non-Gaussian signal Jarque-Bera's power often rejects while KS does not.

Caveat

Because the normal parameters are estimated from the same sample, the naive p-value is anti-conservative (a Lilliefors correction gives an exact test); the \(D\) statistic itself is exact and is what this function certifies.

Examples

set.seed(1); ksNormalityTest(rnorm(500))$p_value    # Gaussian -> large p
#> [1] 0.5003172
set.seed(1); ksNormalityTest(rexp(500))$statistic   # skewed -> large D
#> [1] 0.1432139