/iJ-J-- A, $TI11 Sai!

15 downloads 0 Views 948KB Size Report
Dec 22, 1988 - ASYMPTOTIC DISTRIBUTION OF THE SHAPIRO-WILK W FOR TESTING FOR NORMALITY. 4BY. J. R. LESLIE, M. A. STEPHENS, and S.
%



.i4

r t7

/iJ-J--

.. 'z.oat

-- ,

"W t

A, $TI11 Sai!

2 :xiua -

-. "' ";

ib t

tl

. ......

24

41'

t

*0

'ii V.'

3

ASYMPTOTIC DISTRIBUTION OF THE SHAPIRO-WILK W FOR TESTING FOR NORMALITY

4BY J. R. LESLIE,

M. A. STEPHENS,

and

S. FOTOPOULOS

I

CTECHNICAL REPORT NO.

411

DECEMBER 22, 1988

e

2§_c

13

Prepared Under Contract N00014-86-K-0156

(NR-042-267)

For the Office of Naval Research Herbert Solomon, Project Director

Reproduction in Whole or in Part is Permitted for any purpose of the United States Government

Approved for public release; distribution unlimited.

Accesion For CRA&I TAB

0]

Urlarinotinced

0

NTIS DT!C DEPARTMENT OF STATISTICS STANFORD

UNIVERSITY

STANFORD,

CALIFORNIA

Jistion

y

"I ,

.

O.odes

AV, V9.~

10,0

ASYMPTOTIC DISTRIBUTION OF THE SHAPIRO-WILK W FOR TESTING FOR NOPIALITY

BY J. R. LESLIE,

M. A. STEPHENS,

and

S. FOTOPOULOS

TECHNICAL REPORT NO. 411 DECE.BER 22,

1988

Prepared Under Contract N00014-86-K-0156

(NR-042-267)

For the Office of Naval Research

Herbert Solomon, Project Director

Reproduction in Whole or in Part is Permitted for any purpose of the United States Government

Approved for public release; distribution unlimited.

Accesion For

NTIS

CRA I 0

DTIL" TAB U ano;,nir)0 ed DEPARTMENT OF STATISTICS STANFORD

J'i.tlj

[0

r

UNIVERSITY

STANFORD, CALIFORNIA

0'y..,

.

-

*-

i ...

. .i -

~l

...

~iN .a Iia idlL. N i~llIDv ~m

a ... anlnlnll . ..

in

a

t

-

i

Aw

__

ASYMPTOTIC DISTRIBUTION OF THE SHAPIRO-WILK W FOR TESTING FOR NORMALITY By J. R. Leslie, M. A. Stephens

1.

and S. Fotopoulos

Introduction. A popular test for the normality of a random sample is based on the

Shapiro-Wilk statistic

W.

This -tatistic, which was presented in Shapiro

and Wilk (1965), is the ratio of the square of the BLUE of sample variance, where

a2

a

to the

is the variance of the normal population from

which the sample is assumed, under the null hypothesis, to have been drawn.

For convenience we shall work with

W

X 1 < X 2 < ... < X n,

statistics from the sample, vector and As that

W 1 /2

V0

which has the form

n1 -1I-1 1/2 X'V0 m/_(Xi-X)2m'VoiVolm)

=

X = (X1 ,... ,Xn)',

where

W1 / 2

X

is the vector of order

is the sample mean, and

m

is the mean

the covariance matrix of standard normal order statistics.

is location and scale invariant we can assume from henceforth

X 1 ,... ,X

,

are order statistics for a sample from a

N(O,1)

population. A number of authors (for example, Sarkadi (1975),

(1977) and Gregory

(1977)) have (correctly) guessed at the form of the asymptotic distribution for

W

as well as predicting that the test should be consistent.

no rigorous proofs have been possible due to the presence of Neither

V

0

nor

-i V0

0

However

-i V0 1

can be found explicitly and until recently no

1

reasonably accurate asymptotic approximation for

V0

was available.

A

paper by one of the authors (Leslie 1984) has now remedied the situation; in that paper can be found an approximation for of asymptotic properties of

V09

together with a number

one of which is of particular importance V0 1

is approximately an eigenvector of

m

It states that

to this work.

V0

in the following sense:

(-1 m-

where

C

2m

//

C(logn) -1/2

is a constant independent of

b = (bl,...,bn)'.

n,

and

,

//b/ /2 -

= 7b21.

for

This latter result formalises a similar one appearing

in Stephens (1975). The asymptotic distribution of

W,

after appropriate normalizing,

has been assumed to be the same as that of the De Wet and Venter (1972) statistic W*

here H

r(X,Y)

is the

=

r 2 (X,H)

is the sample correlation coefficient between X

nx 1

th vector whose i

element is

C 1(i/(n+l)}

and and

Y, - (1)

is the inverse function for the standard normal distirbution function ('),

that is

-l (x))

= x.

The rationale behind this assumption was that firstly, known to behave like

2m (see Stephens (1975)), secondly,

approximates the i th

element of

m

and thirdly, as

V0

V.lI

was

t-1{i/(n+l)} is a doubly

stochastic matrix (the sum along any row or column is 1) we may write

2

W = r 2 (X,V 0 1m)

De Wet and Venter (1972) showed that the asymptotic distribution of W

has the form *1/2

(2)

where

=

C (Y -1)/i, 3 i

D

2n(l-W */2

-

{Yi,i> a

is a sequence of i.i.d.

1}

a n

-

N(0,1)

variates, a

(3)

= (+l)1

{

an

-1.

j

=

i/(n+l)

and

is the

-2

3

(j)))

i=l

(')

}

N(O,1)

2j(-j)('{4 2

density function.

Beyond the De Wet and Venter result the first step towards the asymptotic distribution for (1972) statistic

Wt

W

was to show that the Shapiro-Francia

given by r2 (Xm)'

behaves in the same way as

W.

This was done independently and via

different routes by Verrill and Johnson (1983) and by the authors in Fotopoulos, Leslie, and Stephens (1984), henceforth expression (2) was established with

W

called FLS, where

in place of

show in FLS the equivalent result that

(4)

n(W*

2 -W1 11/2

0

3

in probability.

W*.

In fact we

Our task in the present paper is to show that

n(W / 2 _Wtl/ 2 )

(5)

0

in probability

We note that Verrill and Johnson (1983) contains a result (Theorem 3) which should eventually cover the asymptotic distribution of -1 V0 m

certain properties of

However

need to be established before it can be

Inequality (1) does not appear to be enough.

applied.

2.

W.

Asymptotic Properties of W and a . The following theorem presents one version of the asymptotic distriW - in fact the asymptotic distribution for

bution for

the corollary offers the complementary form in terms of Theorem.

WI / 2

-

whilst

W.

Under the hypothesis that the observed sample is from a normal

population the asymptotic distribution of the Shapiro-Wilk

W

takes the

form:

/ 2n(l-W 1/2 ) _ 2En(-W

where

=3

N(0,1)

(Y -l)/i,

and

{Y., i > 3}

1/2

)

D -

is a sequence of i.i.d.

variables.

From the lemma below and from the theorem we have

Yn(l-W

/2

)

-

in probability, which leads to

2n(l-W1/2) - n(l-W) = (/n(l-wl/2)) 2 _, 0

Again applying the lemma below we obtain,

4

in probability.

0

An equivalent form for the asymptotic distribution of

Corollary.

W

is:

D n(W-EW)

-+

-.

It is not obvious from their definition just how the constants behave as

n

an

will

gets large.

The following lemma should shed some light on

an

defined in (3) have the following properties:

this matter.

Lemma.-- The constants (i)

a n - 2nE{l-r(X,b)} . ..

1/2V- 1rm (ii) (iii)

or

-

0,

where

m,

H, 0,

a -nE(I-W) n1

a -n(l-n- m'm) + 3/21

n

can be any of

b

< C(logn)-

_

and (iv)

C 1 loglog(n) < an
3. 1 1 1 _ of

g,_m

and

in section 1).

H

by respectively Further, as

r

gi, m.

and

=n -2 d. , in = L. I .th Denote the i element where

H.i

The consistency of

Theorem 1 of Sarkadi (1981). it

appears to be the case that

and

1H are given

is scale and location invariant we assume

without loss of generality that our sample is from a

Proof of Consistency.

(m

W

N(0,1)

population.

follows directly from

There is a small difficulty in that whilst Volm

is a vector whose elements, as you

0

move down the vector, are monotonic increasing, we are unbale to prove it.

6

W1 / 2

This means we cannot establish that exploits the fact that +1/2

on for of

W

W

1 /2

(note: we distinguish between

W).

+ W .

are equivalent to those based on

W, W

Sarkadi

is always positive to argue that tests based

etc.; it is true that

1/2

WI

W = (W/)

, W

We need to argue likewise

tl/2

2

etc. and the square roots

but in view of what has just

W I /2

been said, we are unable to say whether of

is always positive.

is the positige square root

We overcome this difficulty by showing below that

(6)

W

>

-C(logn)-/2

C

,

independent of

n.

From the theorem and the lemma, the lO0a% critical region for the test based on it is

W

1/2

is:

W

1/2


1

vector of

and from (1),

is consistent. l's and writing

X'm}/(nS G )

provided only that the components of

Sarkadi (1975), Lemma 2),

W

By (6) the two critical regions are asymptoti-

We extablish (6) by setting

As

-

-i/2(c(a)+a

maxlgi-miJ

X

are increasing (see < C//(logn),

we have,

with the help of (21) below,

W1/2> -CEnIXi/{nS G V(logn)} > -C//(logn) n n1 1 -

We turn now to Theorem 1 of Sarkadi (1981). states that

W1 / 2

Applied to our context, it

will determine a consistent test of

7

H0:

that the random

sample is normal, versus

H1 :

that the observations are not normal (Sarkadi

also allows the observations under

H

to be m-dependent with common

non-normal marginal) providing

(7n 1

where

I(A)

n

I(i-l < nu < i) -l(u)du = 1 + 0(l)

is the indicator function for

theorem is framed in terms of a statistic

A. T

n

n -- 1/2-1_ . 2 T n = -7 {(X.-X)n S-c n 1n in" 1

where

c inVn = gi/Gn'

Note that Sarkadi's which here takes the form

112

2(1-W

To establish (7) we require results contained in

the proof of both our lemma

and theorem, therefore we will leave the

derivation of (7) till the end of the article.

Proof of Lemma. We start by showing (iii);

N 2 n(l-M ) = 2{ Var(X i)} 1

observe that

-

(2 N-n)Var(XN)

-

E4(si+Wi)}

1

We can write

Var(Xi) = E{4,(si+Wi)

Expanding Wi

'

in

W.

2

up to third order terms, using the properties of

1I

given in the section on notation and together with results in Leslie

(1984) (in particular, Lemma 6 and the properties of we can show that

8

'

given in section 3)

-1 (exp(-s))

fexp(-s.1(

whiere 4'(s)

and

-2

d.i

=

n -2 Z. v

This yields

N

N

1Var(X)

(8)

~~o~/)

2d

Var(X)-

)s)}d< < C(logn) 2

-

Using the Euler-Maclaurin summation formula (Knapp (1951), p. 534)

(9)

s.

0

log((n+l)/i)

-

1/2(i -- (n+l) 1) < q{2 -(n+l)- 21/12

and

(10)

0 0,

for

being a covariance matrix, is positive definite.

11l

mn'& Set

=

e

at V- I

0

and

V

0

to be the angle

1i

between

m

and

g

cos 6 > 0

then

triangle formed by vectors AC and CB.

Let

CD

/a//2 > (CD2 As

m, g

and

a

r-g,

be the perpendicular from

//g//2 sin 2 O

r.

0 < e
0

for all

H

and

H

As

is shown in Lemma 2 of De Wet and

(22) and an analogue of (22) with

= /((H'H)Inl

i, m. and ii12

1-(3/4)n - 1 + 0(n -2).

H.

g

and

(this analogue holds because

always having the same sign) we have,

12

En{l-r(X,b)}

1

l-03/4)n1 -n

n(ESn)

1

=n(ES n) -1(1-n )

-0(/4)

=n(1-Mn)

+ O(logn)-

-0(/4)

+ O(n 1

m b//b// -1

+ O(logn) -

the latter expression resulting from the fact that

n(l-M n

Q(loglogn)

(using (iii)

M n-~ 1).

This

and (iv) of the lemma and recall that

establishes (23).

(24)

Analogous to (23) for

InE{l-r 2(X,~) I

To show this we note that as

b

have

=we

n(l-M)n21 + 3/21 f C(logn)1

nES

2 n

n-l

=

we can write

EX'f)2 /{(-l'c

nEr 2 (X,')

2

}

n with

R

A

InM 2 nn

using (1) and (22). we obtain (24).

(25)

2

IV'r +

-W

+ (na GM)2-M

n

(1)2

= 1 '

nn

+(a

+O(n/logn)

Again using the property that

n(l-M n

O(loglogn),

As

n(l-M 2

2n(l-M) n

it is clear from (23),

-

V'n(l-M ))2 =O{(loglogn) 2/n

n

n

(24) and (iii) of the lemma that

13

Wi

and (ii) hold.

a -E {l-r 2(X,b)}

Undoubtedly it is true that

and

H.

However this entails showing that

lVp2m/l

-

0

approximation

-

0,

/ VoH-2H//

both of which will follow once

V0

for

0

b = m

and

is replaced by the

V

given in Leslie (1984): corollary 1 in Leslie (1984)

permitting this.

These two results will involve a quantity of tedious

analysis and it seems unnecessary to set it down here.

Proof of Theorem.

The theorem follows from (2),

(4) and (5) together

with the lemma; therefore to prove the theorem it remains to establish (5).

Now

n

) =

nS (W1/2_W

1

(Xi-mi)(gi-mi)Gn + Sn

S

As

Sn n

1

a.s.

Xi(g G -m nM) i n in (Xi-mi)mi(G l-M ) + (m' n n

I//

-

)Gn I n

and with (21) and (22), expression (5) will follow from

Markov's inequality once we demonstrate that

(26)

ELZ(Xi-m i)(gi-mi)1-

(27)

EII(Xi-mi )(G n-M n)

0,

and

0

Result (26) follows from Schwarz inequality:

E jZ (X -m

) (g

- m i)

{n (l-M 2))}

I/ 2

-m



With (1) and with (iii) and (iv) of the lemma we have (26).

14

To deal with (27) we note that in Lemma 11 of FLS we show

EX

(28)

i-j