Omnibus test statistics for normality. Modified Jarque–Bera test statistic. Power study. Monte Carlo simulation. Alternative distributions are contaminated normal ...
A Power Comparison for Testing Normality .
. Shigekazu Nakagawa, Hiroki Hashiguchi, and Naoto Niki Kurashiki University of Science and the Arts Saitama University Tokyo University of Science
COMPSTAT2010, Paris-France, Aug. 22-27
Nakagawa, Hashiguchi, and Niki ()
A Power Comparison for Testing Normality
COMPSTAT2010, Paris
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Table of Contents
Background · Motivation · Problem Omnibus test statistics for normality Modified Jarque–Bera test statistic
Power study Monte Carlo simulation Alternative distributions are contaminated normal distributions
Summary
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Background Jarque–Bera (1987) pointed out that a Lagrange multiplier test is equivalent to JB test against the Pearson distributions.
Jarque–Bera
0.4
(n → ∞) under H0 .
0.3
.
0.0
However, the χ2 approximation does not work well. Unfortunately, a normalizing tr. has not been given yet.
Histgram of JB: n = 100 curve in blue: pdf of χ2 2
0.2
Motivation
0.1
JB ∼ χ22
0.5
(√ 2 ) (b2 − 3)2 b1 JB = n + , 6 24 √ 3/2 where for a random sample (X1 , X2 , . . . , Xn ), b1 = m3 /m2 , b2 = m4 /m22 , ∑n ¯ j , j = 2, 3, 4. mj = (1/n) i=1 (Xi − X) .
0
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A Power Comparison for Testing Normality
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Proposed test statistic Nakagawa et al. (2007) proposed a modified version of the Jarque–Bera test.
Modified Jarque–Bera test ′
JB =
√ 2 b1 6
+
b22
.
24
A normalizing tr. of the null dist. for J B ′ has been derived. .
2 0
1
When is the power of J B ′ test superior to that of JB? As H1 , the contaminated normal distributions are considered.
3
4
Our goal
Nakagawa, Hashiguchi, and Niki ()
A Power Comparison for Testing Normality
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Power study 1 Let X1 , X2 , . . . , Xn be i.i.d. rv with a cdf F . Φ: the cdf of the standard normal distribution
Omnibus testing for normality (two sided) (
H0 : F (x) = Φ H1 : F (x) ̸= Φ
x−µ
)
σ ) ( x−µ σ
(∀x ∈ R) (∃x ∈ R)
µ and σ may be known or unknown
.
We consider J B ′ , J B, and Shapiro–Wilk SW tests. As H1 , contaminated normal distributions are considered: F = (1 − p)N (µ0 , σ02 ) + pN (µ, σ 2 ) . asymmetric CN covers a broad range of distributions, symmetric and ones. Nakagawa, Hashiguchi, and Niki ()
A Power Comparison for Testing Normality
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Power study 2 i.i.d.
H0 :X1 , X2 , . . . , Xn ∼ N (0, 1) i.i.d.
H1 :X1 , X2 , . . . , Xn ∼ (1 − p)N (0, 1) + pN (µ, σ 2 ) Ex. 1 2 3 4 Ex. 5 6 Ex. 7
n 100 100 200 50 n 50 50 n 50
µ 0 0 0 3 σ 4 1 µ 0
p σ 0.10 1, 2, 3, 4, 6 Sym. 0.50 1, 2, 3, 4, 6 Sym. 0.80 1, 2, 3, 4, 6 Sym. 0.50 1, 2, 3, 4, 6 Asym. p µ 0.05 0, 1, 2, 3, 4 Asym. 0.50 0, 1, 2, 3, 4 Sym. σ p 4 0, 0.1, . . . , 1 Sym.
significance level: α = 0.1 the number of replications: 104 Omnibus test statistics: J B, J B ′ , and SW Nakagawa, Hashiguchi, and Niki ()
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Ex.1: n = 100, µ = 0, p = 0.1, σ = 1, 2, 3, 4, 6 √
−4 −2
0
2
4
σ=2
0.3 0.2 0.1 0.0
0.1
0.2
0.3
0.4
6 19.33
0.4
1 3.00
0.0
0.0
0.1
0.2
0.3
0.4
σ β2
β1 = 0 2 3 4 4.44 8.33 12.72
−4 −2
0
2
4
σ=4
−4 −2
0
2
4
σ=6
curves in red: pdf of N (0, 1). curves in blue: pdf of F = (1 − p)N (0, 1) + pN (µ, σ 2 ). Nakagawa, Hashiguchi, and Niki ()
A Power Comparison for Testing Normality
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Ex. 1: Powers of J B ′ , J B, SW
1.0
n=100, p=0.1,µ=0.0
0.0
0.2
0.4
power
0.6
0.8
SW JBP JB
1
β2 = 3.0 Nakagawa, Hashiguchi, and Niki ()
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3
4
σ A Power Comparison for Testing Normality
6
−→heavy tails COMPSTAT2010, Paris
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Ex. 2: n = 100, µ = 0, p = 0.5, σ = 1, 2, 3, 4, 6 √
−6
−2 0
2
4
6
0.3 0.0
0.1
0.2
0.3 0.2 0.1 0.0
0.0
0.1
0.2
0.3
0.4
6 5.68
0.4
β1 = 0 2 3 4 4.08 4.92 5.34
1 3.00
0.4
σ β2
−6
σ=2
−2 0
2
4
6
σ=4
−6
−2 0
2
4
6
σ=6
curves in red: pdf of N (0, 1). curves in blue: pdf of F = (1 − p)N (0, 1) + pN (µ, σ 2 ). Nakagawa, Hashiguchi, and Niki ()
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Ex. 2: Powers of J B ′ , J B, SW
1.0
n=100, p=0.5,µ=0.0
0.0
0.2
0.4
power
0.6
0.8
SW JBP JB
1
β2 = 3.0 Nakagawa, Hashiguchi, and Niki ()
2
3
4
σ A Power Comparison for Testing Normality
6
−→heavy tails COMPSTAT2010, Paris
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Ex. 3: n = 200, µ = 0, p = 0.8, σ = 1, 2, 3, 4, 6 √
−6
−2
2 4 6
0.3 0.0
0.1
0.2
0.3 0.2 0.1 0.0
0.0
0.1
0.2
0.3
0.4
6 3.70
0.4
β1 = 0 2 3 4 3.37 3.56 3.64
1 3.00
0.4
σ β2
−6
σ=2
−2
2 4 6
σ=4
−6
−2
2 4 6
σ=6
curves in red: pdf of N (0, 1). curves in blue: pdf of F = (1 − p)N (0, 1) + pN (µ, σ 2 ). Nakagawa, Hashiguchi, and Niki ()
A Power Comparison for Testing Normality
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Ex. 3: Powers of J B ′ , J B, SW
1.0
n=200, p=0.8,µ=0.0
0.0
0.2
0.4
power
0.6
0.8
SW JBP JB
1
β2 = 2.04 Nakagawa, Hashiguchi, and Niki ()
2
3
4
σ A Power Comparison for Testing Normality
6
−→heavy tails COMPSTAT2010, Paris
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Ex. 4: n = 50, µ = 3, p = 0.5, σ = 1, 2, 3, 4, 6 3 0.92 3.72
4 0.96 4.37
0
2
4
6
8
0.3 0.0
0.1
0.2
0.3 0.0
0.1
0.2
0.3 0.2 0.1 0.0 −4
−4
σ=2
6 0.83 5.11
0.4
2 0.65 2.85
0.4
1 0.00 2.04
0.4
σ √ β1 β2
0
2
4
6
8
σ=4
−4
0
2
4
6
8
σ=6
curves in red: pdf of N (0, 1). curves in blue: pdf of F = (1 − p)N (0, 1) + pN (µ, σ 2 ). Nakagawa, Hashiguchi, and Niki ()
A Power Comparison for Testing Normality
COMPSTAT2010, Paris
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Ex. 4: Powers of J B ′ , J B, SW
1.0
n=50, p=0.5,µ=3.0
0.0
0.2
0.4
power
0.6
0.8
SW JBP JB
√ β1 = 0 β2 = 2.04
1
Nakagawa, Hashiguchi, and Niki ()
2
3
4
σ A Power Comparison for Testing Normality
√ 6 β1 = 0.83 β2 = 5.11 COMPSTAT2010, Paris
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Ex. 5: n = 50, σ = 4, p = 0.05, µ = 0, 1, 2, 3, 4 1 0.90 14.12
2 1.71 15.75
3 2.35 17.65
−5
0
5
10
0.3 0.0
0.1
0.2
0.3 0.0
0.1
0.2
0.3 0.2 0.1 0.0 −10
−10
µ=0
4 2.84 19.24
0.4
0 0.00 13.47
0.4
µ β1 β2
0.4
√
−5
0
5
10
µ=2
−10
−5
0
5
10
µ=4
curves in red: pdf of N (0, 1). curves in blue: pdf of F = (1 − p)N (0, 1) + pN (µ, σ 2 ). Nakagawa, Hashiguchi, and Niki ()
A Power Comparison for Testing Normality
COMPSTAT2010, Paris
15 / 24
Ex. 5: Powers of J B ′ , J B, SW
1.0
n=50, p=0.05,σ=4.0
0.0
0.2
0.4
power
0.6
0.8
SW JBP JB
√ 0 β1 = 0 β2 = 13.47 Nakagawa, Hashiguchi, and Niki ()
1
2
3
µ
A Power Comparison for Testing Normality
√ 4 β1 = 2.84 β2 = 19.24 COMPSTAT2010, Paris
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Ex. 6 : n = 50, σ = 1.0, p = 0.5, µ = 0, 1, 2, 3, 4 √
−10
−5
0
5
10
0.3 0.0
0.1
0.2
0.3 0.2 0.1 0.0
0.0
0.1
0.2
0.3
0.4
4 1.72
0.4
β1 = 0 1 2 3 2.92 2.50 2.04
0 3.00
0.4
µ β2
−10
µ=0
−5
0
5
10
µ=2
−10
−5
0
5
10
µ=4
curve in red: pdf of N (0, 1). curves in blue: pdf of F = (1 − p)N (0, 1) + pN (µ, σ 2 ). Nakagawa, Hashiguchi, and Niki ()
A Power Comparison for Testing Normality
COMPSTAT2010, Paris
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Ex. 6: Powers of J B ′ , J B, SW
1.0
n=50, p=0.5,σ=1.0
0.0
0.2
0.4
power
0.6
0.8
SW JBP JB
0
β2 = 3.0 Nakagawa, Hashiguchi, and Niki ()
1
2
3
µ
A Power Comparison for Testing Normality
4
β2 = 1.72 COMPSTAT2010, Paris
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Ex. 7: n = 50, σ = 4.0, µ = 0, p = 0.0, 0.1, . . . , 1.0 √
−10
−5
0
5
10
0.3 0.0
0.1
0.2
0.3 0.2 0.1 0.0
0.0
0.1
0.2
0.3
0.4
1.00 3.00
0.4
β1 = 0 0.20 0.50 0.70 9.75 5.34 4.07
0.00 3.00
0.4
p β2
−10
p = 0.0
−5
0
5
10
p = 0.5
−10
−5
0
5
10
p = 1.0
curves in red: pdf of N (0, 1). curves in blue: pdf of F = (1 − p)N (0, 1) + pN (µ, σ 2 ). Nakagawa, Hashiguchi, and Niki ()
A Power Comparison for Testing Normality
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Ex. 7: Powers of J B ′ , J B, SW
1.0
n=50,µ=0.0,σ=4.0
0.0
0.2
0.4
power
0.6
0.8
SW JBP JB
0
β2 = 13.47 Nakagawa, Hashiguchi, and Niki ()
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
p
A Power Comparison for Testing Normality
0.9
1
β2 = 3.29 COMPSTAT2010, Paris
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Summary Ranking: Ex. JB J B′ SW 1 2
1 2 1 3 S HT
2 3 1 2 S HT
3 2 1 1 S HT
4 2 3 1 A HT∼ST
5 2 1 3 A HT
6 2 3 1 S ST
7 3 1 2 S HT
JB ′ test is the best for symmetric dist. with heavy tails. JB ′ test is superior to JB test except for dist. with short tails. The power of JB ′ and JB is poor for dist. with short tails.
1 2
S:Symmetric, A:Asymmetric HT:Heavy tails, ST:Short tails
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Normalizing tr. of the null dist. for JB ′ Normalizing tr. of the null dist. for JB ′ Under the null hypothesis H0 , let 8 A=6+ √ β1 (JB ′ )
[
2 + √ β1 (JB ′ )
√
4 1 + (√ )2 β1 (JB ′ )
] ,
and then, a transformed variate )1/3 √ ) ( ( 1 − (2/A) / 2/9A 1− 2 √ − 9A 1 + Z 2/(A − 4)
is asymptotically distributed as a standard normal distribution, where Z =
√
β1 (JB ′ )
=
JB ′ − µ′1 (JB ′ ) √ µ2 (JB ′ )
and
. √ √ √ ( ) 8 6 2750 6 1 33968 6 1 1 + − + O , 9 9 n1/2 n3/2 n5/2 n7/2
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Shapiro–Wilk test
SW =
(∑
( ))2 . ai X(n−i+1) − X(i) ) ∑n ( ¯ 2 i=1 Xi − X
[n/2] i=1
.
0
20
40
60
X(1) ≤ X(2) ≤ · · · ≤ X(n) are the ordered statistics.
80
Shapiro–Wilk test
0.95
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A Power Comparison for Testing Normality
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0.97
0.98
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Related works
In 1987, Jarque and Bera proposed the J B test. Jarque, C. M. and Bera, A. K.A test for normality of observations and regression residuals. Inter. Statist. Rev., 55(2), 163–172, 1987.
In 2007, Nakagawa et. al. proposed a modified version of the Jarque–Bera test J B ′ . Nakagawa, S. and Niki, N. and Hashiguchi, H. An Omnibus test for normality. Proc of the ninth Japan-China sympo. on statist., 191–194,2007.
.................................................................. In 2007, a power comparison was conducted through Monte Carlo simulation assuming the contaminated normal distributions as H1 . Thadewald, T. and Buning, H. Jarque-Bera test and its competitors for testing normality-A power comparison J. of Appl. Statist., 34(1), 87–105, 2007.
Nakagawa, Hashiguchi, and Niki ()
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