Tests for Normality Based on Skewness and Kurtosis Measures

A C T A U N I V E R S I T A T I S L O D Z I E N S I S FO L IA OECONOMICA 48, 1985

C ze sław O om ański*, W ie s ła w Wagner * *

TESTS FOR NORMALITY BASED ON SKEWNESS ANO KURTOSIS MEASURES

1 , In t r o d u c t io n

I n th e th e o r y of s t a t i s t i c a l in f e r e n c e a w ide c l a s s of goodness o f f i t t e a t s in c lu d e s t e s t s f o r n o r m a lit y . They a llo w to v e r i f y the goodness o f f i t of norm al and e m p i r ic a l d i s t r i ­ b u tio n s o f the te s t e d random v a r i a b l e . The problem o f th e v e r i ­ f i c a t i o n o f n o r m a lit y assu m p tio n s o f a d i s t r i b u t i o n i s o f v i ­ t a l Im p o rta n ce f o r tho m a th e m a tic a l s t a t i s t i c s s in c e m a jo r it y of the methods a re based on t h i s a ssu m p tio n .

T h ie p a p e r p r e s o n ts a c la s s o f t e s t s f o r n o r m a lit y based on m easures o f the d i s t r i b u t i o n sh a p e . Those m easures in c lu d e skew ness (a s y m m e try ) measure and k u r t o s ls m easure. On the b a s is o f th e se m easures the d e p a rtu r e o f the c o n s id e re d d i s t r i b u t i o n from the norm al d i s t r i b u t i o n can be d e te rm in e d . They assume ¡ o r each d i s t r i b u t i o n f ix e d v a lu e s i f th e re a re f i n i t e v a lu e s of the f i r s t fo u r c e n t r a l moments o f the d i s t r i b u t i o n . F o r in s t a n ­ c e , the m easures o f skewness in the case of sym m e tric d i s t r i » b u tio n s assume th e v a lu e o f z e r o .

* D r . , L e c t u r e r , I n s t i t u t e o f E c o n o m e tric s and S t a t i ­ s t i c s , U n i v e r s i t y o f Ł ó d ź .

2. M easures o f Asymme t r y and K u r t o s lo L e t ( 1 ) X - ( x a ... xp ) ' be a p - d lm e n s io n a l random v e c t o r w it h f i n i t e d i s t r i b u t i o n p a r a ­ m eters

EX - R - ( * V

Pp)'

(

2

)

OX - £ ■ ( where E i s a p o s i t i v e d e te rm in e d m a tr ix . O

In th e ca se when p * 1 we s h a l l use X , y and ct , r e s p e c t i v e l y . L e t

( 3 ) ( X j ... Xn ) - { X j }

d eno te an n-eiem ant random sam ple o f p - d lm e n e io n a l in d ep e n d en t v e c t o r s w it h a u n ifo rm d i s t r i b u t i o n , w hich a re th e r e a l i z a ­

t io n s of random v e c t o r X.- I f P • 1 the random sam ple I s de-oted as ( x . , X ) ■ f x . } . U n b ia se d e s t im a t o r s o f the

pa-i ' n J 2

ra m a te r sam ple £ , 2 j as w e l l as (j and d a re denoted as

. X - ( X x ... X ) ' ( 4 ) J L S - ( s u 4 - £ £ ( X j - x x x j - x ) ' J - i — 2 and X , S , r e s p e c t i v e l y . We assume t h a t the d i s t r i b u t i o n o f v e c t o r X i e d e te rm in e d by d i s t r i b u t i o n f u n c t io n F p (x_), w h ile o f v a r i a b l e X - d i s t r i b u ­ t io n f u n c t io n F ( x ) , where x e R*3 and x e R ; R* d e n o te s an 1- - d im e n s io n a l r e a l s p a c e . We In tr o d u c e n o t a t io n e i>p ( x ) and $ ( x ) f o r d i s t r i b u t i o n f u n c t io n p - v a r ia t e and u n i v a r i a t e norm al d i ­ s t r i b u t i o n . Next by HQp ; F p ( x ) - $ p ( x ) we d eno te a n u l l hypo­ t h e s is s t a t i n g t h a t v e c t o r X has a p - d im e n s lo n a l norm al d i s t r i ­ b u tio n w h ile f o r v a r i a b l e X vie have HQ s F ( x ) - $ ( x ) .

'.Vo e h a l l d o f ln e naxt d i s t r i b u t i o n paroraotora f o r p » 1 - c e n t r a l moment o f the r- th o rd er ( 5 ) ■ E C (X “ ^/"1» r ■ 0# 1« 2 , . . . w# - asym m etry (s k e w n e s s ) c o e f f i c i e n t ( e ) - e 3/ e ? /?‘ o r P i ■ t*3 ^ 2 * - k u r t o s is c o e f f i c i e n t

(

7)

|

32

-Thfe f o llo w in g i n e q u a l i t i e s o c c u r among the abovo m entioned cot f i c i e n t s

( 8 ) (S2 > 1 ♦ (3±

(32 < 3 ♦ 1 .5 (3A

The m easures and (3

2

o re a p p lie d m a in ly to

1) a c h o ic e o f r e p r e s e n t a t i v e s in a f a m ily o f d i s t r i b u t i o n s ( o . g . in th e f a m ily o f P e a r s o n 's d i s t r i b u t i o n s ) ,

2

) a d e te r m in a tio n o f t e s t s f o r n o r m a lit y ( e . g . the t e s t ba­ sed on the s ta n d a rd iz e d f o u r t h c e n t r a l sam ple moment) ,

3) s tu d y in g the ro b u s tn e s s o f some t e s t in g p ro c e d u re s f o r d e p a rtu r e from norm al d i s t r i b u t i o n ( e . g . u sin g the c o e f f i c i e n t

in s tu d y in g r o b u s tn e s s o f t- S tu d e n t t e s t in th e v e r i f i c a ­ t io n o f h y p o th e s is * yo , whore i s an e x p e cte d v a lu e in the p o p u la t io n , and i t s h y p o t h e t ic a l v a l u e ) .

A d e c i s i v e p o in t in in t r o d u c in g the d i s t r i b u t i o n o f t- S tu d e n t s t a t i s t i c o f a q u o t ie n t fo rm , i s indep en d en ce o f th e num erato r from the d e n o m in a to r w hich o c c u rs a t the h y p o th e s is HQ. I f the sam ple comes from a p o p u la t io n w it h non-norm al d i s t r i b u t i o n , then from the c e n t r a l l i m i t theorem , e s p e c i a l l y from th e L ln d e n b e rg - - L e v y theorem , i t f o llo w s t h a t th e mean from th e sample ( x ) and u n b ia sed v a r i a n c e e s t im a t o r ( s ^ ) has an a s y m p to tic norm al d i s t r i ­ b u tio n ( c f . [ 4 ] ) .

L e t k d eno te th e r - t h cum ulant in a p o p u la t io n , where k2 ■ r

p

" ^

2

*

^3

” f

3

*

k4

* t

*4

"

3

Tho In f lu e n c e o f n o n n o rra a lity on t s t a t i s t i c used In t e s t i n g tho h y p o th e s is p ■ I s e x p re s s ­

ed by the c o r r e l a t i o n c o e f f i c i e n t between the v a r i a b l e s X and 5 o f tho form

( 9 )

COV ( X . S 2 ) k3/n

k ( » * * S i * l

)]1/2

M u *

* 1)У/г

because a t n —>oot ~ x —► 1 . I f the non-norm al p o p u la t io n l e sym­ m e t r ic , k j ■ 0 and hence g - 0 , thon X and S 2 a re a s y m p to ti­ c a l l y Ind ep end en t w hich a llo w s to a p p ly the th e o r y o f norm al d i ­ s t r i b u t i o n f o r la r g e n. F o r k^ 4 0 , g ta k e s s m a ll v a lu e s when k

4

i s l a r g e , but £ * 0. E q u a tio n ( 9 ) I s now w r it t e n in the form

(

10

)

Ц

•

T 2 Щ ")1'2

( ł P l

)V2

assum ing t h a t k^ »

0

.

As a r e s u l t , under the above a s s u m p tio n s , th e c o r r e l a t i o n c o e f f i c i e n t ç can be t r e a t e d a s skewness m easure. Assuming th a t V fil “

0

, we have £ » 0 and th u s , th e v a r i a b l e s X and S

2

a re u n c o r r e la t e d . The c o e f f i c i e n t ( 3 i s a p p lie d , f i r s t o f a l l , in the v e r i f i c a t i o n o f h y p o th e s is t h a t the e x p e cte d v a lu e o f j o i n t v a r i a b l e s becomes z e ro when th e re i s no a ssu m p tio n o f n o rm a lity . B o x

8

nd A n d e r s o n [ 2 ] u s in g P it m a n 's p e rm u ta tio n t e s t , showed t h a t the sq u a re o f t s t a t i s t i c used in the v e r i ­ f i c a t i o n o f the above m entioned h y p o th e s is o f the e x p e cte d v a ­ lu e , has F d i s t r i b u t i o n w it h end (n - l)t^ d e g re e s o f freed o m ,

(1 1) - 1 +

P * - 3 » a - 3 /1 S

--- . , --- . <>(,-). n ( l - ( 3 ,) / " ♦ 2

T h is r e s u l t has been d e r iv e d under tho la c k of th e n o r m a lit y a s ­ sum ption f o r th e d i s t r i b u t i o n from w hich the sample { x j } was drawn.

L e t us d e f in e now the b a s ic p a ra m e te rs f o r m u l t i v a r i a t e d i s ­ t r i b u t i o n s p > 1 :

- the mixed c e n t r a l moment o f v a r i a b l e s ( s i p )

1 . s of th e ( r j + . . . ♦ r 6) - t h o rd e r

(

1 2

)

( 1. # . . . • e r r " 1 r l ... r 0 _k=*l

where ( i . . . . . . 1 ) i s an a r b i t r a r y s-elom ent subooquence from

1 9 _

the sequence ( l , . . . » p ) and r j * . . . » r 0 * 0» 2 * •** - the asym m etry c o e f f i c i e n t [ l l ]

C13) 'v \ A

«

1 2 * 3 1 ’ 2* 3 where 2 ” ^ » ( d ^ ) , - k u r t o s ia c o e f f i c i e n t [ l l ] U 4 ) A . , A . , 1* 2 1* 2

B e s id e s , we In tr o d u c e p a ra m e te rs from tho sam ple f o r p ■ 1: - the r - t h o r d e r c e n t r a l moment ( 1 5 ) _{mr * Ï Ï Z CXJ “ X ) r *} J - l r ■ 2 , 3* - th e aeym m otry c o e f f i c i e n t /

( 1 6 ) "y/b^ ■ o r • m^/m^ ,

- the k u r t o s is c o e f f l c i o n t

(1 7 ) b2 ■ m^/m,, .

S i m i l a r l y , f o r p > 1 , we have

- the mixed c e n t r a l moment o f v a r i a b l e s X. , X. o f the

l l e ( r a ♦ . . . ♦ r 8) - t h o r d e r (1 6 ) m _{r i ...} * ’ r ₈ “ n _{n Z_,} 5 ^ _{I I} F I ( X 1 _{l k j} " X i ^_{l k} J - l L k -1 - th e asym m etry c o e f f i c i e n t [ l l ] P P _{* 111213 i l 12l} <1 9 ) bi . P Z £ s V l s l 2 ‘ 2 s l 5 ‘ 3 * m - i n l j , 12 , i 3" l i l f i 2 , i 3» l where S ^ ■ ( S ^ ) - the k u r t o s ie c o e f f i c i e n t [ l l ]

(

2 0

)

2,p

E

Z

1l ' i 2-1 1l i 2 "1 S S mi l l l

The m easures d e te rm in e d by fo rm u la e (1 9 ) and (2 0 )c a n be g iv e n in th e form o f c e r t a i n pow ers in t w o - lin e a r and sq u a re form s

( 2 1 )

J . i - l n

<2 2 > b

2

.p ■ 7; Y , K ' ‘ *3 ' - } ] J - l

« 3 and (3, • (p ♦ 2 )p . H**nco thoeo hypotho .38 c an bo p ro - sen te d In the e q u iv a le n t fo rm s : h; , (3X - 0 A (J2 - 3 and Hop : (31#p - O * 0 2 fP " P ( P + 5?)« ■>« F u r t h e r on we s h a l l c o n s t r u c t t e a t f u n c t io n s f o r tho v e r i < i c n t i o n of h yp o th e se s Hq and Ho p . 3. T e s t Based on -j/b^

Now, we s h a l l d is c u s s th e a tte m p ts o f d e te rm in in g the d i s t r i b u t i o n under the assu m p tio n o f th e h y p o th e s is HQ. The b e at r e s u l t s have been o b ta in e d u sin g Dohnson s system of c u r v ej

[ 9 ] . Such a r e s u l t i s p re s e n te d by D 'A g o

0

t 1 n o

who reduced th e s t a t i s t i c to a random v a r i a b l e w it h N (C ,1 ) d i s t r i b u t i o n assum ing th e h y p o th e s is HQ and n > n.

L e t [ ( n * l ) ( n * 3 ) l 1/2

Y " V^l L 6

(n-

2 )

j

*

„ /i->\ 3 (n 2* 2 7 n - 7 0 )(n + l)(n + 3 )

P2 T V ‘ fn^TCn+5)

TnV7)(n+9T^ •

w2 ■

- 1

♦

[ 2

( p 2( V ^ ) - O ] 1^“ * S - 1/ [ i n w ]a/2 , t - [

2

(vv2 - l ) ] ^ , then the v a r i a b l e ( 2 3 ) Z - i In [ y/ T + V ( Y / T ) 5' ♦ l ] has a p p ro x im a te ly th e N ( 0 , l ) d i s t r i b u t i o n .

The h y p o th e s is HQ i e r e j e c t e d i f | z | > u^. where $ ( u a ) • - 1- « / 2 , and a i s a g iv e n s i g n i f i c a n c e l e v e l .

c o m p a r . T t i v o o t v d i e s o f v a r i o u s a p p ro x im a tio n s o f th e d l-f > * n b u ; i o n t a k i n g i n t o a c c o u n t t h e f o llo w in g r e s u l t s (c f . T a b ls l ) : a ) t b o i r own s im u la t io n r e s u l t s , b ) t h e c u r v e s (th o a p p ro x im a tio n o f O 'A g o s t 1 n o [

5

]), c ) th e ap p ro xim ated t- S tu d e n t d i s t r i b u t i o n , d ) C o r n is h - r ic h e r e x p ro c e io n [

3

] , e ) t h o m o d ifie d C o r n is h - F ls h e r e x p r e s s io n [ 8 ] , f ) t h e a p p ro x im a tio n by norm al d i s t r i b u t i o n .

The a p p ro x im a tio n by t - d i s t r l b u t i o n o r V U - t y p o c u r v e s from f \ > a r 3 o n " ' 3 system i s os f o ll o w s :

( 2 4 )

The s t a t i s t i c g iv e n in fo rm u la (2 4 ) has t- S tu d e n t d i s t r i b u t i o n .v ith v d e g re e s of freed o m , w ith

(.15) V = - , ti2 ( ) - 6 (n-2 )/ft [3 ],

where y5 £ i s d e te rm in e d in fo rm u la ( 1 6 ) , n ^ - n ( n - l ) . . . ( n - k * l ) . The a p p ro x im a tio n by norm al d i s t r i b u t i o n ta k e s In t o s c c o u n t as a v a r i a b l e w ith norm al d i s t r i b u t i o n and w it h i t s e x p e c­ ted v a lu e e q u a l zo ro and v a r ia n c e t*

2

(V ^ l )*

On th e b e s is o f T a b le 1, we can n o te th a t tho a p p ro x im a tio n o f v a r i a b l e y ? £ by norm al d i s t r i b u t i o n i s o f r e l a t i v e l y s m a ll a c­ c u r a c y . I n o t h e r c a s e s s l i g h t d i f f e r e n c e s o c c u r In q u a n t l le s o f Vb7 d i s t r i b u t i o n .

The c r i t i c a l v a lu e s f o r n > 25 were g iv e n by P e a r s o n and H a r t 1 3 y [ i 5 ] arid f o r n < 2 5 by M u l h o l l a n d [ l 4 j (cf. T a b le 2), ivho found them on th e b a s is o f some a n a l y t i c a l s t u d ie s on the s i n g u l a r i t y o f the d e n s it y f u n c t io n of d i ~ s t r l b u t i o n .

O 'A g o s t i n o and T i e t J e n [

7

] ( c f . T a b le 3 ) e ls e gave the c r i t i c a l v a lu e s f o r n - 5 (1 )1 1 ,1 3 ,1 5 ,1 7 ,2 0 ,2 3 ,2 5 ,3 0 ,

the-T a b l e 1

Q u a n t ilo s o f ap p ro xim ated d i s t r i b u t i o n s

n Approxim a­ t io n а

_{____ ________}

I

0.1 0 0 .0 5 0.0 1 с . оси 8

C e )

0.7 60 0.991 1.455 1.073 ( Ь ) 7 -1 -34 56 _I ( с ) 7 -1 -34 56 ( d ) 8 14 -14 -17 ( в ) -4 17 19 1 ( f ) 12 1 -52 -10 15 ( в ) 0 .6 40 0.8 62 1.275 1.775 ( b ) 2 -12 -13 27 ( с ) 2 -12 -16 27 ( d ) 0 -12 -9 48 ( о ) -1 -12 -7 49 ( f ) 19 -6 -64 -1G7

---20 ( в ) 0 .5 9 3 0.7 7 7 1 .1 52 1.614 ( b ) -4 -5 -2 30

C c

) -4 -6 -4 38 C d ) -6 -9 1 76

C e )

-6 -9 2 76

C f )

13 1 -52 -153 35

C o )

0.4 7 4 0 .6 2 4 0 .9 3 2 1.332 Cb) 1 -3 -9 -13 Cc) 1 -3 -11 -13 -4 C d) 0 -4 -7 Cf) 12 2 -47 -156

T a b l e 2 Q u a n t ile o of d l a t r l b u t l o n o f s t a t i s t i c n a n a 0 .0 5 0.0 1 0 .0 5 0 .0 1 4 0 .9 8 7 1.120 15 0.851 1.272 5 1.049 1.337 16 0 .8 34 1.2 4 7 6 1.042 1.429 17 0 .8 1 7 1.2 22 7 1.018 1.4 5 7 18 0.801 1.199 8 0 .9 9 8 1.452 19 0 .7 8 6 * 1.176 9 0 .9 7 7 1.433 20 0 .7 7 2 1.1 55 ¿0 0.954 1.407 21 0 .7 58 1.134 11 0.9 31 1.381 22 0 .7 4 6 1.1 1 4 12 0 .9 10 1.3 5 3 23 0 .7 3 3 1.096 13 0 .8 90 1.3 25 24 0 .7 2 2 1.078 14 0.8 7 0 1.298 25 0.7 1 0 1.060 S o u r c e s On tho b a s is o f [1 4 T a b l e 3 Q u a n t ile s o f d i s t r i b u t i o n o f - y ^ s t a t i s t i c

1

n 5 6 7 8 9 10 11 0.0 5 1.058 1.034 1.008 0.9 91 0 .9 7 7 0 .9 5 0 0.9 2 9 0.0 1 1.3 42 1.415 1.4 3 2 1.425 1.408 1.3 9 7 1.3 76 n 13 15 17 20 23 25 a L 0 .0 5 0.9 0 2 0.8 6 2 0 .8 2 0 0 .7 7 7 0 .7 4 3 0 .7 1 4 0 .0 1 1.312 1.275 1.188 1.1 52 1.119 1.0 73 S o u r c e : On th e b a s is o f [

5

]. se v a lu e s w it h th e r e s u l t s o f M u lh o lla n d shows s l i g h t d if f e r e n c e s between them.

4 , T o s t Based on

An a c c u r a t e d i s t r i b u t i o n o f b_ f o r n > 4 assum ing t h a t the h y p o th e s is H i s t r u e , has not been known so f o r . T h at i s why v a r io u s a p p ro x im a tio n s f o r b „ by Dohneon's S u

IV - t y p e d i s t r i b u t i o n have boon found. and P e a r s o n 's

d i s t r i b u t i o n The ap p ro ­ x im a tio n by S d i s t r i b u t i o n has the f o llo w in g form [ l ] i

( 2 6 )

r ♦

•vhore c o n s ta n ts

'f ♦ 5 ln

_{• M}

♦ 1

5ln iV x -“

n < 25

S , | and A. w i l l be found u s in g th e method o f moments, p re s e n te d among o t h e r s , by P e a r s o n and H a r t l e y [ l 5 ] . The v o r i a b l e 2 has a p p ro x im a te ly norm al N

H con-( 0 , 1 ) d i s t r i b u t i o n . The v e r i f i c a t i o n of the h y p o th e s is s i e t s in a com p ariso n o f th e v a lu e s o f Z w it h a v a lu e of u a , where i> ( u a ) - 1 - a . o f the d i s t r i b u t i o n o f b. H a r t l e y [ l 5 ] ( t a b l e 3 4 c ) f o r n <200 A ls o f o r the same v a lu o 3 o f a c r i t i c a l va-C r i t l c a l v a lu e s

P e a r s o n and

and a - 0 .0 5 , 0 .0 1 .

c o rre s p o n d in g

w ere g iv e n by

lu e s w ere g iv e n a d d i t i o n a l l y by the a p p ro x im a tio n and V l- t y - pe a t n ■ 5 0 (2 5 ) 150, 200, 400. These v a lu e s do not d i f f e r from each o t h e r up to th e second p la c e a f t e r comma. U sin g the sim u­ l a t i o n method O 'A g o 8 t 1 n

0

and T i e t J e n [ 6 ] gene­ r a te d c r i t i c a l v a lu e s f o r s m a ll sam ple s i z e s n ■* 7 ( l ) l 0 , 1 2 , 1 5 ( 5 ) 50 ( c f . T a b le 4 ) . T a b l e 4 Q u a n t ile s o f d i s t r i b u t i o n o f b,, s t a t i s t i c

01

n 0 .0 5 0.0 1

lo w e r upper lowe r upper

1 2 3 4 5

7 1.41 3.5 5 1.25 4 .2 3

T a b le 4 ( c o n t d . ) 1 2 3 4 5 9 1 .5 3 3.86 1.3 5 4 .8 2 10 1 .5 6 3.95 1 .3 9 5.00 12 1.6 4 4 .0 5 1.4 6 5.20 15 1 .7 2 4 .1 3 1 .5 5 5.30 20 1 .8 2 4 .1 7 1.65 5 .3 6 25 1.9 1 4 .1 6 1 .7 2 5.30 30 1 .9 0 4.11 1.7 9 ,5 .2 1 35 2 .0 3 4 .1 0 1 .8 4 5 .1 3 40 2 .0 7 4 .0 6 1.09 5.0 4 45 2.11 4 .0 0 1 .9 3 4 .9 4 50 2.15 3.99 1.95 4 .0 8 S o u r c e s On th e b a s is o f [ 6 ] . 5. P r o p e r t ie s o f b, S t a t i s t i c * lP Now we s h a l l d i s c u s s t h e p r o p e r t i e s o f t h e g e n e r a l i z e d s k e w -i k b c o e f f i c i e n t b ^ p s ( i ) The b j s t a t i s t i c i s i n v a r i a n t in r e l a t i o n to th e o r th o ­ g o n a l t r a n s f o r m a t io n ^ ■ Ç X . I t r é s u l t a Im m e d ia te ly f r o » the form o f e q . ("21) to w hich vie s u b s t i t u t e X j - X ■ Ç '

( i i ) The s t a t i s t i c " i s I n v a r i a n t in r e l a t i o n to the non- - s in g u la r t r a n s f o r m a t io n X » A X. ♦ ÎL* * t r e s u l t * from th e form

o f e q . (2 1 ) and X j - X - A Y^ - Y.

( i i i ) The b., _ s t a t i s t i c In c lu d e s f ■ p ( p ♦ l ) ( p ♦ 2 )/ 6 d i- s t i n c t o le m e n ts .

I n the summation form o f bA p e t a t i e t i c we have 2P e le m e n t« ( v a r i a t i o n w it h r e p e t i t i o n s ) , but o n ly f ■ (t h r e e - e le m e n t c o m b in a tio n s w it h r e p e t i t i o n s ) o f d i s t i n c t e le m e n ts .

Lot S j - s " 1/ 2( x J - x ) , then n n b 1>p • i Z { s ' s j } 3 I P 3 • K • " j . j ' - i 3 . J -1 2 3 _{p «* np}3

( v ) The b„ s t a t i s t i c oxproseod by moans o f a n g le s and

Moha-1

lP

la n o b ls d istan ce*» assumes the form [1 3 ]

n n 3 ( 2 8 ) b1<p - - ig X ! Z C r j r r c oa e j r ) , n J - l j ' - l / 2 2 ,2 \/2 w here coa r j j ' " v j * rj # " 3J ' d ) J - ‘ ( 2 J ' - 2 j ') end f j ■ ( X j - X ) ' S 1 ( X j - X ).

( v i ) . The e x p e c te d v a lu e o f b, l a e x p re s s e d by the formu-i r

l a

( 2 9 ) E ( b l , p ) " Cn*‘S f e ^ J ^ (n + 1 )(p * 1) " 6 ^‘

T h ia fo rm u la l a g iv e n by M a r d i a [ 1 2 ] f o r n -*■ oo, E ( b 1>p) H i Oua to th e in v a r ia n c e o f th e l i n e a r t r o n e fo r m a tlo n we can p r e s e n t the b j p s t a t i s t i c in the form

-

i

f e w

•

1l ' i 2 , 1 3

-whoros ml l l 12 * 1 * 2* ^ l 1! 1^ {11* 111 m3 w h ile 4 1 ’ - i

X

<*11 - * ! > ’ ■

J-l

n

(

1 2

)

1

m21

r E < \ i - S l ) ( x 2 i - * * ' •

J-l

■ i l l 3 ’ ■ S Z <■*!} - *1> ( X 2 j * * 2 > (x 3 i - V - j - l

Assum ing t h a t the h y p o th e s is th a t the sam ple { * , . } comes from a m u lt id im e n s io n a l norm al p o p u la t io n N ( 0 , I ) , i s t r u e , we have

-1 P ~ r 1

moments up to the n - th o r d e r , o f th e form L U J

(3 1 ) ‘ E ^ i } 12* 3 ) • 0 . . O2 ^ 1 * ) - 6/n, c o v ^ m D2 ( » ‘ J 2 )) . 2 / n ,

0

*(.< ‘ P > ) - »/«.

V j >_ „ . 0> t i _ l 2 t j „

N o te , t h a t the a ssu m p tio n o f ^ « 0 and £ * I i s p o s s ib le due to th e p r o p e r t y ( l i ) .

then I ( 3 3 ) bl j f } • K M On th e b a s la o f fo rm u la e (3 1 ) wo have (3 4 ) E ( M ) » 0 D (M ) - E (M M ') ■ d la g (6 / n , . . . . 6/n, . . . . 6 / n ) - ( C / n ) l . Hence (3 5 ) M ~ Nf ( 0 , 6/n I ) , i w h ile ( 3 6 ) nM'M/6 - n b ^ p / 6 ~ X 2 .

Form ulao (3 5 ) and (3 6 ) o c c u r whon n~+oo. An a c c u r a t e d i s t r i b u t i o n o f th e v a r i a b l e b. _{1 f p} I s not y e t known. _' B e s id e s no o t h e r appro- x lm a tlo n s o f the v a r i a b l e b^ p o re known os in th e u n i v a r i a t e c a s e . F o r p > 7 the f o llo w in g a p p ro x lm o tIo n con bo a p p lie d

( 3 7 ) (2 n b , n/ 6 ) 1/2 ~ N ( 2 f - l , l ) .

_{X ,}

_p

M a r d 1 a [ l 2 ] d o te rm ln ed the c r i t i c a l v o lu e 3 f o r the d i s t r i ­ b u tio n o f b^ u c in g th e M o n te - C a rlo method f o r n « 1 0 (2 )2 0 ( 5 ) , 30(10 ) | P 1 0 0 (5 0 ), 2 0 0 (1 0 0 ), 4 0 0 (2 0 0 ), 1 0 0 0 (5 0 0 ), 3000 (1 0 0 0 ), 5000 and a m 0 .0 0 1 , 0 .0 1 , 0 .0 2 5 , 0 .0 5 , 0 .0 7 5 , 0 .1 0 ( c t . T a b le 5 ) . F o r p - 3 and p - 4 M a rd la d e te rm in e d th e c r i t i c a l v a lu e s , how ever th e y have n ot been p u b lis h e d .

T a b l e 5 Q u e n t ilo a o f d i s t r i b u t i o n o f b. - s t a t i s t i c n 10 12 14 16 10 U

20

25 30 40 50 0< 0 .0 5 3.604 3. 319 3J031 2.7 75 2.556 ... 2. 356 1.9 6 9 1 .6 87 1.319 1.0 69 O .O i 5.9 3 0 4 .9 3 8 4.581 4.231 3.962 3.669 3.106 2.681 i 2.0 87 1.744 s

0

u r c s : On th e b a s is o f [

1 2

] . T a Q u a n t lle s o f d i s t r i b u t i o n o f b , _ s t a t i s t i c J 1 e 6 n 10 12 14 16 18 20 25 30

1

40 50 . -■ a 0 .0 5 4 .8 8 7 9 .2 0 3 5 .0 5 3 9 .5 9 3 5.179 9.7 6 9 5.3 18 9.941 5.3 82 10 .00 5 5.533 10.114 5.689 10.159 5.855 10.156 6.139 10.109 6.239 9 .9 8 7 0 .0 1 4.5 80 10.378 4 .7 3 2 10.881 4 .8 4 2 11.159 4 .9 7 7 11 .38 7 5.0 45 11.478 5.1 75 11.609 5.353 11.628 5.5 18 11 .59 4 5 .7 0 3 11.453 5.909 11.181 S o u r c e : On th e b a s is o f [

1 2

] .

F o r th e b2 p s t a t i s t i c the f o llo w in g p r o p e r t ie s o c c u r .

( 1 ) The b * s t a t i s t i c i s i n v o r ia n t due to the o rth o g o n a l 2|p

t r a n s f o r m a t io n Y ■ Ç X and n o n - s in g u la r X » A Y + b. ( l i ) The e x p ected v a lu e of bg p assumes the lorm [ l l j

\ p ( p * 2 ) ( n - l )

(3 8 ) E ( b 2 ,p ) n+I

( i l l ) The v a r i a n c e o f b0 _ i s d e te rm in e d by tho Tormula [ l 2 ]

(3 9 )

‘ B p (p + 2 )(n - 3 )--- ( n—p—l ) ( n—p * 1 ) (n+-l) ( n * 3 ) ( n * 5 )

S B l B l i ) a t n“ l .

Tho f i r s t fo rm u la was in tr o d u c e d by t a k in g in t o a cco u n t tho m ul­ t i v a r i a t e b e ta d i s t r i b u t i o n , and th e 6econd one by u s in g L a- w 1 e y ' s method [ l o ] ,

( l v ) b „ can be e x p re sse d in the form Z »P

n

J - l

where r . i s M a h a la n o b is d is t a n c e betweon X , and

X*

T a k in g fo rm u la e (3 8 ) and (3 9 ) we can o b t a in two t e s t a v e r i f y ­ in g the h y p o th e s is Ho p , whoso e t a t i s t l c s a re as f o llo w s .

{ ( n » l ) b g p - p ( p + 2 ) ( n - l ) } ‘ {(n * 3 )(n - » 5 )}1^ 2 1 | 8 p (p + 2 )(n - 3 )(n - p - l)(n - p + 1 )}1^2 f o r th e a c c u r a t e v a r i a n c e u (bg and ( 4 2 ) N9 - _Z b2 n ” P< P*2 >_r --- -—_{l i /i} {a p (p + 2 )/ n }1/2 f o r th e ap p ro x im ated v a r ia n c e D2 (b g ^ p ) up to th e n o r d e r .

S t a t i s t i c s (4 1 ) and (4 2 ) hove the N (0 ,1 ) d i s t r i b u t i o n by v i r ­ tue o f the c e n t r a l l i m i t theorem .

An a c c u r a t e d i s t r i b u t i o n o f the v o r i a b l e b2 p under the a s ­ sum ption th a t tho h y p o th e s is Hop i s t r u e , i s unknown. Tho ne- c o s o o ry c r i t i c a l v o lu e 3 f o r the d i s t r i b u t i o n o f b? 2 hove been g e n e ra te d by M o r d 1 o [

1 2

] u sin g the M o nte-C orio method in the some rango o f n as f o r b j 2 , and a - 0 .0 1 , 0 .0 2 5 , 0 .0 5 , 0 .1 0 g iv in g two v a lu e s - upper and lo w e r . T a b le 6 p r e s e n t s th e ­ se v a lu e s f o r n < 50.

BIBLIO G RAPH Y

[ 1 ] B o w m a n K. 0 . , S h e n t o n L . R . , (1 9 7 5 ), Omnibus T e s t C o n to u rs f o r D e p a rtu re s from N o r m a lit y Based on

and b2 , "B io m e t r lk a * 62, p . 243-250.

[ 2 ] B o x G. E . P . , A n d o r s o n S . L . (1 9 5 5 ), Perm uta­ t io n T h e o ry in the D e r iv a t io n o f Robust C r i t e r i a and the S tu d y o f D e p a rtu re s from A s s u m p tio n s , O . R . S . S . B . 17, p. 1-34. [ 3 ] C o r n i s h E. A . , F i s h e r R. A. (1 9 3 7 ), Moments

and C um ulants in the S p e c i f i c a t i o n o f D i s t r i b u t i o n s , R e v. I n s t . I n t . S t a t i s t . 5 , p . 307.

l4 ] C r o m e r H. (1 9 5 8 ), Metody m otem atyczne w s t a t y s t y c e , PWN, W arszaw a. [ 5 ] D 'A g o s t 1 n p R. B . , T 1 e t

3

e n G. L . (1 97 3), Ap­ p ro a c h e s to the N u ll D i s t r i b u t i o n o f -j/b^, “ B io m e t r lk a " 60, p . 169-175. [ 6 ] D 'A g o s t l n o R. B . , T i e t j e n G. L . (1 9 7 1 ), S i ­ m u la tio n P r o b a b i l i t y P o in t s o f b2 f o r S m a ll S am p le s, " B io - m o trik a " 58, p . 669-672. [ 7 ] D A g o s t i n o R. B . (1 9 7 0 ), T r a n s fo r m a tio n to Norma­ l i t y o f the N u ll D i s t r i b u t i o n o f g1 , " B io m e t r lk a " 57, p. 679- -681. [ 8 ] G e a r y R. C. (1 9 4 7 ), The F re q u e n c y D i s t r i b u t i o n o f f o r Sam ples o f a l l S iz e s Drawn a t Random from a Normal Po­ p u l a t i o n , " B io m e t r lk a " 34, p . 68-97.

[ 9 ] 0 o h n 8 o n N. L . (1 9 4 9 ), S ystem s o f F re q u e n c y C u rv e s G e n e ra te d by Methods o f T r a n s l a t i o n , " B io m e t r lk a “ 36, p. 149- -176.

[ 1 0 ] L a w 1 e y D. N. (1 9 5 9 ), T e s t o f S i g n i f i c a n c e in Cano­ n i c a l A n a l y s i s , “ B io m e t r ik a " 46, p . 59-66.

[1 1 ] M a r d 1 a K. V . (1 9 7 0 ), M easures o f M u l t i v a r i a t e Skew­ n ess and K u r t o s is w it h A p p l i c a t io n s , "B io m e trik a * ' 57, p. 519- -530.

[1 2 ] M a r d i a K. V . (1 9 7 4 ), A p p lic a t io n s o f Some M easuros o f M u l t i v a r i a t e Skewness and K u r t o s is in T e s tin g N o r m a lity and R o b u stn e ss S t u d ie s , "Sa n k h y o " 36, s e r . B . , p . 115-128. [1 3 ] M a r d i a K. V . (1 9 7 5 ), Assessm ent o f M u lt in o r m a lit y and

the R o b u stn e ss o f H o t e l l i n g 's T?' T e s t , A p p l, S t a t i s t . 24, p . 163-171.

[1 4 ] M u l h o l l a n d H. P . (1 9 7 7 ), On the N u ll D i s t r i b u ­ t io n o f f o r Sam ples o f S iz e a t Most 25, w it h T ab les, " B io m e t r ik a ” 64, p . 401-409. [ 1 5 ] P e a r e o n E. S . , H a r t l e y H. 0. (1 9 6 6 ). Biom e­ t r i k a T a b le s f o r S t a t i s t i c i a n s , V o l. 1 , Cam bridge U n i v e r s i t y P r e s s . [1 6 ] P e a r s o n E. S . , H a r t l o y H. 0 . (1 9 7 2 ), Biom e­ t r i k a T a b le s f o r S t a t i s t i c i a n s , V o l. 2, Cam bridge U n i v e r s i t y P r e s s .

C z e sław Dom ański, W ie s ła w Wagner

TESTY NORMAljNOŚCI

OPARTE NA MIARACH SKOSNOSCI I SPŁASZCZEN IA

W a r t y k u le p rz e d s ta w io n o t e s t y w e r y f ik u ją c e h ip o te z ę o n o r­ m a ln o ś c i ro z k ła d u zarówno Jednow ym iarow ego, Ja k i w ie lo w y m ia ro ­ wego, o p a r te na m ia ra c h s k o ś n o ś c i i s p ła s z c z e n ia . Do w ię k s z o ś c i omawianych te s tó w podano n ie k t ó r e k w a n ty le ro zk ła d ó w f u n k c j i t e ­ s to w yc h . Zam ieszczono ró w n ie ż podstawowe w ła a n o ó c i u o g ó ln io n e j n l a r y sk o ó n o ó ci - b^ o ra z m ia ry k u r t o z y - b2 .