Dec 22, 1988 - ASYMPTOTIC DISTRIBUTION OF THE SHAPIRO-WILK W FOR TESTING FOR NORMALITY. 4BY. J. R. LESLIE, M. A. ..... David, H.A. (1981).
%
•
.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