为什么 scipy.stats.anderson_ksamp 为这些数据提供超过一百万的 p 值？

机器算法验证 p 值 Python kolmogorov-smirnov 测试 scipy 安德森宠儿测试
2022-03-16 13:50:38

当我跑

from scipy.stats import anderson_ksamp

a = [-1.8, -2.4, -2.4, -0.0, -1.5, -2.7, -1.8, -3.0, -1.8, -1.2, -3.0, -3.0, -2.8, -3.0, -2.1, -0.0, 0.6, -2.5, -2.4, -0.0, -2.7, -0.0, -2.5, -2.1, -0.9, -3.0, -0.6, -0.6, -1.5, -2.2, -1.2, -2.4, -2.4, -3.0, 1.5, -1.8, 1.5, -2.7, -3.0, -2.5, -2.5, -1.5, -1.5, -2.1, -2.1, -3.0, -0.6, -2.7, -3.0, -1.5, -0.6, -0.0, -2.1, 0.6, -2.0, -3.0, -3.0, -2.4, -3.0, -1.8, -0.0, -0.0, -3.0, -1.5, -3.0, -3.0, -1.5, -2.5, -3.0, -2.8, -3.0, -2.2, -0.6, -0.0, -1.5, -2.7, -2.1, -2.1, -2.2, -2.1, -0.6, -0.0, -2.5, -2.1, -1.5, -3.0, -2.2, -1.8, -2.7, -2.4, -1.5, -2.1, -2.9, -2.4, -0.9, -0.0, -0.0, -2.4, -2.7, -0.6, -2.2, -3.0, -1.5, -0.9, -3.0, -3.0, -0.0, -2.7, -2.7, -1.5, -2.2, -3.0, -0.0, 1.5, -3.0, -2.7, -2.2, -2.9, -2.2, -3.0, -1.8, -0.0, -3.0, -1.5, -2.7, -3.0, -3.0, -2.9, -3.0, -3.0, -3.0, -3.0, -2.5, -0.0, 0.9, -3.0, -0.0, -3.0, 3.0, -3.0, -3.0, -1.2, -2.1, 1.5, -0.0, -0.9, -3.0, -2.7, -1.5, -2.4, -2.1, -3.0, -0.9, -3.0, -0.8, -1.5, -2.1, -2.7, -0.0, -0.0, -2.2, -1.8, -2.1, -2.2, -3.0, 0.6, -2.4, -2.2, -2.4, -2.5, -1.5, -0.0, -2.7, -3.0, -3.0, -2.1, -0.0, -2.4, -2.4, -0.0, -2.0, -0.9, -2.4, -3.0, -1.4, -2.7, -2.7, -3.0, -3.0, -2.7, -1.2, -2.1, -3.0, -0.0, -3.0, -2.7, -2.7, -3.0, -3.0, -2.5, -3.0, -1.8, -1.5, -2.7, -2.4, -1.8, -3.0, -2.7, 2.1, -3.0, -2.2, -2.2, 0.6, -0.9, 6.0, -3.0, -2.1, -3.0, -2.1, -2.5, -3.0, -1.5, -2.5, 3.0, -2.1, -3.0, -3.0, -1.5, -2.1, -2.7, -2.5, 1.5, -2.1, -0.0, -0.0, -3.0, -0.6, 1.5, -2.7, -2.4, -2.1, -3.0, -2.7, -3.0, 9.0, -3.0, -1.7, -3.0, -0.0, -3.0, -2.2, -0.0, -0.6, -2.7, -2.7, -3.0, -3.0, -1.7, -2.1, -2.0, -3.0, -2.1, -3.0, -1.1, -3.0, -3.0, -2.4, -1.5, -3.0, -2.2, -3.0, -1.5, -2.7, -3.0, -3.0, -3.0, -2.2, -3.0, -2.1, -2.1, -2.4, -3.0, -3.0, -0.0, -3.0, -3.0, -2.1, -2.0, 1.5, -3.0, -3.0, -2.1, -2.9, -2.4, -3.0, -3.0, -1.5, -2.2, -0.9, -1.8, -2.1, -1.8, -1.5, -3.0, -1.5, -3.0, -1.5, -3.0, -2.4, 1.5, -2.7, -3.0, -1.8, -1.8, -1.5, -2.1, -2.7, -2.7, -2.7, -3.0, -1.5, -2.7, -3.0, -2.7, -3.0, -1.5, -1.5, -3.0, 0.6, -0.6, -3.0, -2.1, -2.4, -2.1, -3.0, -2.2, -3.0, -1.8, -1.2, -3.0, -0.8, -2.4, -2.5, -3.0, -1.5, -1.2, -0.0, -2.7, -2.4, -3.0, -3.0, -2.1, -2.1, -2.1, -3.0, -2.7, -2.4, -2.1, -1.5, -2.1, -0.6, -3.0, -3.0, -3.0, -3.0]
b = [-1.2, 6.0, -3.0, 1.5, 12.0, 1.5, 3.0, -0.0, -1.5, -0.0, 6.0, -0.0, 1.8, -1.5, -3.0, -3.0, 1.5, -0.0, 1.5, -3.0, -3.0, -0.0, -1.2, -3.0, 22.5, -0.0, -3.0, -3.0, -2.1, 3.0, 2.4, 1.5, -2.1, 4.5, -3.0, -3.0, 12.0, 6.0, -3.0, 3.0, -3.0, 12.0, -2.8, 0.6, 4.5, 3.0, -0.0, -3.0, -1.0, -3.0, -0.0, 3.0, 2.1, -0.6, -3.0, -3.0, 1.5, 1.5, -0.6, -0.0, 1.5, 6.0, -2.2, -2.1, -0.0, -0.8, 6.0, -3.0, -3.0, 9.0, -3.0, -0.9, 15.0, 1.5, -2.9, 19.5, 4.5, -3.0, 1.5, -1.8, 1.5, 0.9, -3.0, -3.0, 9.0, -0.3, 9.0, -2.1, -2.2, -3.0, 12.0, -2.1, -1.2, -3.0, 4.5, -1.2, -0.0, 12.0, -0.9, -3.0, -0.6, -3.0, -1.5, -0.0, 27.0, 4.5, -2.4, -2.7, -0.0, 3.0, 1.5, 3.0, -3.0, 16.5, -2.1, -0.6, -3.0, -1.5, -1.2, -3.0, 4.4, -2.5, -2.1, 3.0, -1.2, -3.0, 12.0, -1.8, -0.9, -3.0, -3.0, -0.0, -3.0, -1.2, -3.0, -3.0, -0.0, 1.5, -3.0, 7.5, -3.0, -2.1, 3.0, 1.5, 3.0, -3.0, 1.5, -3.0, 19.5, -3.0, -2.2, 27.0, 3.0, -1.5, -3.0, -3.0, 4.5, -1.2, 12.0, 3.0, 3.0, -2.1, 6.0, -2.2, -3.0, -2.4, 6.0, -3.0, -1.9, -0.6, -0.0, 3.6, 15.0, -3.0, -3.0, 7.5, -0.0, 4.5, -2.4, -3.0, -3.0, -3.0, -2.0, -3.0, -3.0, -3.0, -2.6, 3.0, -0.0, -3.0, 4.5, -1.2, -3.0, -3.0, 3.0, -3.0, 1.6, 1.5, -3.0, -3.0, -3.0, -3.0, 3.0, -3.0, -1.5, -3.0, -1.5, 12.0, -1.5, 3.0, 9.6, -0.0, -1.8, -3.0, -2.1, -0.6, 9.0, -3.0, 19.5, -2.4, -1.8, 15.0, -3.0, -3.0, 27.0, 7.5, 12.0, -3.0, -2.0, -3.0, 12.0, -3.0, -3.0, 0.9, -3.0, -2.5, 12.6, -1.5, -0.6, -3.0, -3.0, -0.0, -0.8, -3.0, 3.0, -0.0, 4.5, -3.0, -0.0, 0.6, 0.3, -2.7, -0.0, 4.5, -3.0, 3.0, -1.5, 9.0, -3.0, 1.5, -3.0, 6.0, -2.2, -0.0, -3.0, -3.0, 3.0, -3.0, -2.7, -0.0, -3.0, 0.6, 4.5, 1.5, 1.5, 1.5, -2.1, 7.5, -3.0, -3.0, -3.0, -2.1, -3.0, 6.0, -1.5, -2.0, -3.0, -0.6, 6.0, -1.5, -3.0, 6.0, -3.0, 3.0, 3.0, -3.0, 7.5, -3.0, -0.0, -3.0, -0.0, -3.0, -3.0, 0.6, -3.0, -3.0, -3.0, -3.0, -2.4, -0.0, 1.5, 3.0, 2.1, -3.0, 3.0, 4.5, -3.0, -2.4, -3.0, -2.2, 7.5, 2.1, -3.0, -0.0, -2.0, -3.0, -3.0, -1.8, -3.0, 4.5, -1.5, -3.0, 4.5, 3.0, 15.0, 4.5, -2.1, 12.0, 6.0, 4.5, 27.0, -3.0, 3.0, 0.6, -3.0, 0.6, -2.4, 4.5, -1.5, -2.2, 12.0, -2.0, 1.5, 9.0, -1.5, -1.2, -2.0, 1.5, -1.2, -1.8, -3.0, -0.0, -0.0, -3.0, -0.9]

print(anderson_ksamp([a,b]))

我得到以下输出：

Anderson_ksampResult(
    statistic=53.560696338122263,
    critical_values=array([ 0.325,  1.226,  1.961,  2.718,  3.752]),
    significance_level=7238105.535194747
)

“显着性水平”远大于1。这是为什么呢？

我主要想知道这是否是 scipy 实现的问题，或者是 Anderson-Darling 测试的数值实现所固有的问题（因为我不知道实际计算 p 值的细节）。我也对有关如何解决此错误的一些信息感兴趣。

我知道这个问题，但是由于来自相同分布的数据，这似乎是外推的一个小错误（scipy.stats.ks_2samp 给出的 p 值为 0.962305，而 anderson_ksamp 给出了 1.075969）。另一方面，有了上面的数据，我们可以运行

from scipy.stats import ks_2samp
from statsmodels.distributions.empirical_distribution import ECDF
import matplotlib.pyplot as plt
import numpy as np

a = [-1.8, -2.4, -2.4, -0.0, -1.5, -2.7, -1.8, -3.0, -1.8, -1.2, -3.0, -3.0, -2.8, -3.0, -2.1, -0.0, 0.6, -2.5, -2.4, -0.0, -2.7, -0.0, -2.5, -2.1, -0.9, -3.0, -0.6, -0.6, -1.5, -2.2, -1.2, -2.4, -2.4, -3.0, 1.5, -1.8, 1.5, -2.7, -3.0, -2.5, -2.5, -1.5, -1.5, -2.1, -2.1, -3.0, -0.6, -2.7, -3.0, -1.5, -0.6, -0.0, -2.1, 0.6, -2.0, -3.0, -3.0, -2.4, -3.0, -1.8, -0.0, -0.0, -3.0, -1.5, -3.0, -3.0, -1.5, -2.5, -3.0, -2.8, -3.0, -2.2, -0.6, -0.0, -1.5, -2.7, -2.1, -2.1, -2.2, -2.1, -0.6, -0.0, -2.5, -2.1, -1.5, -3.0, -2.2, -1.8, -2.7, -2.4, -1.5, -2.1, -2.9, -2.4, -0.9, -0.0, -0.0, -2.4, -2.7, -0.6, -2.2, -3.0, -1.5, -0.9, -3.0, -3.0, -0.0, -2.7, -2.7, -1.5, -2.2, -3.0, -0.0, 1.5, -3.0, -2.7, -2.2, -2.9, -2.2, -3.0, -1.8, -0.0, -3.0, -1.5, -2.7, -3.0, -3.0, -2.9, -3.0, -3.0, -3.0, -3.0, -2.5, -0.0, 0.9, -3.0, -0.0, -3.0, 3.0, -3.0, -3.0, -1.2, -2.1, 1.5, -0.0, -0.9, -3.0, -2.7, -1.5, -2.4, -2.1, -3.0, -0.9, -3.0, -0.8, -1.5, -2.1, -2.7, -0.0, -0.0, -2.2, -1.8, -2.1, -2.2, -3.0, 0.6, -2.4, -2.2, -2.4, -2.5, -1.5, -0.0, -2.7, -3.0, -3.0, -2.1, -0.0, -2.4, -2.4, -0.0, -2.0, -0.9, -2.4, -3.0, -1.4, -2.7, -2.7, -3.0, -3.0, -2.7, -1.2, -2.1, -3.0, -0.0, -3.0, -2.7, -2.7, -3.0, -3.0, -2.5, -3.0, -1.8, -1.5, -2.7, -2.4, -1.8, -3.0, -2.7, 2.1, -3.0, -2.2, -2.2, 0.6, -0.9, 6.0, -3.0, -2.1, -3.0, -2.1, -2.5, -3.0, -1.5, -2.5, 3.0, -2.1, -3.0, -3.0, -1.5, -2.1, -2.7, -2.5, 1.5, -2.1, -0.0, -0.0, -3.0, -0.6, 1.5, -2.7, -2.4, -2.1, -3.0, -2.7, -3.0, 9.0, -3.0, -1.7, -3.0, -0.0, -3.0, -2.2, -0.0, -0.6, -2.7, -2.7, -3.0, -3.0, -1.7, -2.1, -2.0, -3.0, -2.1, -3.0, -1.1, -3.0, -3.0, -2.4, -1.5, -3.0, -2.2, -3.0, -1.5, -2.7, -3.0, -3.0, -3.0, -2.2, -3.0, -2.1, -2.1, -2.4, -3.0, -3.0, -0.0, -3.0, -3.0, -2.1, -2.0, 1.5, -3.0, -3.0, -2.1, -2.9, -2.4, -3.0, -3.0, -1.5, -2.2, -0.9, -1.8, -2.1, -1.8, -1.5, -3.0, -1.5, -3.0, -1.5, -3.0, -2.4, 1.5, -2.7, -3.0, -1.8, -1.8, -1.5, -2.1, -2.7, -2.7, -2.7, -3.0, -1.5, -2.7, -3.0, -2.7, -3.0, -1.5, -1.5, -3.0, 0.6, -0.6, -3.0, -2.1, -2.4, -2.1, -3.0, -2.2, -3.0, -1.8, -1.2, -3.0, -0.8, -2.4, -2.5, -3.0, -1.5, -1.2, -0.0, -2.7, -2.4, -3.0, -3.0, -2.1, -2.1, -2.1, -3.0, -2.7, -2.4, -2.1, -1.5, -2.1, -0.6, -3.0, -3.0, -3.0, -3.0]
b = [-1.2, 6.0, -3.0, 1.5, 12.0, 1.5, 3.0, -0.0, -1.5, -0.0, 6.0, -0.0, 1.8, -1.5, -3.0, -3.0, 1.5, -0.0, 1.5, -3.0, -3.0, -0.0, -1.2, -3.0, 22.5, -0.0, -3.0, -3.0, -2.1, 3.0, 2.4, 1.5, -2.1, 4.5, -3.0, -3.0, 12.0, 6.0, -3.0, 3.0, -3.0, 12.0, -2.8, 0.6, 4.5, 3.0, -0.0, -3.0, -1.0, -3.0, -0.0, 3.0, 2.1, -0.6, -3.0, -3.0, 1.5, 1.5, -0.6, -0.0, 1.5, 6.0, -2.2, -2.1, -0.0, -0.8, 6.0, -3.0, -3.0, 9.0, -3.0, -0.9, 15.0, 1.5, -2.9, 19.5, 4.5, -3.0, 1.5, -1.8, 1.5, 0.9, -3.0, -3.0, 9.0, -0.3, 9.0, -2.1, -2.2, -3.0, 12.0, -2.1, -1.2, -3.0, 4.5, -1.2, -0.0, 12.0, -0.9, -3.0, -0.6, -3.0, -1.5, -0.0, 27.0, 4.5, -2.4, -2.7, -0.0, 3.0, 1.5, 3.0, -3.0, 16.5, -2.1, -0.6, -3.0, -1.5, -1.2, -3.0, 4.4, -2.5, -2.1, 3.0, -1.2, -3.0, 12.0, -1.8, -0.9, -3.0, -3.0, -0.0, -3.0, -1.2, -3.0, -3.0, -0.0, 1.5, -3.0, 7.5, -3.0, -2.1, 3.0, 1.5, 3.0, -3.0, 1.5, -3.0, 19.5, -3.0, -2.2, 27.0, 3.0, -1.5, -3.0, -3.0, 4.5, -1.2, 12.0, 3.0, 3.0, -2.1, 6.0, -2.2, -3.0, -2.4, 6.0, -3.0, -1.9, -0.6, -0.0, 3.6, 15.0, -3.0, -3.0, 7.5, -0.0, 4.5, -2.4, -3.0, -3.0, -3.0, -2.0, -3.0, -3.0, -3.0, -2.6, 3.0, -0.0, -3.0, 4.5, -1.2, -3.0, -3.0, 3.0, -3.0, 1.6, 1.5, -3.0, -3.0, -3.0, -3.0, 3.0, -3.0, -1.5, -3.0, -1.5, 12.0, -1.5, 3.0, 9.6, -0.0, -1.8, -3.0, -2.1, -0.6, 9.0, -3.0, 19.5, -2.4, -1.8, 15.0, -3.0, -3.0, 27.0, 7.5, 12.0, -3.0, -2.0, -3.0, 12.0, -3.0, -3.0, 0.9, -3.0, -2.5, 12.6, -1.5, -0.6, -3.0, -3.0, -0.0, -0.8, -3.0, 3.0, -0.0, 4.5, -3.0, -0.0, 0.6, 0.3, -2.7, -0.0, 4.5, -3.0, 3.0, -1.5, 9.0, -3.0, 1.5, -3.0, 6.0, -2.2, -0.0, -3.0, -3.0, 3.0, -3.0, -2.7, -0.0, -3.0, 0.6, 4.5, 1.5, 1.5, 1.5, -2.1, 7.5, -3.0, -3.0, -3.0, -2.1, -3.0, 6.0, -1.5, -2.0, -3.0, -0.6, 6.0, -1.5, -3.0, 6.0, -3.0, 3.0, 3.0, -3.0, 7.5, -3.0, -0.0, -3.0, -0.0, -3.0, -3.0, 0.6, -3.0, -3.0, -3.0, -3.0, -2.4, -0.0, 1.5, 3.0, 2.1, -3.0, 3.0, 4.5, -3.0, -2.4, -3.0, -2.2, 7.5, 2.1, -3.0, -0.0, -2.0, -3.0, -3.0, -1.8, -3.0, 4.5, -1.5, -3.0, 4.5, 3.0, 15.0, 4.5, -2.1, 12.0, 6.0, 4.5, 27.0, -3.0, 3.0, 0.6, -3.0, 0.6, -2.4, 4.5, -1.5, -2.2, 12.0, -2.0, 1.5, 9.0, -1.5, -1.2, -2.0, 1.5, -1.2, -1.8, -3.0, -0.0, -0.0, -3.0, -0.9]

ecdf1, ecdf2 = ECDF(a), ECDF(b)

xs = np.linspace(min(a+b),max(a+b), num=10000)
plt.figure(figsize=(12,8))
plt.plot(xs,ecdf1(xs), xs,ecdf2(xs))
plt.show()

print(ks_2samp(a,b))

得到 2.6169e-17 的 p 值，下面的 ECDF 图显示从 a 和 b 采样的分布非常不同： ECDF 图

KstestResult(statistic=0.3232876712328767, pvalue=2.8586245432606456e-17)
1个回答

2022 年 2 月 27 日更新，适用于可能遇到相同问题并找到此帖子的人：

注意：此答案旨在指出scipy最初在 2016 年 7 月 25 日提出问题时的错误。此错误导致 p 值大于 1，并在此提交中修复，看起来 scipy 版本scipy>=1.2.0不受此影响。代码示例和输出可在此处获得，以便重现。以下是测试Python 3.9.9和scipy==1.8.0

第一个代码块：

from scipy.stats import anderson_ksamp

a = [-1.8, -2.4, -2.4, -0.0, -1.5, -2.7, -1.8, -3.0, -1.8, -1.2, -3.0,
     -3.0, -2.8, -3.0, -2.1, -0.0, 0.6, -2.5, -2.4, -0.0, -2.7, -0.0,
     -2.5, -2.1, -0.9, -3.0, -0.6, -0.6, -1.5, -2.2, -1.2, -2.4, -2.4,
     -3.0, 1.5, -1.8, 1.5, -2.7, -3.0, -2.5, -2.5, -1.5, -1.5, -2.1,
     -2.1, -3.0, -0.6, -2.7, -3.0, -1.5, -0.6, -0.0, -2.1, 0.6, -2.0,
     -3.0, -3.0, -2.4, -3.0, -1.8, -0.0, -0.0, -3.0, -1.5, -3.0, -3.0,
     -1.5, -2.5, -3.0, -2.8, -3.0, -2.2, -0.6, -0.0, -1.5, -2.7, -2.1,
     -2.1, -2.2, -2.1, -0.6, -0.0, -2.5, -2.1, -1.5, -3.0, -2.2, -1.8,
     -2.7, -2.4, -1.5, -2.1, -2.9, -2.4, -0.9, -0.0, -0.0, -2.4, -2.7,
     -0.6, -2.2, -3.0, -1.5, -0.9, -3.0, -3.0, -0.0, -2.7, -2.7, -1.5,
     -2.2, -3.0, -0.0, 1.5, -3.0, -2.7, -2.2, -2.9, -2.2, -3.0, -1.8,
     -0.0, -3.0, -1.5, -2.7, -3.0, -3.0, -2.9, -3.0, -3.0, -3.0, -3.0,
     -2.5, -0.0, 0.9, -3.0, -0.0, -3.0, 3.0, -3.0, -3.0, -1.2, -2.1,
     1.5, -0.0, -0.9, -3.0, -2.7, -1.5, -2.4, -2.1, -3.0, -0.9, -3.0,
     -0.8, -1.5, -2.1, -2.7, -0.0, -0.0, -2.2, -1.8, -2.1, -2.2, -3.0,
     0.6, -2.4, -2.2, -2.4, -2.5, -1.5, -0.0, -2.7, -3.0, -3.0, -2.1,
     -0.0, -2.4, -2.4, -0.0, -2.0, -0.9, -2.4, -3.0, -1.4, -2.7, -2.7,
     -3.0, -3.0, -2.7, -1.2, -2.1, -3.0, -0.0, -3.0, -2.7, -2.7, -3.0,
     -3.0, -2.5, -3.0, -1.8, -1.5, -2.7, -2.4, -1.8, -3.0, -2.7, 2.1,
     -3.0, -2.2, -2.2, 0.6, -0.9, 6.0, -3.0, -2.1, -3.0, -2.1, -2.5,
     -3.0, -1.5, -2.5, 3.0, -2.1, -3.0, -3.0, -1.5, -2.1, -2.7, -2.5,
     1.5, -2.1, -0.0, -0.0, -3.0, -0.6, 1.5, -2.7, -2.4, -2.1, -3.0,
     -2.7, -3.0, 9.0, -3.0, -1.7, -3.0, -0.0, -3.0, -2.2, -0.0, -0.6,
     -2.7, -2.7, -3.0, -3.0, -1.7, -2.1, -2.0, -3.0, -2.1, -3.0, -1.1,
     -3.0, -3.0, -2.4, -1.5, -3.0, -2.2, -3.0, -1.5, -2.7, -3.0, -3.0,
     -3.0, -2.2, -3.0, -2.1, -2.1, -2.4, -3.0, -3.0, -0.0, -3.0, -3.0,
     -2.1, -2.0, 1.5, -3.0, -3.0, -2.1, -2.9, -2.4, -3.0, -3.0, -1.5,
     -2.2, -0.9, -1.8, -2.1, -1.8, -1.5, -3.0, -1.5, -3.0, -1.5, -3.0,
     -2.4, 1.5, -2.7, -3.0, -1.8, -1.8, -1.5, -2.1, -2.7, -2.7, -2.7,
     -3.0, -1.5, -2.7, -3.0, -2.7, -3.0, -1.5, -1.5, -3.0, 0.6, -0.6,
     -3.0, -2.1, -2.4, -2.1, -3.0, -2.2, -3.0, -1.8, -1.2, -3.0, -0.8,
     -2.4, -2.5, -3.0, -1.5, -1.2, -0.0, -2.7, -2.4, -3.0, -3.0, -2.1,
     -2.1, -2.1, -3.0, -2.7, -2.4, -2.1, -1.5, -2.1, -0.6, -3.0, -3.0,
     -3.0, -3.0]
b = [-1.2, 6.0, -3.0, 1.5, 12.0, 1.5, 3.0, -0.0, -1.5, -0.0, 6.0,
     -0.0, 1.8, -1.5, -3.0, -3.0, 1.5, -0.0, 1.5, -3.0, -3.0, -0.0,
     -1.2, -3.0, 22.5, -0.0, -3.0, -3.0, -2.1, 3.0, 2.4, 1.5, -2.1,
     4.5, -3.0, -3.0, 12.0, 6.0, -3.0, 3.0, -3.0, 12.0, -2.8, 0.6,
     4.5, 3.0, -0.0, -3.0, -1.0, -3.0, -0.0, 3.0, 2.1, -0.6, -3.0,
     -3.0, 1.5, 1.5, -0.6, -0.0, 1.5, 6.0, -2.2, -2.1, -0.0, -0.8,
     6.0, -3.0, -3.0, 9.0, -3.0, -0.9, 15.0, 1.5, -2.9, 19.5, 4.5,
     -3.0, 1.5, -1.8, 1.5, 0.9, -3.0, -3.0, 9.0, -0.3, 9.0, -2.1,
     -2.2, -3.0, 12.0, -2.1, -1.2, -3.0, 4.5, -1.2, -0.0, 12.0, -0.9,
     -3.0, -0.6, -3.0, -1.5, -0.0, 27.0, 4.5, -2.4, -2.7, -0.0, 3.0,
     1.5, 3.0, -3.0, 16.5, -2.1, -0.6, -3.0, -1.5, -1.2, -3.0, 4.4,
     -2.5, -2.1, 3.0, -1.2, -3.0, 12.0, -1.8, -0.9, -3.0, -3.0, -0.0,
     -3.0, -1.2, -3.0, -3.0, -0.0, 1.5, -3.0, 7.5, -3.0, -2.1, 3.0,
     1.5, 3.0, -3.0, 1.5, -3.0, 19.5, -3.0, -2.2, 27.0, 3.0, -1.5,
     -3.0, -3.0, 4.5, -1.2, 12.0, 3.0, 3.0, -2.1, 6.0, -2.2, -3.0,
     -2.4, 6.0, -3.0, -1.9, -0.6, -0.0, 3.6, 15.0, -3.0, -3.0, 7.5,
     -0.0, 4.5, -2.4, -3.0, -3.0, -3.0, -2.0, -3.0, -3.0, -3.0, -2.6,
     3.0, -0.0, -3.0, 4.5, -1.2, -3.0, -3.0, 3.0, -3.0, 1.6, 1.5,
     -3.0, -3.0, -3.0, -3.0, 3.0, -3.0, -1.5, -3.0, -1.5, 12.0, -1.5,
     3.0, 9.6, -0.0, -1.8, -3.0, -2.1, -0.6, 9.0, -3.0, 19.5, -2.4,
     -1.8, 15.0, -3.0, -3.0, 27.0, 7.5, 12.0, -3.0, -2.0, -3.0, 12.0,
     -3.0, -3.0, 0.9, -3.0, -2.5, 12.6, -1.5, -0.6, -3.0, -3.0, -0.0,
     -0.8, -3.0, 3.0, -0.0, 4.5, -3.0, -0.0, 0.6, 0.3, -2.7, -0.0,
     4.5, -3.0, 3.0, -1.5, 9.0, -3.0, 1.5, -3.0, 6.0, -2.2, -0.0,
     -3.0, -3.0, 3.0, -3.0, -2.7, -0.0, -3.0, 0.6, 4.5, 1.5, 1.5, 1.5,
     -2.1, 7.5, -3.0, -3.0, -3.0, -2.1, -3.0, 6.0, -1.5, -2.0, -3.0,
     -0.6, 6.0, -1.5, -3.0, 6.0, -3.0, 3.0, 3.0, -3.0, 7.5, -3.0,
     -0.0, -3.0, -0.0, -3.0, -3.0, 0.6, -3.0, -3.0, -3.0, -3.0, -2.4,
     -0.0, 1.5, 3.0, 2.1, -3.0, 3.0, 4.5, -3.0, -2.4, -3.0, -2.2, 7.5,
     2.1, -3.0, -0.0, -2.0, -3.0, -3.0, -1.8, -3.0, 4.5, -1.5, -3.0,
     4.5, 3.0, 15.0, 4.5, -2.1, 12.0, 6.0, 4.5, 27.0, -3.0, 3.0, 0.6,
     -3.0, 0.6, -2.4, 4.5, -1.5, -2.2, 12.0, -2.0, 1.5, 9.0, -1.5,
     -1.2, -2.0, 1.5, -1.2, -1.8, -3.0, -0.0, -0.0, -3.0, -0.9]

print(anderson_ksamp([a, b]))

输出：

Anderson_ksampResult(
    statistic=53.56069633812226,
    critical_values=array([0.325, 1.226, 1.961, 2.718, 3.752, 4.592, 6.546]),
    significance_level=0.001
)

并生成警告：

UserWarning: p-value floored: true value smaller than 0.001

第二个代码块：

 from scipy.stats import ks_2samp
 from statsmodels.distributions.empirical_distribution import ECDF
 import matplotlib.pyplot as plt
 import numpy as np
 
 a = [-1.8, -2.4, -2.4, -0.0, -1.5, -2.7, -1.8, -3.0, -1.8, -1.2, -3.0,
      -3.0, -2.8, -3.0, -2.1, -0.0, 0.6, -2.5, -2.4, -0.0, -2.7, -0.0,
      -2.5, -2.1, -0.9, -3.0, -0.6, -0.6, -1.5, -2.2, -1.2, -2.4, -2.4,
      -3.0, 1.5, -1.8, 1.5, -2.7, -3.0, -2.5, -2.5, -1.5, -1.5, -2.1,
      -2.1, -3.0, -0.6, -2.7, -3.0, -1.5, -0.6, -0.0, -2.1, 0.6, -2.0,
      -3.0, -3.0, -2.4, -3.0, -1.8, -0.0, -0.0, -3.0, -1.5, -3.0, -3.0,
      -1.5, -2.5, -3.0, -2.8, -3.0, -2.2, -0.6, -0.0, -1.5, -2.7, -2.1,
      -2.1, -2.2, -2.1, -0.6, -0.0, -2.5, -2.1, -1.5, -3.0, -2.2, -1.8,
      -2.7, -2.4, -1.5, -2.1, -2.9, -2.4, -0.9, -0.0, -0.0, -2.4, -2.7,
      -0.6, -2.2, -3.0, -1.5, -0.9, -3.0, -3.0, -0.0, -2.7, -2.7, -1.5,
      -2.2, -3.0, -0.0, 1.5, -3.0, -2.7, -2.2, -2.9, -2.2, -3.0, -1.8,
      -0.0, -3.0, -1.5, -2.7, -3.0, -3.0, -2.9, -3.0, -3.0, -3.0, -3.0,
      -2.5, -0.0, 0.9, -3.0, -0.0, -3.0, 3.0, -3.0, -3.0, -1.2, -2.1,
      1.5, -0.0, -0.9, -3.0, -2.7, -1.5, -2.4, -2.1, -3.0, -0.9, -3.0,
      -0.8, -1.5, -2.1, -2.7, -0.0, -0.0, -2.2, -1.8, -2.1, -2.2, -3.0,
      0.6, -2.4, -2.2, -2.4, -2.5, -1.5, -0.0, -2.7, -3.0, -3.0, -2.1,
      -0.0, -2.4, -2.4, -0.0, -2.0, -0.9, -2.4, -3.0, -1.4, -2.7, -2.7,
      -3.0, -3.0, -2.7, -1.2, -2.1, -3.0, -0.0, -3.0, -2.7, -2.7, -3.0,
      -3.0, -2.5, -3.0, -1.8, -1.5, -2.7, -2.4, -1.8, -3.0, -2.7, 2.1,
      -3.0, -2.2, -2.2, 0.6, -0.9, 6.0, -3.0, -2.1, -3.0, -2.1, -2.5,
      -3.0, -1.5, -2.5, 3.0, -2.1, -3.0, -3.0, -1.5, -2.1, -2.7, -2.5,
      1.5, -2.1, -0.0, -0.0, -3.0, -0.6, 1.5, -2.7, -2.4, -2.1, -3.0,
      -2.7, -3.0, 9.0, -3.0, -1.7, -3.0, -0.0, -3.0, -2.2, -0.0, -0.6,
      -2.7, -2.7, -3.0, -3.0, -1.7, -2.1, -2.0, -3.0, -2.1, -3.0, -1.1,
      -3.0, -3.0, -2.4, -1.5, -3.0, -2.2, -3.0, -1.5, -2.7, -3.0, -3.0,
      -3.0, -2.2, -3.0, -2.1, -2.1, -2.4, -3.0, -3.0, -0.0, -3.0, -3.0,
      -2.1, -2.0, 1.5, -3.0, -3.0, -2.1, -2.9, -2.4, -3.0, -3.0, -1.5,
      -2.2, -0.9, -1.8, -2.1, -1.8, -1.5, -3.0, -1.5, -3.0, -1.5, -3.0,
      -2.4, 1.5, -2.7, -3.0, -1.8, -1.8, -1.5, -2.1, -2.7, -2.7, -2.7,
      -3.0, -1.5, -2.7, -3.0, -2.7, -3.0, -1.5, -1.5, -3.0, 0.6, -0.6,
      -3.0, -2.1, -2.4, -2.1, -3.0, -2.2, -3.0, -1.8, -1.2, -3.0, -0.8,
      -2.4, -2.5, -3.0, -1.5, -1.2, -0.0, -2.7, -2.4, -3.0, -3.0, -2.1,
      -2.1, -2.1, -3.0, -2.7, -2.4, -2.1, -1.5, -2.1, -0.6, -3.0, -3.0,
      -3.0, -3.0]
 b = [-1.2, 6.0, -3.0, 1.5, 12.0, 1.5, 3.0, -0.0, -1.5, -0.0, 6.0,
      -0.0, 1.8, -1.5, -3.0, -3.0, 1.5, -0.0, 1.5, -3.0, -3.0, -0.0,
      -1.2, -3.0, 22.5, -0.0, -3.0, -3.0, -2.1, 3.0, 2.4, 1.5, -2.1,
      4.5, -3.0, -3.0, 12.0, 6.0, -3.0, 3.0, -3.0, 12.0, -2.8, 0.6,
      4.5, 3.0, -0.0, -3.0, -1.0, -3.0, -0.0, 3.0, 2.1, -0.6, -3.0,
      -3.0, 1.5, 1.5, -0.6, -0.0, 1.5, 6.0, -2.2, -2.1, -0.0, -0.8,
      6.0, -3.0, -3.0, 9.0, -3.0, -0.9, 15.0, 1.5, -2.9, 19.5, 4.5,
      -3.0, 1.5, -1.8, 1.5, 0.9, -3.0, -3.0, 9.0, -0.3, 9.0, -2.1,
      -2.2, -3.0, 12.0, -2.1, -1.2, -3.0, 4.5, -1.2, -0.0, 12.0, -0.9,
      -3.0, -0.6, -3.0, -1.5, -0.0, 27.0, 4.5, -2.4, -2.7, -0.0, 3.0,
      1.5, 3.0, -3.0, 16.5, -2.1, -0.6, -3.0, -1.5, -1.2, -3.0, 4.4,
      -2.5, -2.1, 3.0, -1.2, -3.0, 12.0, -1.8, -0.9, -3.0, -3.0, -0.0,
      -3.0, -1.2, -3.0, -3.0, -0.0, 1.5, -3.0, 7.5, -3.0, -2.1, 3.0,
      1.5, 3.0, -3.0, 1.5, -3.0, 19.5, -3.0, -2.2, 27.0, 3.0, -1.5,
      -3.0, -3.0, 4.5, -1.2, 12.0, 3.0, 3.0, -2.1, 6.0, -2.2, -3.0,
      -2.4, 6.0, -3.0, -1.9, -0.6, -0.0, 3.6, 15.0, -3.0, -3.0, 7.5,
      -0.0, 4.5, -2.4, -3.0, -3.0, -3.0, -2.0, -3.0, -3.0, -3.0, -2.6,
      3.0, -0.0, -3.0, 4.5, -1.2, -3.0, -3.0, 3.0, -3.0, 1.6, 1.5,
      -3.0, -3.0, -3.0, -3.0, 3.0, -3.0, -1.5, -3.0, -1.5, 12.0, -1.5,
      3.0, 9.6, -0.0, -1.8, -3.0, -2.1, -0.6, 9.0, -3.0, 19.5, -2.4,
      -1.8, 15.0, -3.0, -3.0, 27.0, 7.5, 12.0, -3.0, -2.0, -3.0, 12.0,
      -3.0, -3.0, 0.9, -3.0, -2.5, 12.6, -1.5, -0.6, -3.0, -3.0, -0.0,
      -0.8, -3.0, 3.0, -0.0, 4.5, -3.0, -0.0, 0.6, 0.3, -2.7, -0.0,
      4.5, -3.0, 3.0, -1.5, 9.0, -3.0, 1.5, -3.0, 6.0, -2.2, -0.0,
      -3.0, -3.0, 3.0, -3.0, -2.7, -0.0, -3.0, 0.6, 4.5, 1.5, 1.5, 1.5,
      -2.1, 7.5, -3.0, -3.0, -3.0, -2.1, -3.0, 6.0, -1.5, -2.0, -3.0,
      -0.6, 6.0, -1.5, -3.0, 6.0, -3.0, 3.0, 3.0, -3.0, 7.5, -3.0,
      -0.0, -3.0, -0.0, -3.0, -3.0, 0.6, -3.0, -3.0, -3.0, -3.0, -2.4,
      -0.0, 1.5, 3.0, 2.1, -3.0, 3.0, 4.5, -3.0, -2.4, -3.0, -2.2, 7.5,
      2.1, -3.0, -0.0, -2.0, -3.0, -3.0, -1.8, -3.0, 4.5, -1.5, -3.0,
      4.5, 3.0, 15.0, 4.5, -2.1, 12.0, 6.0, 4.5, 27.0, -3.0, 3.0, 0.6,
      -3.0, 0.6, -2.4, 4.5, -1.5, -2.2, 12.0, -2.0, 1.5, 9.0, -1.5,
      -1.2, -2.0, 1.5, -1.2, -1.8, -3.0, -0.0, -0.0, -3.0, -0.9]
 
 ecdf1, ecdf2 = ECDF(a), ECDF(b)
 
 xs = np.linspace(min(a + b), max(a + b), num=10000)
 plt.figure(figsize=(12, 8))
 plt.plot(xs, ecdf1(xs), xs, ecdf2(xs))
 plt.show()
 
 print(ks_2samp(a, b))

输出：

在此处输入图像描述

KstestResult(statistic=0.3232876712328767, pvalue=2.8586245432606456e-17)
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