On the power of the chi-square test for multidimensional normality

(1)

! " # $ % & ' ( ) * & " " & * # + , - . & ( % * & * #

/,+& #,(!,%,0&! ##1234#1565#

#

!"#$%!"&'%()"+*

7

4#,-./"&01&23456

77

#

**!"#$%"& '%(" )"#$%"*$+,-./0(%"#%-#""**

) ("1/2#+3+1%!-+ !02"! (102+#4"

#

05-#(0*#6""89#:;9#<=>9;?=:;@A#;:B>@A=CD#E8=F?GH@B9#I::<;9??F:JFJ=C#C9?C#=?#K9AA#L;:K;M#

N9B9# K9# E:;?=<9B# ?:>9# I9;9B@A=O@C=:;# :J# C8=?# C9?C# =;C:# >HAC=<=>9;?=:;@A# E@?9M# $?H@AAD4# C89# E:;?CBHEC=:;# :J# C89# C9?C# =?# P@?9<# :;# C89# E:;C=;I9;ED# C@PA9# =;# K8=E8# C89# @QQB:QB=@C9# QB:P@P=A=C=9?# <9@A?# K=C8# 9R9;C?# C8@C# >HAC=<=>9;?=:;@A# R@B=@PA9# C@L9?# C89# R@AH9# JB:># B9EC@;IHA@B# E9AA?M# &;# C8=?# Q@Q9B#C89#E9AA?#@B9#=;C9B?9EC=:;?#:J#?9C?#<9C9B>=;9<#PD#@QQB:QB=@C9#9AA=Q?:=<?#@;<#:BC8:I:;@A#QA@;9?M# "89#>@=;#QHBQ:?9#:J#C89#Q@Q9B#=?#E:>Q@B=?:;#:J#C89#C9?C#Q:K9B?#9R@AH@C9<#H;<9B#C89#CK:#?D?C9>?# :J#E9AA?M#"89#Q:K9B#=?#@;@ADO9<#:;#C89#P@?=?#:J#E:>QHC9B#?=>HA@C=:;M## 789":;<=>?#>HAC=R@B=@C9#;:B>@A#C9?C4#I::<;9??F:JFJ=C#C9?C4#0:;C9#!@BA:M#

#

**+6"+!#( 3/*#+ !#**

#

,HB# QHBQ:?9# =?# C9?C=;I# C89# 8DQ:C89?=?#

7

₅

S T

@

8 U 4 V

A B # K89B9#

W

4MMM4

4 X

9

₆

9

₁

9

₊

@

:

A

;

U V4

@ &

B @

U V

;

U

@ A @ A

!

VU

!

V M

<

# &C# =?# K9AA#

L;:K;#C8@C#C89B9#9Y=?C?#?HE8#@JJ=;9#CB@;?J:B>@C=:;#

E

D@

!

C

#C8@C#

;

U

E

V

F

4#

G

+

E

B

U

V

#=J#

<9CU

B

V

"

5 M#*:4#C89#8DQ:C89?=?#N

5

#=?#9GH=R@A9;C#C:#C89#J:AA:K=;I#

:;9#

7

₅

S

E

T

8 U

F

4 +

_G

V

M##

+9C#H?#<9J=;9#C89#J:AA:K=;I#?9C?#=;#=

+

#U?99#"@AA=?#ZM#0M46[\2#@;<#"@AA=?#ZM#0M#

6[\3V#

# C89# B=;I#

?

_>

^

H

&

=

+

S

*

_>_'₆

%

"

<

"

$

*

_>

]

& & K89B9&

>

6

4

1

4 MMM4

!

@;<#

(

$

*

_!

*

MMM

5

₅ ₆

&

# C89#?9C#

6 1MMM

54

6 S

64 14MMM4

M

54

5

+ 5 5 + @ @ @ 5 5

"

@

A

=

5 +

"

@

"

#

$

%

$

_&

$

'

(

$

_$

)

$

*

$

*

+

H

#

# C89#?9C#

C

)

>

4 B

*

+

H

&

=

+

S

"

_>₄_B_'₆

$

"

_>

%

"

_>₄_B

4 "

_@

&

=

4 @

"

>

4 @

6

4

1 4MMM4

+

,

4#

>#_#64#14#MMM4+`#B#_#64#14MMM4##

>

4#

'

(

"

>45

$

"

>46

$

MMM

$

"

>4#_>

(

M#

#################################################

7 #a8M-M4#-9Q@BC>9;C#:J#*C@C=?C=E?4#b@C:K=E9#$;=R9B?=CD#:J#(E:;:>=E?M# 77 #aB:J9??:B4#-9Q@BC>9;C#:J#*C@C=?C=E?4#b@C:K=E9#$;=R9B?=CD#:J#(E:;:>=E?M#

(2)

ZBO9I:BO#b:cEO@L4#d@;H?O#+M#eDK=@f#

gg

"89# ?9C#

,

₆ ₁

-

,

-6

4 4MMM4

4

+ + > >

C B B

B

C > B

# K89B9#

B

_>

&

^64 14MMM4 ]

#

_>

# J:B# ># _# 64# 14#

h4# +4# =?# @#+F<=>9;?=:;@A# B9EC@;IHA@BM# +9C# H?# =;CB:<HE9# J:AA:K=;I# ;:C@C=:;?# J:B#

?:>9#QB:P@P=A=C=9?S#

)

U

4 V

*

4

V 4 U

D

C

>

B

E

C >B

H

&

#

)

&

*

-+ > B > C + B B B C

D

C

B

E

+ 6 V 4 U 1 6 V 4MMM4 4 U 6 1

H

U

4 4MMM4

V

4#

)

₊

*

+ @ @ @ > A > @@ @ ?

D

?

A

E

_MMM 1 6 MMM 1 6

&

.

H

4#K89B9#

?

>

.

A

@6@1MMM@+

#=?#@#Q@BC#:J#C89#B=;I#M#

"89# ?@>QA9# <B@K;# JB:># C89# <=?CB=PHC=:;# :J# C89# R9EC:B# @# =?# <9;:C9<# PD#

@FX9

5B

W:& 5&F&G:&H:I:-J& BFG:I:+M# # R@AH9# :J# C89# B@;<:># R@B=@PA9# 9

5B

# =?# @;# 5FC8#

:P?9BR@C=:;# :J# C89# BFC8# B@;<:># R@B=@PA9# 9

B

M# "89# ?@>QA9# R9EC:B# @;<# C89# ?@>QA9#

R@B=@;E9# @B9# <9;:C9<# PD#

@

X

9

₆

4

9

₁

4

111

4

9

_L

W

# @;<#

B

X

K

_=i

W

4# B9?Q9EC=R9AD4#

K89B9##

)

*

)

_5/ _/

*

-5 B 5B

9

9 -K

'

/

6 i?

6

6 #K89B9#B:&/#_#64#14#h4#+M#

"89B9#9Y=?C?#?HE8#@;#:BC8:I:;@A#>@CB=Y#D C8@C#DD

<

F+

+

4#

D

B

D

3

<

#K89B9#

+

#=?#C89#H;=C#>@CB=Y#:J#<9IB99#+M#+9C#

M

1 j 6 '

D3

2 #*:4#

2

<

B

2 +

₊

M

#"89#>@CB=Y#

@#E@;#P9#CB@;?J:B>9<#@?#J:AA:K?S##

#

4I1@4#

_- _- <

--

J

+

1 '

6 #

U6V#

#

E

**1@*3**

'6j1

4#E

<

EI+

+

#

U1V#

#

K89B9# J

;

# =?# C89# H;=C# E:AH>;# K8=E8# @AA# -F9A9>9;C?# @B9# 9GH@A# C:# :;9M# N9;E94# C89#

E:AH>;?#:J#C89#>@CB=Y# E#@B9#;:C#E:BB9A@C9<M#+9C# E

5L

#P9#5FC8#B:K#:J#C89#>@CB=Y# E#

@;<#H

5L&

#P9#C89#R@AH9#:J#E

5L1

#

**++6"#%-#"-#0#+-#+*-"**

#

"B@;?J:B>@C=:;?#U6V#@;<#U1V#A9@<#C:#C89#9R@AH@C=;I#C89#J:AA:K=;I#C9?C#?C@C=?C=E#

:J#C89#8DQ:C89?=?#N

5

M##

#

)

) * ) *

*

) * ) * +

/

,

'

+ + + + # # C C B B # # B B C # # C C

E

2 M

4MMM4 4 4MMM4 1 4MMM4 4 4MMM4 6 6 1 1 6 6

#

U2V##

(3)

g[#

#

,

-.

/

MMM 665MMM56 6 1 MMM 6 1 MMM 6 1 1 ?> A > @ @ @+ > @ @ @+ > @ @ @+ ? A ?A ? A ? A

2 E

M

E

0 0 0 0

!

1 #

UkV##

#

K89B9###

#

_, _-

.

,

-

/

6 7 7 6 1 4MMM4

S

4 4MMM4

+ 5 5 + C # #

C!N "*

"

C B B

#

2 -&

##

#

@;<##

#

.

6 1

/

665MMM56 7

S

7 MMM

M

+ ?> A 5 5 > @ @ @

C!N*

?

A

2

-0

&

0 H

H

#

a@BC=EHA@BAD4#=J# +#_#14#!# _#3#@;<##

6

#_##

1#

_#k#C89#?D?C9>?#:J#E9AA?#J:B#C89#C9?C#

?C@C=?C=E#M

C

&@;<#M

?A

&@B9#?8:K9<#PD#/=IHB9#6#@;<#/=IHB9#14#B9?Q9EC=R9ADM#

#

/=IHB9#6M#"89#E9AA?#J:B#C89#?C@C=?C=E#MC# V 1 4 1 U V 6 4 6 U C C . V 2 4 1 U V 6 4 6 U C C . V k 4 1 U V 6 4 6 U C C . V 6 4 1 U V k 4 6 U C C . V 6 4 1 U V 2 4 6 U C C . V 6 4 1 U V 6 4 6 U C C . CU641V.CU146V V 2 4 1 U V 1 4 6 U C C . V k 4 1 U V 1 4 6 U C C . V 1 4 1 U V k 4 6 U C C . V 1 4 1 U V 2 4 6 U C C . V 1 4 1 U V 1 4 6 U C C . V 2 4 1 U V k 4 6 U C C . V 2 4 1 U V 2 4 6 U C C . V k 4 1 U V 2 4 6 U C C . CU64kV.CU14kV 6 O 1 O 5

(4)

[5

#

/=IHB9#1M#"89#E9AA?#J:B#C89#?C@C=?C=E#M?A#

#

+9C#

6 1MMM

^

]

+ ?A + @ @ @

N

C!N ?*

0 A

#

@;<#

N

_C

C!N C > >*

^ U 4

₆ ₁

4MMM4

>

₊

V]

#

P9#

@QQB:QB=@C9#;H>P9B?#:J#E9AA?M#e89;#C89#8DQ:C89?=?#N

5

#=?#CBH9#@;<#?@>QA9#?=O9#=?#

?HJJ=E=9;CAD#A@BI9#C89;#C9?C#?C@C=?C=E?# M

C

#@;<#M

?A

#8@R9#@?D>QC:C=E@AAD#E8=F?GH@B9#

<=?CB=PHC=:;# K=C8# N

?A

& P& /& P6# @;<# N

C

& P& /& P& 6& <9IB99?# :J# JB99<:># B9?Q9EC=R9AD4#

K89B9#/#=?#C89#;H>P9B#:J#9?C=>@C9<#Q@B@>9C9B?M"

"

+++6"-#/34" )"#$%"#%-#"& '%("

"

**38>K<LMNL;O";P"<QO=;A"RQ<LQST8>"LO"1;ON8"*Q<T;">NU=9"**

"89# ?=O9# @;<# C89# Q:K9B# :J# @;@ADO9<# C9?C?# K@?# 9?C=>@C9<# =;# 0:;C9# !@BA:#

?CH<DM#"89#E:>QHC9B#?=>HA@C=:;?#K9B9#QB9Q@B9<#H?=;I#)#!B@;#?:JCK@B9#UKKKMBF

QB:i9ECM:BIVM#"89B9#@B9#QB9?9;C9<#C89#B9?HAC?#=;#C89#Q@Q9B#J:B#C89#CK:#<=>9;?=:;@A#

E@?9M# "89# ?=O9# :J# C9?C?# K@?# :PC@=;9<# J:B# I9;9B@C9<# CK:F<=>9;?=:;@A# ;:B>@A#

B@;<:>#R@B=@PA9#

@

T

8 U 4 V4

A B #K89B9#

A

U545V

#@;<#

65 g

M

g

65

2

3

4

5

4

5

6

7 B

#"89#Q:K9B#

:J# C89# <9?EB=P9<# C9?C?# K9B9# @;@ADO9<# =;# 0:;C9# !@BA:# ?CH<D# J:B# J:AA:K=;I# CK:#

<=>9;?=:;@A#<=?CB=PHC=:;?S#

5 6 66 ? A . 1 66 ? A . 2 66 ? A . k 66 ? A . 6 56 ? A . 1 56 ? A . 2 56 ? A . k 56 ? A . 6 55 ? A . 1 55 ? A . 2 55 ? A . k 55 ? A . 6 65 ? A . 1 65 ? A . 2 65 ? A . k 65 ? A . 3 66 ? A . 3 65 ? A . 3 55 ? A . 3 56 ? A . 6 Q 1 Q

(5)

[6#

# +%# l# +:IF;:B>@A# <=?CB=PHC=:;# K89B9# P:C8# :J# CK:# R@B=@PA9?# 8@R9# A:IF

;:B>@A#<=?CB=PHC=:;#K=C8#Q@B@>9C9B?#5#@;<#6M#

# mU64# \V4# # mU14# 1V# l# P9C@# <=?CB=PHC=:;# K=C8# C89# ?8@Q9# @;<# C89# ?E@A9#

Q@B@>9C9B?##

# ZU6V4#ZUk5V#l#I@>>@#<=?CB=PHC=:;#K=C8#C89#?8@Q9#Q@B@>9C9BM#

# $#l#H;=J:B>#<=?CB=PHC=:;#:;#C89#CK:F<=>9;?=:;@A#?Q89B9M##

"89# ?D>P:A=E# ;:C@C=:;4# P@?9# <9?EB=QC=:;# @;<# C89# @?D>>9CBD# UC8=B<# E9;CB@A#

>:>9;CV#:J#C89?9#B@;<:>#R@B=@PA9?#@B9#QB9?9;C9<#=;#"@PA9#6M#

#

"@PA9#6M#"89#B@;<:>#R@B=@PA9?#<9C@=A?#=;#0:;C9#!@BA:#?CH<D# -=?CB=PHC=:;# *D>P:A=E# ;:C@C=:;#:J#C89# B@;<:>#R@B=@PA9# -9?EB=QC=:;## :J#@ U9 9₆4 ₁V# ?D>>9CBD# %:B>@A# %UAVBV# U545V A # 65 g g 65 2 3 4 5 4 5 6 7 B # 2 5 # +:IF;:B>@A# +%# 9 5

#

9

4#5#_64#1M# nT%U54#6V# 2 k 8 # ZU6V# 9=#8@?#I@>>@#<=?CB=PHC=:;#U5_64#1V4# K=C8#C89#?8@Q9#Q@B@>9C9B#9GH@A#C:# 6M# 281 # Z@>>@# ZUk5V# 9=#8@?#I@>>@#<=?CB=PHC=:;#U5_64#1V4# K=C8#C89#?8@Q9#Q@B@>9C9B#9GH@A#C:# k5M# 285Mk # mU64#\V# 9=#8@?#P9C@#<=?CB=PHC=:;#U5_64#1V4#K=C8# C89#?8@Q9#Q@B@>9C9B?#9GH@A#C:#6# @;<#\M# 286Mk # m9C@# mU14#1V# 9=#8@?#P9C@#<=?CB=PHC=:;#U5_64#1V#K=C8# C89#?8@Q9#Q@B@>9C9B?#9GH@A#C:#1# @;<#1M# 2 5 # $;=J:B># $# "89#<9;?=CD#JH;EC=:;# 1 1 1 1 6 6 U 4 V 5 6 R 3 S R 3 R 3 " $ ₉ ₎ $ $ ' $ ₉ _% $ $ * # 2 #5

#

"89# 0:;C9# !@BA:# ?CH<D# K@?# QB9Q@B9<# J:B# ># _# 24# k4# 3# @;<# \# O:;9?# J:B# 9@E8#

R@B=@PA9#=;#C89#B9EC@;IA9?#E9AA?#E@?9#@;<#J:B#>&_#14#k4#\#@;<#g#B=;I?#=;#C89#Q@BC?#:J#

B=;I?#E9AA?#E@?9M#"89#CK:#<=>9;?=:;@A#E@?9#K@?#@;@ADO9<M#&C#A9@<?#C:#[4#6\4#13#@;<#

2\# E9AA?# =;# C89# J=B?C# E@?9# @;<# g4# 6\4# 1k# @;<# 21# E9AA?# =;# C89# ?9E:;<# E@?9M# "89#

?@>QA9# ?=O9# K@?# k34# g54# 615# @;<# 155M# "89# E9AA?# K9B9# E:;?CBHEC9<# ?HE8# C89#

9YQ9EC9<# ;H>P9B?# K9B9# 9GH@A# J:B# 9@E8# E9AAM# &C# K@?# C@L9;# =;C:# @EE:H;C# C8@C# C89#

>=;=>H># 9YQ9EC9<# ;H>P9B?# J:B# 9@E8# E9AA# ?8:HA<# P9# EM@M# 3# :P?9BR@C=:;?M# "89#

(6)

[1

<9?EB=QC=:;#:J#?@>QA9#?=O9?#J:B#9@E8#E:;?=<9B9<#;H>P9B#:J#E9AA?#=;#CK:#@;@ADO9<#

E@?9?#@B9#QB9?9;C9<#=;#C@PA9#1M#

#

"@PA9#1M#*=O9?#:J#?@>QA9?#=;#0:;C9#!@BA:#?CH<D#J:B#B9EC@;IA9?#E9AA?#@;<#Q@BC?#:J#B=;I?#E9AA?# )9EC@;IA9?#E9AA?# a@BC?#:J#B=;I?#E9AA?#

%H>P9B#:J#E9AA?# *@>QA9#?=O9?# %H>P9B#:J#E9AA?# *@>QA9#?=O9?#

[# k34#g54#6154#155# g# k34#g54#6154#155# 6\# g54#6154#155# 6\# g54#6154#155# 13# 6154#155# 1k# 6154#155# 2\# 155# 21# 155#

#

**#W8"<8>UTN>";P"1;ON8"*Q<T;">NU=9"**

"89#0:;C9#!@BA:#?CH<D#K@?#QB9Q@B9<#J:B#B@;<:>#R@B=@PA9?#<9?EB=P9<#=;#C@PA9#

6#@;<#J:B#?@>QA9#?=O9?#@;<#;H>P9B#:J#E9AA?#QB9?9;C9<#=;#C@PA9#1M##

6M "89#?@>QA9#:J#?=O9# -#U?99# C@PA9#1V#JB:>#<=?CB=PHC=:;# T#U?99#C@PA9# 6V#K@?#

I9;9B@C9<M#

1M "89# R@AH9?# :J# C89# ?C@C=?C=E?# M

m

# @;<# M

!

# K@?# E@AEHA@C9<# @;<# K9B9#

E:>Q@B9<#C:#@QQB:QB=@C9#EB=C=E@A#R@AH9?M#

2M C9Q?#6l1#K9B9#B9Q9@C9<#8#_#65#555#C=>9?M#*

kM "89#9>Q=B=E@A#QB:P@P=A=C=9?#:J#B9i9EC=:;#7

5

#J:B#C89#Q@BC?#:J#B=;I?#UM

m

V#@;<#

C89#B9EC@;IA9#E9AA?#UM

!

V#K9B9#E@AEHA@C9<M#

"89B9#K@?#@??H>9<#C89#?=I;=J=E@;E9#A9R9A#@#_#5M53#=;#C89#0:;C9#!@BA:#?CH<DM#

"89#9?C=>@C9<#QB:P@P=A=C=9?#:J#B9i9EC=:;#:J#C89#;HAA#8DQ:C89?=?#=;#C89#>HAC=R@B=@C9#

I::<;9??F:JFJ=C#C9?C?#K@?#C@PA9<#J:B#9@E8#E@?9#U?99#C@PA9?#2l\VM##

#

"@PA9#2M##"89#9?C=>@C9<#QB:P@P=A=C=9?#:J#B9i9EC=:;#:J#75#J:B#B9EC@;IA9?#E9AA?#U[#E9AA?V## @;<#Q@BC?#:J#B=;I?#E9AA?#Ug#E9AA?V# )9EC@;IA9?# 2#O:;9?#l#[#E9AA?# "89#Q@BC?#:J#B=;I?# 1#B=;I?#l#g#E9AA?# -=?CB=PHC=:;## :J#B@;<:>#R@B=@PA9# -#_#k3# -#_#g5# -#_#615 -#_#155 -#_#k3# -#_#g5# -&_#615# -#_#155#

%UAVBV# 5M6ogo# 5M6\ok# 5M6o[g# 5M6ooo# 5M12g\# 5M1213# 5M11g2# 5M113[# +%# 6M5555# 6M5555# 6M5555# 6M5555# 5M[[g[# 6M5555# 6M5555# 6M5555# ZU6V# 5M[o53# 5M[[[6# 6M5555# 6M5555# 5M[o[5# 6M5555# 6M5555# 6M5555# ZUk5V# 5M1512# 5M1153# 5M12\[# 5M1g\g# 5M1o15# 5M1ogo# 5M1[2g# 5M2231# mU64\V# 5Mgog5# 5M[g2o# 6M5555# 6M5555# 5Mgogg# 5M[o1[# 6M5555# 6M5555# mU141V# 5M166o# 5M131# 5M1og3# 5M2\\\# 5M2135# 5Mk133# 5M3k52# 5Mok36# $# 5M221\# 5Mkg21# 5M\kg\# 5Mg\2[# 5M\323# 5Mg[5k# 5M[ogk# 5M[[[\#

*:HBE9S#0:;C9#!@BA:#?CH<DM# #

(7)

[2#

"@PA9#kM#"89#9?C=>@C9<#QB:P@P=A=C=9?#:J#B9i9EC=:;#:J#75#J:B#B9EC@;IA9?#E9AA?#U6\#E9AA?V## @;<#Q@BC?#:J#B=;I?#E9AA?#U6\#E9AA?V# )9EC@;IA9?#k#O:;9?#l#6\#E9AA?# "89#Q@BC?#:J#B=;I?#k#B=;I?#l#6\#E9AA?# -=?CB=PHC=:;## :J#B@;<:>#R@B=@PA9# _{-#_#g5#} _{-#_#615#} _{-#_#155#} _{-#_#g5#} _{-&_#615#} _{-#_#155#}

%UAVBV# 5M5[k5# 5M5[1g# 5M5go[# 5M65\1# 5M656o# 5M6556#

+%# 6M5555# 6M5555# 6M5555# 6M5555# 6M5555# 6M5555#

ZU6V# 5M[[[6# 6M5555# 6M5555# 6M5555# 6M5555# 6M5555#

ZUk5V# 5M61o1# 5M6kk1# 5M6ok[# 5M62o1# 5M6\1g# 5M15g3#

mU64\V# 5M[gok# 6M5555# 6M5555# 5M[[\6# 6M5555# 6M5555#

mU141V# 5M6[13# 5M1k23# 5M2\51# 5M1565# 5M1\o2# 5Mk56g#

$# 5M2kk[# 5M31[3# 5Mg1k[# 5Mg5[o# 5M[23o# 5M[[o6#

*:HBE9S#0:;C9#!@BA:#?CH<DM# # # "@PA9#3M#"89#9?C=>@C9<#QB:P@P=A=C=9?#:J#B9i9EC=:;#:J#75#J:B#B9EC@;IA9?#E9AA?#U13#E9AA?V## @;<#Q@BC?#:J#B=;I?#E9AA?#U1k#E9AA?V# )9EC@;IA9?#3#O:;9?#l#13#E9AA?# "89#Q@BC?#:J#B=;I?#\#B=;I?#l#1k#E9AA?# -=?CB=PHC=:;## :J#B@;<:>#R@B=@PA9# -#_#615# -#_#155# -#_#615# -#_#155#

%UAVBV# 5M5o63# 5M5o3\# 5M5g12# 5M5gk6#

+%# 6M5555# 6M5555# 6M5555# 6M5555#

ZU6V# 6M5555# 6M5555# 6M5555# 6M5555#

ZUk5V# 5M655[# 5M62g2# 5M61[5# 5M6ooo#

mU64\V# 6M5555# 6M5555# 6M5555# 6M5555#

mU141V# 5M11[6# 5M2gk3# 5M1g[\# 5Mk363#

$# 5Mkog[# 5Mo[1\# 5M[oo5# 6M5555#

# *:HBE9S#0:;C9#!@BA:#?CH<DM# # # "@PA9#\M#"89#9?C=>@C9<#QB:P@P=A=C=9?#:J#B9i9EC=:;#:J#75#J:B#B9EC@;IA9?#E9AA?#U2\#E9AA?V## @;<#Q@BC?#:J#B=;I?#E9AA?#U21#E9AA?V# )9EC@;IA9?#\#O:;9?#l#2\#E9AA?# "89#Q@BC?#:J#B=;I?#g#B=;I?#l#21#E9AA?# -=?CB=PHC=:;## :J#B@;<:>#R@B=@PA9# -#_#155# -#_#155# %UAVBV# 5M5\65# 5M5o3k# +%# 6M5555# 6M5555# ZU6V# 6M5555# 6M5555# ZUk5V# 5M612g# 5M6ko\# mU64\V# 6M5555# 6M5555# mU141V# 5M2g\\# 5Mk[g6# $# 5Moog5# 5M[[[[# # *:HBE9S#0:;C9#!@BA:#?CH<DM#

(8)

[k

"89#0:;C9#!@BA:#?CH<=9?#8@R9#?8:K;#C8@C#C89#C9?C#=;#K8=E8#C89#Q@BC?#:J#B=;I?#

E9AA?# @B9# H?9<# =?# >:?C# Q:K9BJHAM# &C# E@;# P9# ?99;# =;# $;=J:B>4# m9C@# U141V# @;<#

Z@>>@#Uk5V#<=?CB=PHC=:;#E@?9?#U?99#C@PA9#2l\VM#"89#B9?HAC?#:J#0:;C9#!@BA:#?CH<D#

@B9#QB9?9;C9<#=;#C89#J=IM#2#@;<#kM#

#

/=IHB9#2M#"89#9?C=>@C9<#QB:P@P=A=C=9?#J:B#B9i9EC=:;#:J#75#J:B#C89#B9EC@;IA9#E9AA?#U[#E9AA?V# @;<#C89#Q@BC?#:J#B=;I?#E9AA?#Ug#E9AA?VM# *:HBE9S#"@PA9#2M#

"

#W8"K;AMQ<L>;O";P"KWL,>XUQ<8"C;;=O8>>,;P,PLN"N8>N>"QO="-WQML<;,'LTGY>"

O;<AQTLN9"N8>N"

"89#Q:K9B#:J#@;@ADO9<#;:B>@A=CD#C9?C?#P@?9<#:;#C89#E8=F?GH@B9#?C@C=?C=E#K@?#

E:>Q@B9<# C:# C89# Q:K9B# :J# C89# >HAC=R@B=@C9# 9YC9;?=:;# U):D?C:;# aM4# 6[g14#

-:>@c?L=# !OM4# 6[[gV# K9AA# L;:K;# *8@Q=B:Fe=ALp?# ;:B>@A=CD# C9?CM# "89# E@?9# :J#

(9)

[3#

?@>QA9# :J# ?=O9# -# _# 155# K9B9# @;@ADO9<# =;# 0:;C9# !@BA:# ?CH<DM# "89# 9?C=>@C9<#

QB:P@P=A=C=9?#:J# B9i9EC=:;# :J# C89# ;HAA#8DQ:C89?=?# J:B# C89?9# C9?C?# K9B9# :PC@=;9<# =;#

8#_#65#555#?=>HA@C=:;?M#"89#B9?HAC?#@B9#QB9?9;C9<#=;#C@PA9#oM#

#

/=IHB9#kM#"89#9?C=>@C9<#QB:P@P=A=C=9?#:J#B9i9EC=:;#:J#75#J:B#C89#B9EC@;IA9#E9AA?#UHQQ9B#Q@BCV# @;<#C89#Q@BC#:J#B=;I?#UP:CC:>#Q@BCV#E9AA?# *:HBE9S#"@PA9#2l\M#

(10)

[\

"@PA9#oM#"89#9?C=>@C9<#QB:P@P=A=C=9?#:J#B9i9EC=:;#:J#75#J:B#B9EC@;IA9?#E9AA?#U13#E9AA?V4## Q@BC?#:J#B=;I?#E9AA?#U1k#E9AA?V#@;<#>HAC=R@B=@C9#*8@Q=B:Fe=ALp?#C9?C#U-&_#155V# -=?CB=PHC=:;## :J#B@;<:>#R@B=@PA9# )9EC@;IA9?# \#O:;9?#l#2\#E9AA?# "89#Q@BC?#:J#B=;I?# g#B=;I?#l#21#E9AA?# *8@Q=B:Fe=ALp?#C9?C# %UAVBV# 5M5\65# 5M5o3k# 5M5[\[# +%# 6M5555# 6M5555# 6M5555# ZU6V# 6M5555# 6M5555# 6M5555# ZUk5V# 5M612g# 5M6ko\# 5Mk63k# mU64\V# 6M5555# 6M5555# 6M5555# mU141V# 5M2g\\# 5Mk[g6# 5M5\1[# $# 5Moog5# 5M[[[[# 5M[325# # *:HBE9S#0:;C9#!@BA:#?CH<DM#

#

"89#B9?HAC?#:J#E:>Q@B=?:;#C89#Q:K9B#:J#@;@ADO9<#E8=F?GH@B9#I::<;9??F:JFJ=C#

C9?C?#@;<#>HAC=R@B=@C9#9YC9;?=:;#:J#*8@Q=B:Fe=ALp?#C9?C#@B9#QB9?9;C9<#=;#C89#J=IHB9#oM#

! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # ,-."/0 1, 2-#0 2-&!0 3-#"(0 3-$"$0 4 56789:;<6= >9?8= @A ?B:;= CD9EB?@FGB<HI= # /=IHB9#3M#"89#9?C=>@C9<#QB:P@P=A=C=9?#:J#B9i9EC=:;#:J#75#J:B#C89#B9EC@;IA9#E9AA?4#C89#Q@BC#:J#B=;I?# E9AA?#@;<#J:B#C89#*8@Q=B:Fe=ALp?#C9?CM# # *:HBE9S#C@PA9#o# # #

e9#E@;#;:C=E9#C8@C#J:B#@;@ADO9<#?@>QA9?#C89#9>Q=B=E@A#Q:K9B#:J#C89#*8@Q=B:F

e=AL# C9?C# =?# 8=I89B# C8@;# C89# Q:K9B# :J# C89# E8=F?GH@B9# I::<;9??F:JFJ=C# C9?C?# J:B#

I@>>@#<=?CB=PHC=:;#@;<#J:B#H;=J:B>#<=?CB=PHC=:;M#&;#C89#E@?9#:J#P9C@#<=?CB=PHC=:;#

C89#Q:K9B#=?#8=I89B#=;#C89#E8=F?GH@B9#I::<;9??F:JFJ=C#C9?C#E@?9#UC@PA9?#oVM#

(11)

[o#

**+B6"* !*2/3+!D"(%10(7-#**

#

"89# I9;9B@A=O@C=:;# :J# C89# E8=F?GH@B9# I::<;9??# :J# J=C# C9?C# =;C:# >HAC=R@B=@C9#

E@?9# K@?# E:;?=<9B9<M# "K:# ?9C?# :J# E9AA?# =;# C89# >HAC=R@B=@C9# E@?9# K9B9# @;@ADO9<M#

"89# ?9C?# :J# B9EC@;IA9?# E9AA?# @;<# C89# ?9C# :J# C89# Q@BC?# :J# C89# B=;I?# E9AA?# K9B9#

E:;?=<9B9<M#"89?9#CK:#E@?9?#K9B9#E:>Q@B9<#H?=;I#?=>HA@C=:;#?CH<DM#"89#Q:K9B#

:J# C89# E8=F?GH@B9# I::<;9??F:JFJ=C# C9?C# =;# C89?9# CK:# E@?9?# K9B9# @;@ADO9<M# "89#

?=>HA@C=:;# ?CH<D# 8@R9# ?8:K;# C8@C# C89# E8=F?GH@B9# I::<;9??# :J# J=C# C9?C# =?# >:?C#

Q:K9BJHA#K89;#C89#Q@BC?#:J#B=;I?#E9AA?#@B9#H?9<M##

#

(%)%(%!*%-" "

-:>@c?L=# !OM# U6[[gV4# ef@?;:qE=# C9?CH# K=9A:KD>=@B:K9i# ;:B>@A;:qE=# *8@Q=B:Fe=AL@# =#i9I:# O@?C:?:K@;=9M#C!*)%4&M-5U#!/5@3&%S&;)%-%V5)/&=#)@%!W/&0#)@.!#/4#%:M#2oM#

):D?C:;#aM#U6[g1V4# AI:B=C8># *#6g6S#<>#&2&<#/@&S%!&8%!V*X5@3M#?EEX5#N&Y@*@5/@5)/4#264#6o\l6g5M#

)HP=;?C9=;# )M# rM4# bB:9?9# -M# aM# U155oV#Y5V.X*@5%-& *-N& @>#& Z%-@#& C*!X%& Z#@>%N4# e=A9D# l#

&;C9B?E=9;E9M#%9K#d9B?9DM#

"@AA=?# ZM# 0M# U6[\2V4# ;XX5E@5)*X& *-N& !*N5*X& @!.-)*@5%-& 5-& -%!V*X& E%E.X*@5%-M# ?--*X/& %S&Z*@>#V*@5)*X&Y@*@5/@5)/#R:AM#2k4#QQM##[k5l[kkM#

"@AA=?# ZM# 0M# U6[\3V4#DX*-#& @!.-)*@5%-& 5-& -%!V*X& E%E.X*@5%-M# ,%.!-*X& %S& @>#&=%3*X& Y@*@5/@5)*X& Y%)5#@3#*9B=9?##m4#R:AM##1o#QQM#256l25oM# # # # !"#$%!"&'%()"*+:&,*-./"&01&2345*6& "

"1 *4"#%-#/"*$+,7'03(0#"320"$+& #%E4" "! (102! Z*+"( E7[03/" '+%2 '41+0( '%D "

#

,IsA;=9# O;@;D# C9?C# OI:<;:qE=# E8=FLK@<B@C# i9?C# KDL:BOD?C@;D# <:# K9BDJ=L@Ei=# 8=Q:C9OD# :#;:B>@A;:qE=# B:OLf@<H# QB@K<:Q:<:P=9c?CK@# O>=9;;9i# A:?:K9i# K=9A:KD>=@B:K9iM# %@iEOtqE=9i# E9A9# C9?CH# L:;?CBHHi9# ?=t# K# L?OC@fE=9# QB:?C:LuCsKM# e# @BCDLHA9# B:OK@v:;:# 9A=Q?:=<D4# LCsBDE8# K?QsA;D# qB:<9L# >@# K?QsfBOt<;9# KDO;@EO:;9# QBO9O# :E9;D# O# QBsPD# K@BC:qE=# qB9<;=E8# O>=9;;DE8# A:?:KDE8M# ;@A=Ot# >:ED# C9?CH# QBO9QB:K@<O:;:# O# KDL:BOD?C@;=9># ?D>HA@Ei=# L:>QHC9B:K9iM# a:BsK;DK@;:#>:E#C9?CH#<A@#Bsv;DE8#A=EO9P;:qE=#QBsPD#:B@O#<A@#Bsv;DE8#:<#;:B>@A;9I:#@AC9B;@CDK;DE8# B:OLf@<sK# QB@K<:Q:<:P=9c?CK@M# aBO9QB:K@<O:;:# BsK;=9v# Q:BsK;@;=9# O# K=9A:KD>=@B:KD># C9?C9># *8@Q=B:Fe=AL@M#

On the power of the chi-square test for multidimensional normality

! " # $ % & ' ( ) * & " " & * # + , - . & ( % * & * #

/,+& #,(!,%,0&! ##1234#1565#

#

#

#

#

!"#$%!"&'%()"*+

4#,*-./"&01&2345*6

#

#

!"#$%"& '%(" )"#$%"*$+,-./0(%"#%-#""

) ("1/2#+3+1%!-+ !02"! (102+#4"

#

#

#

+6"+!#( 3/*#+ !#

#

,HB# QHBQ:?9# =?# C9?C=;I# C89# 8DQ:C89?=?#

7

S T

@

8

U 4 V

A B # K89B9#

W

4MMM4

4

X

9

9

9

@

:

A

;

U V4

@ &

B @

U V

;

U

@ A @ A

!

VU

!

V M

# &C# =?# K9AA#

L;:K;#C8@C#C89B9#9Y=?C?#?HE8#@JJ=;9#CB@;?J:B>@C=:;#

E

D@

!

C

#C8@C#

;

U

E

V

F

4#

+

E

B

U

V

#=J#

<9CU

B

V

"

5

M#*:4#C89#8DQ:C89?=?#N

#=?#9GH=R@A9;C#C:#C89#J:AA:K=;I#

:;9#

7

S

E

T

8

U

!"#$%!"&'%()"+*

4#,-./"&01&23456

**!"#$%"& '%(" )"#$%"*$+,-./0(%"#%-#""**

**+6"+!#( 3/*#+ !#**

_&