This document discusses and compares several statistical tests for assessing multivariate normality: Mardia's tests, Small's tests, Royston's W, Henze-Zirkler, and Wmin(5). It provides details on the mathematical formulas and null distributions of each test. It also describes a simulation study that applies these tests to different multivariate distributions varying the dimension (p) and sample size (n) to evaluate and compare the performance of the tests.
6. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Mardia (1)
Mardia (1970)
,
7. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Mardia (1)
Mardia (1970)
,
Mardia’s Skewness
n
1
b1,p = {(Xr − X ) S −1 (Xs − X )}3 ,
¯ ¯ 1 ≤ r, s ≤ n
n2
r ,s=1
H0 : X ∼ Np (µ, Σ)
n p(p + 1)(p + 2)
b1,p → χ2 ,
v v=
6 6
8. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Mardia (2)
Mardia’s Kurtosis
n
1
b2,p = {(Xj − X ) S −1 (Xj − X )}2
¯ ¯
n
j=1
H0 : X ∼ Np (µ, Σ)
b2,p → N(µ, σ 2 )
p(p + 2)(n − 1)
µ=
n+1
8p(p + 2)
σ2 =
n
9. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Small (1)
Small (1980) Q1 , Q2
Q1 Q2 Q3
10. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Small (1)
Small (1980) Q1 , Q2
Q1 Q2 Q3
Small’s Q1
−1
Q1 = y1 U1 y1
x1
y1 = δ1 sinh−1 ( )
λ1
x1 Johnson’s Su (1949)
p×1 , 3 ) , 1≤j ≤k ≤p , ρ
U1 = (ρjk jk
Xj Xk
11. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Small (2)
Small’s Q2
−1
Q2 = y2 U2 y2
x2 − ξI
y2 = γ2 I + δ2 sinh−1 ( )
λ2
x2 Johnson’s Su (1949)
p×1 , U2 = (ρ4 ) , 1 ≤ j ≤ k ≤ p , ρjk
jk
Xj Xk
12. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Small (2)
Small’s Q2
−1
Q2 = y2 U2 y2
x2 − ξI
y2 = γ2 I + δ2 sinh−1 ( )
λ2
x2 Johnson’s Su (1949)
p×1 , U2 = (ρ4 ) , 1 ≤ j ≤ k ≤ p , ρjk
jk
Xj Xk
Small’s Q3
Q3 = Q1 + Q2
13. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Small (3)
29 ≤ n ≤ 100 2≤p≤8 , H0 : X ∼ Np (µ, Σ)
Qi = yi Ui−1 yi → χ2 (p), i = 1, 2
14. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Small (3)
29 ≤ n ≤ 100 2≤p≤8 , H0 : X ∼ Np (µ, Σ)
Qi = yi Ui−1 yi → χ2 (p), i = 1, 2
, Small Q1 Q2 ,
H0 : X ∼ Np (µ, Σ)
Q3 = Q1 + Q2 → χ2 (2p)
15. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Royston’s W
Royston (1983) Shapiro-Wilk W
,
p
1 1
G= {Φ−1 [ Φ(−zj )]}
p 2
j=1
zj = f (Wj ) Wj , Φ(x)
H0 : X ∼ Np (µ, Σ)
H = eG → χ2
e
e = p/(1 + (p − 1)¯)
c
16. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Henze-Zirkler
Henze-Zirkler (1990) (consistent)
,
dβ (P, Q) = ˆ ˆ
|P(t) − Q(t)|2 ϕβ (t)dt
Rp
ˆ
P(t) ˆ
Q(t) X ,
1
1
1 2p+1 p+4
ϕβ (t) Np (0, β 2 Ip ) , β= √
2 4 n p+4
H0 : X ∼ Np (µ, Σ)
dβ (P, Q) → lognormal(µ, σ 2 )
17. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Wmin (5) (1)
Malkovich and Afifi (1973) Roy’s (1953) union
intersection principle (UIP) Shapiro-Wilk W
,
Wmin = minp W (c)
∀c∈R
c c=1
W (c) cX Shapiro-Wilk W
18. Mardia
Small
Royston’s W
Henze-Zirkler
Wmin (5)
Wmin (5) (1)
Malkovich and Afifi (1973) Roy’s (1953) union
intersection principle (UIP) Shapiro-Wilk W
,
Wmin = minp W (c)
∀c∈R
c c=1
W (c) cX Shapiro-Wilk W
Wang and Hwang (2009) W (c)
c ,
Wmin (5)
20. p = 2,3,4,5,10 n = 10,20,30,40,50,75,100 ,
10000 ,
MVN
1 Multivariate Normal Distribution
2 Multivariate T Distribution with df = 3
3 Multivariate T Distribution with df = 10
4 Multivariate Uniform Distribution
5 Khintchine (KHN) Distribution
6 Mixed Multivariate Normal Distribution
7 Multivariate Skew Normal Distribution (SNp (Ω, α = 2))
8 Multivariate Skew Normal Distribution (SNp (Ω, α = 4))
