Anderson-Darling 검정¶

import numpy as np
from scipy import stats

# 정규 230개에 로그정규 70개를 섞은 자료다.
rng = np.random.default_rng(0)
x = np.concatenate([rng.normal(0, 1, size=230),
                    rng.lognormal(0, 0.6, size=70)])

# A^2 은 경험분포와 이론분포의 차이를 제곱해 적분한 값인데, 꼬리 쪽에
# 더 큰 무게를 싣는다. 그래서 KS 보다 꼬리 이탈을 잘 잡는다.
res = stats.anderson(x, dist="norm")
print(f"Sample size n = {x.size}")
print(f"Anderson-Darling A^2 = {res.statistic:.4f}")
print("Critical values vs significance levels:")
for cv, sl in zip(res.critical_values, res.significance_level):
    flag = " REJECT" if res.statistic > cv else ""
    print(f"  {sl:.1f}% -> {cv:.4f}{flag}")

reject_5 = res.statistic > res.critical_values[
    list(res.significance_level).index(5.0)]
print(f"Decision at 5%: {'Reject' if reject_5 else 'Fail to reject'} normality")

Sample size n = 300
Anderson-Darling A^2 = 1.0034
Critical values vs significance levels:
  15.0% -> 0.5690 REJECT
  10.0% -> 0.6480 REJECT
  5.0% -> 0.7770 REJECT
  2.5% -> 0.9060 REJECT
  1.0% -> 1.0780
Decision at 5%: Reject normality

import numpy as np
from scipy import stats
from statsmodels.stats.diagnostic import lilliefors

rng = np.random.default_rng(0)
x = np.concatenate([rng.normal(0, 1, size=230),
                    rng.lognormal(0, 0.6, size=70)])
res = stats.anderson(x, dist="norm")
print(f"A^2 = {res.statistic:.4f},  1% 임계값 = {res.critical_values[4]:.4f}"
      f"  (차 {res.critical_values[4] - res.statistic:.4f})")
print(f"Shapiro-Wilk  W = {stats.shapiro(x)[0]:.4f}, p = {stats.shapiro(x)[1]:.4g}")
print(f"Jarque-Bera  JB = {stats.jarque_bera(x)[0]:.4f}, p = {stats.jarque_bera(x)[1]:.4g}")
print(f"DAgostino K^2  = {stats.normaltest(x)[0]:.4f}, p = {stats.normaltest(x)[1]:.4g}")
print(f"Lilliefors    D = {lilliefors(x)[0]:.4f}, p = {lilliefors(x)[1]:.4g}")

# 이 혼합 대립에서 네 검정의 검정력 (n = 300, alpha = 0.05)
rg = np.random.default_rng(13)
R, n_norm, n_ln = 1500, 230, 70
cnt = dict(AD=0, SW=0, JB=0, LF=0)
for _ in range(R):
    y = np.concatenate([rg.normal(0, 1, n_norm), rg.lognormal(0, 0.6, n_ln)])
    r = stats.anderson(y, dist="norm")
    cnt["AD"] += r.statistic > r.critical_values[2]
    cnt["SW"] += stats.shapiro(y)[1] < 0.05
    cnt["JB"] += stats.jarque_bera(y)[1] < 0.05
    cnt["LF"] += lilliefors(y)[1] < 0.05
print(f"\n혼합 대립에서의 검정력 (n = 300, R = {R}, MC SE 최대 {0.5 / np.sqrt(R):.4f})")
for k in ("AD", "SW", "JB", "LF"):
    print(f"  {k}: {cnt[k] / R:.4f}")

# 정규 자료에서의 크기도 확인
cnt0 = dict(AD=0, SW=0, JB=0, LF=0)
for _ in range(R):
    y = rg.standard_normal(300)
    r = stats.anderson(y, dist="norm")
    cnt0["AD"] += r.statistic > r.critical_values[2]
    cnt0["SW"] += stats.shapiro(y)[1] < 0.05
    cnt0["JB"] += stats.jarque_bera(y)[1] < 0.05
    cnt0["LF"] += lilliefors(y)[1] < 0.05
print(f"정규 자료에서의 크기 (명목 0.05, MC SE {np.sqrt(0.05 * 0.95 / R):.4f})")
for k in ("AD", "SW", "JB", "LF"):
    print(f"  {k}: {cnt0[k] / R:.4f}")

A^2 = 1.0034,  1% 임계값 = 1.0780  (차 0.0746)
Shapiro-Wilk  W = 0.9723, p = 1.519e-05
Jarque-Bera  JB = 58.5069, p = 1.974e-13
DAgostino K^2  = 26.6879, p = 1.602e-06
Lilliefors    D = 0.0470, p = 0.1617

혼합 대립에서의 검정력 (n = 300, R = 1500, MC SE 최대 0.0129)
  AD: 0.4313
  SW: 0.4987
  JB: 0.4033
  LF: 0.3260
정규 자료에서의 크기 (명목 0.05, MC SE 0.0056)
  AD: 0.0420
  SW: 0.0493
  JB: 0.0520
  LF: 0.0373

import numpy as np
from scipy import stats

rng = np.random.default_rng(0)
x = rng.normal(0, 1, size=300)

res = stats.anderson(x, dist="norm")
print(f"A^2 = {res.statistic:.4f}")
print(f"15% critical value: {res.critical_values[0]:.4f}")
print("Below 15% critical value:", res.statistic < res.critical_values[0])

A^2 = 0.3329
15% critical value: 0.5690
Below 15% critical value: True

import numpy as np
from scipy import stats

rng = np.random.default_rng(0)
x = rng.standard_t(df=4, size=300)

# Anderson-Darling 검정 — 꼬리에 민감하다
ad = stats.anderson(x, dist="norm")
print(f"A^2 = {ad.statistic:.4f}")
for cv, sl in zip(ad.critical_values, ad.significance_level):
    flag = " *" if ad.statistic > cv else ""
    print(f"  {sl:.1f}%: {cv:.4f}{flag}")

# KS 검정
D, p_ks = stats.kstest(x, 'norm', args=(0, 1))
print(f"\nKS: D = {D:.4f}, p = {p_ks:.4f}")
print(f"Sample variance = {x.var(ddof=1):.4f}")

A^2 = 2.0862
  15.0%: 0.5690 *
  10.0%: 0.6480 *
  5.0%: 0.7770 *
  2.5%: 0.9060 *
  1.0%: 1.0780 *

KS: D = 0.0496, p = 0.4386
Sample variance = 1.6985

import numpy as np
from scipy import stats

rng = np.random.default_rng(0)
x = rng.exponential(2.0, size=200)

for dist in ["norm", "expon", "logistic"]:
    res = stats.anderson(x, dist=dist)
    print(f"\n{dist}: A^2 = {res.statistic:.4f}")
    for cv, sl in zip(res.critical_values, res.significance_level):
        flag = " REJECT" if res.statistic > cv else ""
        print(f"  {sl}%: {cv:.4f}{flag}")

norm: A^2 = 7.8017
  15.0%: 0.5650 REJECT
  10.0%: 0.6440 REJECT
  5.0%: 0.7720 REJECT
  2.5%: 0.9010 REJECT
  1.0%: 1.0710 REJECT

expon: A^2 = 0.5851
  15.0%: 0.9190
  10.0%: 1.0750
  5.0%: 1.3370
  2.5%: 1.6010
  1.0%: 1.9510

logistic: A^2 = 16.1719
  25.0%: 0.4250 REJECT
  10.0%: 0.5620 REJECT
  5.0%: 0.6590 REJECT
  2.5%: 0.7680 REJECT
  1.0%: 0.9050 REJECT
  0.5%: 1.0090 REJECT