The Kolmogorov-Smirnov test

Draws a sample from a Student t distribution with 3 degrees of freedom and compares its empirical CDF with the standard normal CDF. The KS statistic is the largest vertical gap between the two curves.

import matplotlib.pyplot as plt
import numpy as np
from scipy import stats

from mathematicskit.statistics import kolmogorov_smirnov_test

One sample against a model

rng = np.random.default_rng(0)
data = np.sort(rng.standard_t(df=3, size=800))
result = kolmogorov_smirnov_test(data, "norm")
print(f"vs. N(0,1):  D = {result.statistic:.4f}, p = {result.p_value:.4f}")
print(f"vs. t(3):    p = {kolmogorov_smirnov_test(data, stats.t(df=3).cdf).p_value:.4f}")

ecdf = np.arange(1, data.size + 1) / data.size
fig, ax = plt.subplots()
ax.step(data, ecdf, where="post", label="empirical CDF")
ax.plot(data, stats.norm.cdf(data), label="normal CDF")
model = stats.norm.cdf(data)
above, below = ecdf - model, model - (ecdf - 1 / data.size)
k = int(np.argmax(np.maximum(above, below)))
top = ecdf[k] if above[k] >= below[k] else ecdf[k] - 1 / data.size
ax.vlines(data[k], model[k], top, color="C3", lw=3, label=f"D = {result.statistic:.3f}")
ax.legend()

vs. N(0,1):  D = 0.0539, p = 0.0184
vs. t(3):    p = 0.6969

<matplotlib.legend.Legend object at 0x11d82f8c0>

Two samples

other = rng.normal(size=300)
two = kolmogorov_smirnov_test(data, other)
print(f"two-sample:  D = {two.statistic:.4f}, p = {two.p_value:.4f}")

two-sample:  D = 0.0708, p = 0.2139