ASYMPTOTIC DISTRIBUTION THEORY FOR DEGENERATE U-STATISTICS
FROM STATIONARY SEQUENCES
E. Carlstein
University of North Carolina
at Chapel Hill
SUMMARY
This paper initiates the study of degenerate U-statistics in the
case of dependent data.
Although the asymptotic distribution of non-
degenerate U-statistics has been extensively investigated under various
mixing assumptions, the study of degenerate U-statistics has until now
been restricted to the iid case.
Our asymptotic distribution theory
covers * -mixing, <p-mixing_, and strong-mixing.
The limiting
distrib2
ution which we arrive at is a weighted sum of (dependent) X random
variables; this is intuitively analogous to the well-known iid result.
Moreover, our general treatment of degenerate U-statistics includes
several important special cases:
X2 goodness-of-fit statistic
Cramer-von Mises statistic
The asymptotic distributions of the
(under strong-mixing) and of the generalized
(under
*-mixing)
are obtained as appli-
cations of our theory.
Supported by NSF Grant DMS-8400602.
AMS 1980 subject classification:
Key words and phrases:
Running heading:
Primary 62E20, secondary 60GIO.
dependence, strong-mixing, goodness-of-fit,
Cramer-von Mises statistic, chi-squared.
Degenerate U-statistics.
1.
Introduction
Let {Z. :
_00
1
< i < oo} be a strictly stationary sequence of
RP-valued ran?om vectors
Cl.:: p < (0);
the marginal distribution of Zo
r
is F.
X RP
i =1
and assume that ¢ is symmetric in its r vector arguments.
Let ¢ (z 1 ,z2 " .. zr) be a real-valued function defined on
Cl.:: r < (0) ,
!, based on data
Then the corresponding U-statistic with kernel
(Zl'Z2, ... ,Zn)
U
n
=
(n~r),
is:
2
l<i <i <···<i <n
-
1
2
~
For O<k<r denote <P k (zl,z2"",zk)
and
=J· .. f
r
<P(zl,zi, ... ,zr)
<P
If
i =k+l
1
Observe that <P (-) = <PC-)
k (zl,z2,···,zk) =<P k (zl,z2,···,zk) -<PO'
and <PO = constant; write ;p(.) =¢r(-) .
II dF(z.)
r
f
-2
<Pl(z)dF(z) =0, then we shall
say that Un is degenerate; otherwise Un is nondegenerate .
Asymptotic distribution theory for Un has been studied in great
depth.
Hoeffding (1948) introduced the concept of U-statistics and
established asymptotic normality fornondegenerate U-statistics when {Z.}
1
is iid.
Asymptotic normality of nondegenerate U-statistics (r ~ 2) has
been extended to the dependent case in the following steps:
Sen (1963)
allowed for m-dependence in {Z.}; Sen (1972) treated *-mixing sequences;
1
Yoshihara (1976) considered absolutely regular sequences.
Asymptotic
normality for nondegenerate U-statistics has not been established for
strong-mixing sequences.
*-mixing
~
¢-mixing
~
(Recall the hierarchy:
absolutely regular
~
iid
~
m-dependent
~
strong-mixing (see Yoshihara
(1976) and Sen (1972)).)
On the other hand, the asymptotic distribution of degenerate U-statistics
has only been studied under the assumption that {Z.}
is iid.
1
Serfling (1980)
-2-
and Gregory (1977) have shown (independently) that the asymptotic
distribution in this case is equivalent to that of a weighted sum of
2
independent X random variables.
Our objective is to extend the
asymptotic distribution theory for degenerate U-statistics to cases
where {Z.}
is dependent.
1.
As a first logical step (analogous to Sen's (1972) work for nondegenerate U ) , Theorem 1 (below) treats the case of *-mixing in
n
{Z.}.
1.
Next, by imposing an additional constraint on the kernel function
¢, we obtain an asymptotic distributional result (Theorem 2) which
allows {Z.} to be strong-mixing.
This is particularly surprising since,
1.
as mentioned above, no strong-mixing result exists. for nondegenerate
U-statistics.
These two new results are not mere mathematical curiosities,
filling in obscure voids in..our,
to several well-known statistical problems.
and Chanda (1981) hav.e
2
~
they can be applied
knowled~.:;Rather,
For eXample, Moore (1982)
studi;~d t,~e .~~ymJ>t;otic:,dirStr,~bution
,'8 .
, ..
I" "'.J <:'i
\. (" ..... , . I
~
•
c.
'" (' ...
~
.::
of the standard
l ~ - ~ ; . . ; ..
-
•
X goodness-of-fit statistic when the data come from a strong-mixing
~_;5,!~_'~_':~) l)'3l1~-~:):
sequence.
. :~,
crni.!';"J.;,·~'t-·~
l.f~.tV.;::"I.1~i.C:~·'
Their basfc-"lIiiilting distribution can actually be obtained as
)~'~iJf'~,:-"iU (.<".J
g::1'\',-12i .~rs~
", ~"";'
29!}.fF,,:-v;;e~~!_ f.-J
;
..
As another exami;1.e~' we-consider the
~~ :::~'~)iJf}t
I ,.~ ',~ .t:~t. .:<1: 1: -:./=J
;- l =i ,~ J:o~ (= ~ ""Ib ;
generalized Cramer-von Mises statistics. Their asymptotic distributions
a special case of our Theorem 2.
t
.b'jt;c~;·l(;x·:,
sci
\~m
have not previously been studied in the
.-~; fVHf;'
•
~esence
<", ,...
.·.i) ..
of dependent data, since
the sample variance, and to the cross-product statistic .
•
-
.." t
,~
+c
,,'. . .
In Section 2 we present and discuss'uur main theoretical results
(Theoremsl and 2).
Applications of these results are found in Section 3.
The proofs of Theorems 1 and 2 are deferred to Section 4.
•
-3-
2.
Main Results
We are interested in the asymptotic distribution of the standardized
Un =n(V n
-¢O),
·
V-statistic
assume that r~2 and that
where Vn is degenerate.
JJ
To avoid trivialities,
¢;(zl,z2) dF(zl) dF(z2)
(0,00).
E
Theorems 1
and 2 will actually assume r =2, but this restriction is evidently not of
much practical consequence:
All of Gregory's (1977) distribution theory
and applications for degenerate V-statistics in the iid case
have r =2;
all of Serfling's (1980, Sec. 5.5.2) examples of degenerate V-statistics
2
have r =2; our examples in Section 3 (e.g. the X goodness-of-fit
statistic and the generalized Cramer-von Mises statistic) only involve
r = 2.
Nevertheless, Theorem 4 (in Section.3) shows that Theorem 1 actually
can be extended to the case r >3: as· weIr. ,"';
We shall make use of the:foLtowUig 'ftp:te.sentation 'of ¢2 (see Gregory
(1977)).
The
equati~n~',.
I')
has distinct nontrivial solutions {g.} (called eigenfunctions), with
~)I
correspondi~g
.. .:
f
.-~~.
·,.l.!.e;}.~':... t';""E~.< ..: r;.8.LJ~Jd1:r~iJ~.LL ~jnJ".:·ilrr~):.r ::.t;,
eigenvalues {A.}, satisfying the
.,:~t\'.;J
gi (z) gj (z) ~F5z)
'';;'.,.,
It,
S
I'~fxi;,J) 'T~};iJo.n.i.:: 2i~..
orth~normality
t~. f~<.· "J.J)t1rl.i. .~; ....
~,I{.i=!.~,' .'......r~~. fixe.,d; ~Ko~_,~,'
;:de~ot;~, gI«Z}
.r . . .
) .... .) "~~'1.,,:\-,_.,
,j ....
t
.. ' - _ . -
,J ....
".~:t.-
'l.~_
.... i
= (gl (z) , ... ,gK (z)) ,
i".·,~
-
condition
, ••• ,A }. Then ~ may be expressed (in the limit) in terms
lC, ',. c"~",,c"'''''~l "J{-' 1~'J r'~'l' h·,.., c,.. .,n,~n..........,. .
of {g.} and {A.} by virtue of:
.
'"
.
and !I.
K
1
=diag{A
·.J"q··'f'·c..(,
..'.~.~u ~.,., _,:
.j~ ..
L:~_~·,1::~r:~j :~<:O
j.',_~
•.
,._·;;"<., ... 1
"'1'~.l;~~')
0,;
.• ,,
...: ........
1:'3til'G
1.
.... -":.1 .
...,1,.; ...
n..i taD9~11 ~l.
j
".-.
The joint distributio.n of (2 ,2 i).. (~~ 1) will be denoted by Fi ( • , .) •
0
..'
..'
Let 0+ =o{Z. : i > n} and 0- =o{Z. : i < n}. We say that {Z.} is *~mixing if
n
].
n].
].
,,-
';;'
lim 1J;(n) =0, where 1J;(n) =sup{lp{BIA}/P{B}-l!: AE:J~, BEr]+, P{A}>O, P{B}>O}.
~oo
We say that {Z,}
is
].
n
~-mixing
if lim
~(n)
=0, where ¢(n) =
n-+ oo
= sup{!P{B!A} - P{B}! : A€ o~, BE o~, P{A} > O} .
We say that {Z.}
is
].
-4-
strong-mixing if
lim
~
a(n)
0, where
n+ co
= sup{ IP{A (l B} - P{A}P{B} I
a(n)
We assume that
-2
E{<P (ZO,Zi)}<co Vi>l.
2
THEOREM 1.
Let {Z.} be
1
and let r=2.
*-mixing
If
co
I
(l.a)
< co
n=O
and
00
l:IA.\<OO
(l.b)
i=I
then
1
co
u ~
~
n
A.
1
i =1
(W~1 -1)
as n +00
The joint distribution of any
•
finite set of W. 's (with A. FO) is'multivariate normal with means 0 and
1
1
00
L
i =-00
E{gr (Zo) gJ(Z)}.
-<- .•
~): :J L
(P~t, Wi ::
'," 'I:I
~~
i:' ;';J:;
?
~._
if A. = 0) •
j ;..: ...,,;...~-cr.. . "
This result is analogous to Gregory's (1977) Theorem 2.1 for iid
The W. 's in our
data (with Q :: Po in his notation).
nI
1
concl~ion
are not
necessarily iid, because our Z.1 's. _,,_..
~r~2D!P.hne~~~s1l,riJ.f.JIU!i,
'__..
.. ,_'__... .. _. {Note, however,
°"
- ..
----~---
that if {Z.}
are iid p Dtir i'!1--noiTT.t-,~
di.'st.15
1
..... ,....-1"......
I Jbutton,
.. ::is ·:i:tlentical to his.)
~-;
When the set of eigen£unct.ibnS. is;: fim.}t&',-t:l'Ie
'canli-}l:eJrax- the conditions
J
J
on d epend ence.
-,~,
.. -.
Th e' X,.2.~igo-~ues:s:rlOJ::l"l.:L:.Ltrl~~~tli::C):~':l[S:~~
- ...;...:1...'
J:. .~
...... -~..
••
':]important
example
where {gil is finite and Jthe.prem, 2 ,~~lie.s F~:ee.SectiQn :3)r.
THEOREM 2.
#{g.}~:
1
Let {Z.} be strong-mixing, let r=2, and assume that
1
K<oo.
•
If
36> 0
s.t.
I
E{ gi (Zo)
2+6
1
} < co
ViE {l, ... ,K} and s.t.
-500
(a(n))O/(2+0)
<00,
and if F.« [FXF]
1.
~
"
U
n
~>
K
2
A. (W. -1)
I
1.
i=l
as n
1.
-+ 00
Vi> 1, then
The joint distribution of the W. 's is
•
1.
as in Theorem 1 .
Note that, for nondegenerate Un (with r.::. 2), there are no asymptotic
distributional results allowing strong-mixing.
If we make the more
restrictive ¢-mixing assumption, we can eliminate the moment conditions
on g .•
1.
THEOREM 3.
Let {Zi} be ¢-mixing, let r =2, and assume that #{gi} =: K<oo •
00
If
I
and
Fi« [F~F] .iV hi:)" then the conclusion of Theorem 2
n=l
holds.
I
,Vi
.."
;-~. ~
.\ ,
'I,
;
-'
The proofs of Theorems 1, 2, and 3 are defer~ed to Section 4.
3.
;" •. 1.
l'
.C
Example I:
X2 Go6dnes!;';'o:f"'Fit~Btat,istl~J
Suppose
J'
'; i.,
Y~;'':~
thatF~e cIlallgfi:bi1Z\j:::iJ:S; pa~d u.nto ~1'::' 2
mutually
exclusive and 6:lGhiaustive 'S'e~s~im-d E:t i::~dl:}.r;rwillw3i:'):;::.P{ZO€A.} > 0
].
{p. : 1 < i < I})
1.
-
-
-
-
aJidth~"data,s(-Zli.?i.) ,:~ .') ;, w:e'xr(1nsi~
n
goodness-of-fit statistic:
I
Xn =
1.
n
L( L
i =1 j =1
2
I {Z . EA.} - np.) /np .
1.
].
1.
J
Vi.
1.
2
the standard X
-6I
where <PCzl,z2) =
I
V. =
J i=l
L
L
i =1
I{zlE A.}I{ z 2 E A.}/p. ,
l.
l.
l.
<P
0
=1, and
Hz. E A.} Ip . • Observe that the implicit U-statistic with
. J
1.
.
1.
n
kernel ¢ is degenerate.
I
The Ergodic Theorem implies that
V./n a.s·~I ,
j =1
provided {Z.} is strong-mixing (say).
l.
Un +
the asymptotic distribution of
J
Thus it suffices to determine
•
I-I .
Without loss of generality, the equation defining the eigenfunctions
•
-
-+
-+-+-+
-+
and el.genvalues of ¢>2 may be expressed in matrix form as y - 1 p'y = A Y •
i
Here
is an I-dimensional column vector of l's,
I
-+
and y = (y 1 , ... , y I) , E R
p'
l
=(Pl, ... ,PI)' AE R ,
charac teriz es the eigenfunction g ( .) by the
I
g(z)
relation
=
\'
v
L.
i=l
Y1 = i.
•i
I{zEA.}.
Corresponding to Al = 0 is the solution
1.
The remaining eigenvalues tA ,A , ... ,AI} are each equal to 1, and
2 3
a
their corresponding eigenfurictiorls~atglgfriiPl~ set of I-I vectors
I
{Y ,Y , ... ,Y } in R
I
2 3
(Le.
the
y.l. 's
sa:ti~fYirlgTp'~i=:6J
ii.'ild-hYr-dtiig{Pl'''',PI}Yj =I{i=j}
are muttiafCiVL8PlhJftdnfial'\~~t.i;;i:?he::-Wfhiirts{ p.}).
l.
~Tli2,n
We now wish to
<p
Z!l.
q~~~X;3~hr<J~el~L.21 51~:~~ ~~;atr -lrt~li ~~t~B\)
is
the~men~lcondib~h?onTgi
for some \)E (0,1);
= (I {Z. E AI} , ... ,
l.
here.
00
Since
[F x F]
Hz.l. E AI})' the relevant absolute continuity condition
Vi>
- l,
' where
-.....
is the marginal.
F.l.
is the j oint distribution of
.~ :;•.' .~ . - -7,,-
.. -.
--_..
~_.
.~; -:'. •
..~,
~
-
-_._._-_.
Hence we may conclude that X -~
n
1. 2
.I
l. =2
(Zo ,ll..)
and
F
-":' ~ t·:.~
Since {Z.} are discrete r.v.s
l.
,"
.i';~.-~ :.;);)'r,;.
D
t
trivial
<
may be rewritten i.n,~terJJi&jJofii..:tbe l1aIidoll:twectofsl;l{.aiJ ;"!whelleJI\.
is F.«
l.
f
c;:
'-'.'
this condition is satisfied.
.
00
-+
Wi" where (W 2 ,· .. ,W I )- NI_l(OI_l'
t
is a square matrix of dimension I-I with (u,v)th entry
L
f t) ,
=_00
•
-7-
P -+ (Z _u_'
< < I Z.-:v_
< I) ' an d p
. W1.t
. h (.1.,J.) -th entry
t '1.S an I x I matr1.X
u t Yv
-+,
Y
p{ZO
Ai' Zt
E
E
Aj }
Note that when {Zi} is iid the asymptotic distribution
reduces to the familiar X~I-l).
Also note that the asymptotic distribution
I
2
L
is in general equivalent to that of
s.w.,
1. 1. where {s.}
1. are iid N(O,l)
i=Z
00
and {w z ,... ,w r } are the eigenvalues of the matrix
r
l: rt
=
t
.
=_00
To see that our asymptotic distribution is the same as that in Chanda's
(1981) Theorem Z.l, let T =diag{O,L ,L , .•. ,L } be the diagonal matrix
Z 3
r
-+
-+
-+
containing the eigenvalues of his matrix A, and let IT =(TIl,TI ' ... ,TI ) be
r
Z
the corresponding matrix of eigenvectors.
~
-+
-+
IT = (TI Z '·" ,TIl)
and
T = diag{TZ " •. ,LI } .
,
o -- (Y-+Z , .. • ,Y- +
r)
diag{p~ , .•.
1
1
~:~ce
,p:p.
square orthonormal matrix
of
'.
dimen~iQn
"
~.
-J ,d .
r','
'.:,;.f :".
Then A =IT TIT' = IT TIT' , where
Observe that r
2,r .=..
where
= 01\ 0',
_
B T ~.'
, where
B
=0 IT is a
I-I.
This7:'::-1"_
shows that the eigenvalues
i.; ~ u 1.. ~: ~; 1"
of r (Le. the W,i ~~) ~f.e: equal b79 ('52'.; . "TIJ},.;· \~J}d;t~at Chanda's (1981)
..
Theorem Z.l is
jus~ [Ijl" ,sp.~Gia~
J
no proof of his
..... .,...
."'...
,......,
<;,ase f9~
.. o~~.:I;.Q.~or.E;m
.?!. ..
~
,,,J.L ,-' ;
'of.:.
....;
_
....
I...);,
•. "
I" ....
.l. '.)
~handa
oj t";;n
(1981) provides
resul~.
Moore's'(198ZY m~nri*£C: rff~C~~e lSot~~rg:~he6;em 2.1, is identical
to Chanda's (1981) lmatrtix AJ•.J ~ThusrMOatehs:;li!m~t;ing~,distributionalso is
obtained
from;outlrrbeQrem2~11JNote:)Dhatwai:-docnat::L&ssume
Gaussian.
Oi);· __ :~:'r~;
0'
;.iJiJ-:·J.i.,
:'ni; 'J';;.i
s':'
~;;,'.
,(:
~A
{Z.} to be
1.
:; .::.
j
:;,
Example 2:
-
Generalized Cramer-von Mises Statistic.
.
::',
f
,"
,)"'
Suppose that {Z.} are *-mixing real-valued r.v.s with absolutely
l.
continuous strictly increasing distribution function F.
Cramer-von Mises goodness-of-fit statistic is
A
The generalized
A
2
n J_00 w(F(z)) [F(z) -F n (z)] dF(z),
00
where F is the empirical c.d.f. of (Zl, ... ,Zn) and w is a non-negative
n
-8-
weight function.
n
Without loss of generality, we consider
n
=
n
L
<'<'<
1_1
y
¢(U. ,U.) /(n ) + L ¢(U. ,U.) /n, where {U. : _00< i <+oo} is a strictly
Z
1
J
. 1
1
1
1
J_n
1=
stationary sequence of *-mixing Uniform [0,1] r.v.s, and
¢(ul,u Z) =~l w(u)(u-I{u l 2. u})(u-I{u Z2. u})du.
so ¢1 :: ¢O = 0
Observe that ~l ¢(ul,uZ)du Z =0,
and the corresponding U-statistic is degenerate.
Two commonly used weight functions are wI (u) :: 1 and wZ(u) = l/u(l-u)
(0 < u < 1).
It is straightforward to verify that all the integrability
conditions on ¢ are satisfied for these weight functions.
Hence we can
apply Theorem 1 and the Ergodic Theorem to the first and second terms
(respectively) in Y.
(The limiting value of the second term is 1/6 for
n
De Wet and
Ven~er
'. ',.:.~
(1973)
have studied the eigenvalues
)
) : ' -~
'-."
'-,'
and eigenfunctions of ¢ ,in g~r~~ l~rfl~JtfX. ~fRLparticular, their work
Z
shows that \ =l/rrZi (no~.1!ji;~1~?rf;J~h[fydJrhff~jJi::=:~j~.,(i+l) for Wz .
Thus condition (l.b) is satisfied.
Note that {g.} are based on the
1
infinite set of Jacob~,~~~~~mia~~~:<~~.T~~~re~.~:.21~~1 ~icannot be
applied to the Cramer-von Mises;::~tatistl~•. I~o~eo~er, cmt,trffiatment of
for any nontriviald~p~Bdl:ml;~l'
.,Q·:tc,n-'.."t "n.c'
rr',
~.>..T .... :Il ...... '•
C!-' ...-:.. ..... 11.
.L ....
....
::.6:;"
Example 3:
j
c
J"
Y
..,J
...
v ~"'''''~m..i...,J:.;,J ...
c.r{~t_4 .....
,/0__•.
A Measure of Dependence.
~
J,
"Icr":_'...r
_~~ .....
; ,j
~J.'vJ.:~. . . . (.
.... --
'.'.,
c . ~. {~ J £;ti ~,j
,~~
Suppose Zo has discrete distribution F con~entrating on the set
'~r"(_1
8 2 ::_~)§'Iq ~?_~~
~:
2JJL :..'s,,-\ ~_
t""
The sequence {Z.} is supposed
1
to be independent, but, upon observing (Zl, ••. ,Zn)' the practitioner
suspects the presence of nontrivial dependence within ranges of m
•
-9-
consecutive observations (2..2.;/1..2. n).
If in fact no dependence is present,
m
then p{Z. 1 =z.
1+
1.
for any i and any
1-
j =1
1
(i ,i , ... ,i ).
l 2
m
p.
II
=
J
We can measure the degree of departure from this null
-+i
hypothesis by comparing how many subseries Zm := (Z.1.+ l'Z,1.+ 2' ..• 'Z,1.+m'1
(0,,:::, i..:::. n-m) in our sample equal (z.
1.
,z.
1
1.
vs
, ••. ,z.)
1.
m
2
how many were
expected to equal (z. ,z. , .•. ,z. ) assuming independence in {Z.} • This
1.
1.
1.
1.
2
1
m
is the standard observed vs expected concept, but with the following
important distinction:
Even if we assume independence in {Z.}, the
1.
random vectors {Zi: 0 < i < n-m} are still nontrivially (m-I) -dependent.
m
--
(Of course, one could look only at nonoverlapping subseries
{zim: 0 < i < [n/m]-l}, which are
m
--
".
in.depe·nd~ht:under H'O'.
'-I""': ;. ;
£
--e-c..
t'
'? (.
.... ".,:,. ~I
I:',
('"!
But in doing so,
_-
the majority of the available in£ormation aDout 'serial dependence would
be thrown away.)
Calculation'
~f:::the,'~symptb~i'C nuti
distJi.ibut10n of the
statistic
P.J2 /(n-m+ 1)
1..
J
now fits
m
II p.
j =1
easily<'fnto~tJl~,1fr~e~8fR£~t:rBxaDi~lec;E';:S~ri.ce" ft i } are (m-I)m
dependent, suuunability of the mixing coefficierttsblsfl~ediate. For
\t\2:.m we have f
t
=0.
For It I <m the matrix P
t
can be calculated explicitly
. ,~: ~;rJ''3 L.'r;s~') .:)(t
~.:
~
in terms of the original p. 's, because tne-preClse' d'ependence structure in
.
.
{Zi} is known. "0 ;,c:iz"<r:
m
1.
::.>nc:
q
nct~~dlTJ~ib
3!~I~~'
1. j
-10Example 4:
The Sample Variance.
The standard sample variance
s
2
n
=
2
(Z. -Z ) I(n-l) may be expressed
n
I
i =1
n
1.
as the V-statistic with <P(z 1 ,z2) = (z l-z2) 2 12 and <PO =Var{ZO}'
This
(.
statistic is a useful estimator even under dependence, because
2
E{;;n}
•
00
-+
L
Var{ZO} whenever
Cov{ZO'Z'} converges.
i =1
Since <PI (z) =
,1.
Iz o - E{Zo}1
[(Z-E{ZO})2 -Var{ZO}]/2, V is degenerate if
n
=s.d.{Zo} a.s.
Note that this condition does not necessarily imply that Zo is a degenerate
In particular, consider Zo - Binomial (1,1).
r.v.
shows that, in this case, K=l, gl(z) =2z-l, and Al
results to the case of a strong-nlixing sequence.
Serfling (1980, p. 194)
=-1.
We now extend his
Since the moment condition
and the absolute continuity condition of Theorem 2 are trivial here, we
00
may conclude that
n ( sn2- ,4)
D,( l-W 2 ),
-;> 4
provided
.",....
I
(a(n)) v <
00
n=l
.
~.I.i.
for some V €(O,l).
The r.v.
W-is normal with mean 0 and variance
00
i
If '{'Zi. (t'i's!
I
1
=_00
i~epeb£J'ent'/
tha.:r;ths.tilt
\.:-=11
!
~::
:reduces to that
given by Serfling (l.980J<l" 1QJ ~;; . '{! 1fleUV)('!1oJ
Example 5:
Cross-Prodto:~,·S:tat'.if'UcHi·f(Serflini;
( 1980)(; -E~,.\ ±5.5 .,2B) .
2
Let Hz l ,z2) = z l z2' E{ZO} = 0, and E{ZO} > O.
so
Vn = 1 .L
.
'::1. <J.::n
Z1.. ZJ. I {nl:'
deg~rne..J'.p;~~_.
.2 .~s_LOI~j,)~'
..r._~<:. .. ,-,
.
gl (z) = z IE l{z~},
E{Z6} < 00;
"'V
and
A~:~ it~}":,!
we also must
ass~e
Li
,'Ro
" .' ~~ ..., (
Then <Pa..(.zl.:¥. <PO = 0,
:Iq,,~,i::.,.,.:r'\'l'''_
):h_ius ,~~!,p.t~t~on;,.
~=l ,
15_,J.•.~ ...
.
fiJ
Jp~l~ JTh~~~i~k ::f~'rj3,
i) - ~. U.~ ~
__
-"L
we assume
31.
that the :kbsolute continuity condition
and the summability condition (on the' 'iniiing coefficients) are satisfied.
Note that, in Theorem 2, the moment condition reduces to simply
•
-11-
2
2
The limiting distribution of Un is E{ZO}(W -1) , where
00
W is normal with mean 0 and variance
i
~
If
{Z.}
is
1
=_00
independent, this reduces to Serfling's result.
0
Lastly, we show that Theorem 1 can in fact be extended t,o degenerate
U- statistics with r -argument kernels (r ~ 3).
technique (see Sen
We use the usual proj ection
(1972)) to write
r
U
n
= U
n
+ n
'" : = l2r) n
where Un
L l~) u~h)
h=3
¢2(Zi,Zj)/(~)
L
l~i<J::n
Sen (1972, eq. 3.25).
degeneracy.
Note that
and
u~h)
is as defined in
Un =n(t~),l~~l) ,+ l~)U~2j)
because of Un's
The assumptions and notatidniof:Sect-ions 1 and 2 are retained.
THEOREM 4.
Let {Z.} be 'I.-mixing and let r> 3 •
1
- '"
P
Then (U -U ) ~ 0
n
,
n
as n -+ 00
•
Assume that
Consequently, if {ll.b:)"J(aJ.'sb holds, then
00
Un ..Q..>
(~) ';:l~~:~i (.wit-!Eg~.asr :~""_M'h~~§lf!1e;~~id'J~veJ":1l~:.~ame
distribution
1=1
as in Theorem; '1.£/"
PROOF.
[L.:.
TherequfZ:i<t~ohVeii~Vce'ln ·dl?~t\:1:1Ut~ol·ioi~ n'
\, will
immediately by
a~plli~~ lR~fs~~ 'h qt () U'li. ~o~_:S:f, ~ha,t Pn approximates
U
(in probability), consider E{(U
n
:='; i «,. (f
By equation (3.27) of Sen
o(l In). 0
follow
~u
)2} < n 2J (r-2)
P .:,J\
(1~72),
J .E7.
..'
I
h =3
(r) 2 E{ [u(h)] 2}
h
n
the r.h ._s. of the above inequality is
-12-
4.
Proofs
-
We will show that T : = 2
n
Proof of Theorem 1.
<P(Z. ,Z.)
1.
converges in distribution to the required r. v •
J
In
:::l
U
n
For K > 1 define
T
= 2
L gK'(Z1..)AK gK(ZJO) In, and let Tn and TnK be the characteristic
nK
l.-:i< j..:n
functions of T
and T K respectively.
n
Let
n
Furthermore, (Tn-TnK)/(n-l)
SE
R1 be arbitrary but fixed.
is itself a 2-argument U-statistic with
-
+
+
-
kernel G (zl,z2) =<P(zl,z2) - gy«(zl)A K gK(z2)'
K
and E{gi(ZO)} =0 whenever \
rO,
so that
f
Observe that <P (ZO) =0 a.s.,
1
G (zl,z2) dF(zl) =0
Hence, (T -T K) I(n-l) is a degenerate
(see Serfling (1980, pp. 196-197)).
n
n
U-statistic and therefore is precisely of the form
Sen (1972, eq. 3.25)).
have E{ (Tn-T nK ) 2} ..::' c'
a.e. [F]
K
U(2) (as defined by
n
No:wapplyihgeq~:tiion' (3.27) of S~n (1972), we
'HJJG-~~f'Z2J)W{41'~W(tti2).i·~Vtii:Fwhe¥e;:c
<00
depends
TheirE!£dr-er?lb~f(.,q·.!}El(itb.,¥:
Yf.'}<'t.. Dfp,x)·K:>K(E:),
ItfI- n nJq1
.",-
only on {ljJ(n)}.
uniformly in n .
For fixed K->
CwequM~;1fJnK:;:~k ~L
~m[h-.~f1""l-g'
\[\r.~:,~';E J.'~ where
~l
ke
kn:f':i'k~ .-... -- ... ' r:;-'1,J0f!T
n
gkn =) gk (Zi) In
1. =1
~kn ~. 1
as
n+
l ,
u
and
00
~ld
r:,-·'l-iJ 2 . 1
n
.m~ll.·:)-::;rlT
~kn = i ~1 g~ (Z i) In.
for each k.
(:0(:',1
(:'''/0(;':.;..
~
~_.
By the Ergodic Theorem,
By the multivariate analog of Ibragimov' s
1
(1962) Theorem 1.5, the asymptotic joint distribution of those nigkn's
with A
k
W ' s.
k
T
r0
Thus,
o
nK
(1..: k..: K) is precisely the joint distribution of the corresponding
->Y: =
K
•
K
L
k =1
2
A (W -1)
k k
as
n+ oo
•
-13-
IAiIEl{(W~-1)2)t,
Now, for any M>N we have E{(y -y )2) "- [il.!
M N
with E{(W~_1)2} <3E2{W~}+ 1.
1
-
2
< 1+4
E{W.}
1
_.
For each i S.t. A. fO, we find
1
1
00
I
(the first inequality follows from Lemma 1.1
n=l
of Ibragimov (1962); the second follows from our condition (l.a)).
Thus, condition (lob) implies that E{(y -y )2}-+O
M N
in M > N.
as
N-+oo, uniformly
By the Cauchy convergence criteria, we may conclude that an
00
.
y .--
A.(W~-l)
exists s.t.
1
1
I
i=l
lim E{ (YK-Y) 2} = O.
K-+oo
Let 8 and 8
K
be the characteristic functions of Y and Y respectively. Again we have
K
12
2
18K(s)-8(s) I ..s. !s\E {(y -y) }.
Combining the results of this paragraph
K
with those of the previous 2 paragraphs; we see that
,SinQ.e.:{*)n~!I1~~p ;F"h,~~1*it{z'Y,;z2)
Proof of Theorem 3"
L
n
(s) -+ 8(s) as
-+
-+
=gK(zl)l\ gK(z2)
a.e. [FX:F], and s:LJ1P~:,,e3cql\.(Fii.~.,p·~!S,oJJ.I~,ly:,cpntj;nupus:
w.r.t.
_
follows that
Tn ;;::TnKi
The 3
rd
...
11'
.
a._s~ t'fn,,~d.:1I'-htf~r~:as,r<:<i~f"i-n~)'ii.1Ythe
~
Theorem 1.)
•
;
.'.
.
FX:F, it
proof of
).
paragraph in the proof of Theorem 1 now applies.
Proof of
Theor~l'Il- 2.; ;;AExactl1,;~h& ;~f a~ th~ .rrop'&:pf"J'heorem 3, but
replace
Ibragimov's (1962) Theorem 1.5 with his Theorem 1.7.
_.'
..-'
1'=.,
;1
0
0
REFERENCES
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Dordrecht, Holland.
de Wet, T. and Venter, J.H. (1973) Asymptotic distributions for quadratic
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f
Gregory, G.G. (1977) Large sample theory for U-statistics and tests of
fit. Ann. Statist. 5, 110-123.
Hoeffding, W. (1948) A class of statistics with asymptotically normal
distribution. Ann. Math. Statist. 19, 293-325.
Ibragimov, I.A. (1962) Some limit theorems for stationary processes.
Theory Probab. Appl. 7, 349-382.
Moore, D.S. (1982) The effect of dependence on chi squared tests of
fit. Ann. Statist. 10, 1163-1171.
Sen, P.K. (1963) On the properties of U-statistics when the observations
are not independent. Part one: estimation of non-serial parameters
in some stationary stochastic process. Cal. Statist. Assoc. Bull. 12,
69-92.
Sen, P.K. (1972) Limiting behavior of regular functionals of empirical
distributions for stationary *-mixing processes. Z. Wahrsch. verw.
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Serfling, R.J. (1980)
Wiley, New York.
Approximation Theorems of Mathematical Statistics.
Yoshihara, K. (1976) Limiting behavior of U-statistics for stationary,
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Department of Statistics
321 Phillips Hall 039A
University of North Carolina
Chapel Hill, NC 27514
•
•