THE THEORY OF UNBIASED ESTlfvIATION FOR SOME
NONPARAlviETRIC FAMILIES OF PROBABILITY IvIEASURES*
by
Nicholas I. Fisher**
Deparbnent of Statistias
University of North Carolina
Chapel HilZ~ North CaroZina 2?514
Institute of Statistics rUmeo Series No. 1051
January, 1976
* Ph.D. dissertation t.m.der the direction of Wassily Hoeffding.
** This research was supported by the C.S.I.R.O. 7 and partially supported
by the National Science Fot.m.dation tmder Grant XA324.
NICHOLAS I. FISHER.
The Theory of Unbiased Estimation for Some Non-
Parametric Families of Probability Measures
(Under the direction of
WASSILY HOEFFDING.)
The research of this dissertation, outlined in the second paragraph,
is based on the following known results.
Let
Po be the family of all
probability measures concentrated on finite subsets of a set X, and let
g(xl, ... ,x ) be a function symmetric in its arguments.
n
for every PEP ' then g must be identically zero.
O
If jgdpn = 0
Let P be the subk
family of probability measures P in Po which satisfy Ju.dP
= c.1 ,
1
I
~
i
~
k (*),
and let P be a convex family of probability measures
which satisfy (*), such that P ~ P .
k
If jgdp
n
= 0, all PEP, g need not
be identically zero; however, there exist symmetric functions
hI (xl'" .,x _l ),· .. ,hk(x l ,·· .,xn _l ) such that
n
k
n
g (x I' .. . , xn ) = L L {u. (x.) - c.} h. (x I ' . . . , x. I' x. 1"" x ).
i=l j=l 1 J
1
1
JJ+
n
Further, if g is bounded, and every non-trivial linear combination of
u ' ... ,uk is unbounded, then g must be identically zero.
l
results hold for analogous families,
Po(~)' Pk(~)
ility measures dominated by some a-finite measure
It
and
Corresponding
P(~),
of probab-
~.
is shown in this dissertation that if the conditions (*) are
replaced by the single condition (t) judp
2
= c, where u(x,y)
= u(y,x)
then, provided u satisfies certain conditions, a symmetric function
g(xl, ... ,x ) satisfying jgdp
n
have the representation
n
= 0 for all PEP (conditioned by'n will
•
L
15i~j5n
{u(x.,x.)-dh(xl,···,x. l'x. ], ... ,
J
1
1-
1+
x.J - I'X.J + 1""'X)
n
for some symmetric function h, and if gis hounded and u unhounded, then
g must be identically zero.
Corresponding results are obtained for
families dominated by a a-finite
measure~.
The class of all functions
(of a random sample) which are uniform minimum variance unbiased estimators of their expectations for all PEP (conditioned by (*)) is shown to
comprise those sYmmetric functions
t(x , ... ,x ) which satisfy
l
n
Similarly, if P is
conditioned by (t),
the class of such functions comprises those sYmmetric
TABLE OF CONTENTS
Acknowledgements-------------------------------------------------iii
Notation--------------------------------------------------------- iv
Chapter 1:
SUFFICIENCY, COMPLETENESS AND UNBIASED ESTIMATION FOR
SOME NON-PARAMETRIC FAMILIES OF PROBABILITY MEASURES
1.1 Introduction------------------------------------------1.2 On Sufficiency, Completeness and Unbiased Estimation--1.3 Completeness and Bounded Completeness of Some Non~
parametric Families of Probability Measures-----------1.4 Characterization of Minimum Variance Unbiased
Estimators in Some Non-parametric Families------------Chapter 2:
1
3
7
12
CANONICAL FORMS FOR SYMMETRIC UNBIASED ESTIMATORS OF
ZERO
.
2.1 Introduction------------------------------------------- 16
2.2 Some Functional Forms for u(x,y)----------------------- 17
2.3 Canonical Forms for Symmetric Unbiased Estimators
of Zero------------------------------------------------ 36
Chapter 3:
BOUNDED COMPLETENESS OF SOME NON-PARAMETRIC FAMILIES
OF PROBABILITY MEASURES------------------------------ 69
References------------------------------------------------------- 86
ii
ACKNOWLEDGEMENTS
I wish to record my gratitude to my advisor, Professor Wassily
Hoeffding, for introducing me to the problems considered herein, and
for his guidance in their solution.
worked under his direction.
It has been a privilege to have
I am also grateful to the chairman of
my committee, Professor Gordon Simons, and to the other committee
members, Professor Stamatis Cambanis, Professor Fred Rademaker and
Professor Edward Wegman for their assistance in this project.
I wish to thank the instructors of the many courses I attended
in the Departments of Statistics and Biostatistics; in particular I
wish to thank Professors Hoeffding, Simons and P.K. Sen for the
several excellent courses I took from each.
June Maxwell has done a typically excellent job of typing the
manuscript, for which I thank her very much.
Finally, I am indebted to the late Dr. E.A. Cornish for his
assistance in obtaining for me the C.S.I.R.O. Divisional Studentship
which has supported me during my studies at Chapel Hill.
iii
NOTATION
r r,s - set of all s-tuples of specified indices r l , ... ,r s ' say, such
that r
l
~
O, ... ,r
s
0, rl+ ... +r
2
s
=
r.
rril,· .. ,i s _
set of all distinct partitions of {il, ..• ,i } into k parts,
s
rl,· .. ,r k
with r. in the i'th part, 1 ~ i ~ k.
1
set of all permutations of il, ... ,i .
s
LE
_ summation over all even integers i in the set S; similarly
iES
refers to odd values.
r
(s)
- r(r+l) ... (r+s-l), r, s natural numbers.
natural numbers.
res) - r(r-l) ... (r-s+l), r, s
leA) - indicator function of the set A.
Rk
- k-dimensional Euclidean space.
k
R+
k
- positive orthant of R .
_ g(x. , ... ,x. ).
11
In
- g(xl,···,xl,···,xN,···,xN)
'--v----'
'---v------'
il
iN
iv
LO
CHAPTER 1
SUFFICIENCY, COMPLETENESS AND UNBIASED ESTIMATION
FOR SOME NON-PARAMETRIC FAMILIES OF PROBABILITY MEASURES
1.1
Introduction
Consider a random sample Xl"",Xn drawn from a population with
underlying probability measure P, where P belongs to some family
probability measures.
P of
One of the standard problems of statistical in"-
ference is that of finding an "optimum" estimator SeXl, ... ,Xn ) of some
characteristic Sep) of the population.
In many situations, "optimum"
is taken to mean "unbiased with minimum variance among all unbiased
estimators, for all PE: P," and P is a family indexed by a subset of
finite-dimensional Euclidean space. ep is then said to be a parametric
family.)
Apart from the fact that its use is traditional, estimators
satisfying this criterion of optimality may be attractive because they
sometimes arise when standard estimation procedures are applied to parametric families.
"-
Since the estimator SeXl, ... ,X ) is to be based on a random sample,
n
A
it is only reasonable that S be symmetric, that is, invariant under permutation of its arguments.
Our purpose in this dissertation will be to
investigate the relationship between the symmetry of an estimator and its
optimality in the sense described above, for some non-parametric families
of probability measures which, lacking the structure of parametric
do not lend themselves to some of the standard estimation methods.
f~nilies,
We begin, in Section 1.2, with the definitions of sufficiency of
a statistic and completeness and bounded completeness of the family
of probability measures induced by the statistic.
(For brevity, we
shall say that a statistic is complete, rather than refer to its
associated probability measures).
Section 1.2 continues with a review
of the results of Rao, Blackwell, and Lehmann and Scheffe, on the relationship between sufficient, and possibly complete, statistics and
minimum variance unbiased estimation.
We also take note of the rela-
tionship of bounded completeness to similar tests.
In Section 1.3, we present the original results on unbiased
estimation in nonparametric families, published by Halmos in 1946.
He
considered the family of all probability measures concentrated on
finite subsets of a given set, and showed that the order statistic
based on a random sample from one of these probability measures was
a complete sufficient statistic.
(Note that the requirement of sym-
metry of an estimator based on a random sample is equivalent to the
requirement that the estimator be a function of the order statistic of
the sample.)
We indicate the analogous results obtained by Fraser and
by Bell, Blackwell and Breiman for families of absolutely continuous
distributions on the real line, and other families of probability
measures on more general spaces.
Next, we review the results of Hoeffding for those sub-families of
some of the above-mentioned families for which a finite number of
"generalized moments" are given.
The property of completeness assoc-
iated with the order statistic of a random sample is lost; however, the
order statistic may still be boundedly complete, and there is a canonical
2
representation for those symmetric statistics which are unbiased estimators of zero for all members of the sub-family,
Section 1.3 concludes
with mention of the analogues of these results when the univariate conditions are replaced by a single bivariate condition.
Finally (Section 1.4), we characterize all functions e(p) of
probability measures
P
in these conditioned sub-families which admit
unbiased estimators with minimum variance for all
P
in the particular
sub-family chosen.
Chapters 2 and 3 contain the derivations of the bivariate analogues
of Hoeffding's results.
1.2 On Sufficiency, Completeness and Unbiased Estimation.
Let
P
space (X,S).
be a family of probability measures
Write
Xn
on a measurable
for the n-fold Cartesian product
for the n-fold product Sx ... xS
field containing Sn, and
Xl'" "Xn
P
pn
and
Sen) for
Xx ... xX, Sn
o(Sn), the smallest
0-
for the product measure on (Xn,S(n)). If
comprise a random sample drawn from (X,S,P), the order
T
= t(X I , ... ,Xn )
is just the set of observations {Xl'" .,X },
n
T
and has induced probability measure P say, on its domain space (XT,ST)'
statistic
The concept of a sufficient statistic for P, or more precisely,
for
P, was introduced by R.A. Fisher in order to summarize the inform-
ation (in some sense) available about P from Xl"" ,Xn .
shall say that a statistic
Formally, we
W - W(X) on (XW'SW) is sufficient for
P
if. for every w, there exists an Sw-measurable function Q(Alw=w) such that
J
Q(A/W=w)dpWrw)
B
3
(That is, Q does not depend on which P is chosen.)
AES, BES '
W
for all
Informally, we can see the way a sufficient statistic summarizes knowledge about P by looking at a special case, where
A
set.
A
partition
X is a collection of sets {Ay,YEf: Ay dis-
A is a sufficient partition for
U A = X}.
Y
joint,
of
X is a countable
P
if the condi-
yEf
tional probabili ty
P(oIA) is independent of
(assumed to be a sub-a-field of S).
P
for all
In this setting, a statistic
W
P if the partition of X generated by
would be sufficient for
W, {(XEX: W(x)=w), WEX }, were sufficient.
W
The following result, discovered by Rao, and independently, by
Blackwell, shows the relevance of sufficiency to minimum variance unbiased estimation.
(All estimators mentioned from here on will be
assumed to have finite variance for all
THEOREM 1.1
Suppose that
P
P.)
in
(Rao (1945, Blackwell (1947)).
an unbiased estimator of the real-valued function
= Ep [8(X)jW]
e(W)
"-
varpe
~
P
in
e(x)
If
is
e(p), then
is also an unbiased estimator of
varpe, equality holding for all
is a function of
P.
W is a sufficient statistic for
e(p)
with
P if and only if e
W a.e. [Pl.
Before Theorem 1.1 can be used to establish the existence of minimum variance unbiased estimators, we require another condition - uniqueness of
"-
e(W)
as an unbiased estimator of
unbiased estimator of zero for all
unbiased estimator of
P
in
e(p).
P,
For, if
e + g
g(W) is an
is still an
e(p), and may have smaller variance than
'"
e.
One way of eliminating the problem caused by unbiased estimators of
zero is to use the notion of completeness of a family of probability
measures.
4
A family
is said to be
(Y,T)
g
Q of probability measures
if
[boundedly] complete
0 a.e. [Q], where
Q on a measurable space
EQg
==
0, all QEQ, implies
is a [bounded] T-measurable function. Some
W
implications of this property when applied to the family p
of
==
g
probability measures of a sufficient statistic are enunciated in the
next theorem.
THEOREM 1.2
If
(Lehmann and Scheffe (1950)).
W is a sufficient statistic for
any function
P, and
W
p
is complete, then
e(p) which admits an unbiased estimator of finite variance
for all PEP has a unique uniform minimum variance unbiased estimator
A
(UMVUE)
SeW), and every measurable function of
iance for all PEP)
W (with finite var-
is the unique UMVUE of its expectation.
["Uniform"
W
means "for all PEP," and "unique" means "unique a.e. [p ]".]
If
pW is not complete, we have recourse to the following result,
which is a generalization of Theorem 1.2.
THEOREM 1.3
Let
V be the class of all measurable functions of
==
Epvlv a
W with finite
PEP, let
V
o
comprise those
VaEV
such that
a, all PEP, and let
VI
comprise those
VIEV
for which
variance for all
Epv a
(Lehmann and Scheffe (1950)).
==
a, all VaEV
a
and all PEP.
Then
UMVUE's, and we can find all functions
applying the operator
Ep
VI
e (P)
to all members of
is the class of all (unique)
which admit UMVUE's by
VI'
We now point out the relationship between bounded completeness and
similar tests.
Consider a sample
underlying probability measure
level
Ct,
X drawn from a population with
PEP, and suppose we wish to test, at
the (composite) hypothesis that
5
PEP* (cP).
Then a test
¢(x) must satisfy
f
¢(x)dP(x)
~
a,
It is sometimes possible to simplify the testing problem by requiring
the stronger condition of similarity (of size a) for the test:
r
J
¢ex) dP (x)
a,
In many cases, the testing problem reduces under this condition to one
of testing a simple hypothesis, and there may exist a test which is
most powerful among all similar tests (of a given size).
Now suppose that there exists a sufficient statistic
,
with induced family of probability measure
A test
W for
P*,
is said to
be of Neyman structure (W,a) if
a
this implies that
similar of size o..
Ep[¢(X)]
=a
for all
PEP*, so such a test is
Note that the left-hand side of the equation is
independent of P, as W is sufficient; thus the testing problem is reduced to testing a simple hypothesis for each value of w.
We may then
be able to obtain a most powerful test among all those of Neymann
structure by solving the testing problem for each
w.
The resulting
test will then be most powerful among all similar tests which have the
Neyman structure, and Lehmann and Scheffe (1950) proved .that if
is boundedly complete, all similar tests of size
structure (W,o.).
6
a
must be of Neyman
1.3
Completeness and Bounded Completeness of Some Non-parametric
Families of Probability Measures.
In 1946, Halmos published the first paper on the problem of un-
biased estimation under fairly general conditions.
we denote by
sx
X,
the smallest a-field which contains the one-point
X, and by Px
sets of
Given a space
the family of all probability measures concen-
X.
trated on finite subsets of
based on a random sample,
Halmos showed that
T
p
T(X , ... ,Xn )
l
Xl"" ,Xn ' drawn from
is complete.
x
is the order statistic
(X,Sx,P~,
some PEP X '
We shall give the proof of this
result, because the general principle used in the proof is the basis
for the proof of related results presented later in this chapter.
Lemma 1.1
(Halmos (1946)).
If Q == Q(PI' ... ,Pn) is a homogeneous polynomial in n
real variables, such that whenever
then
= 1
Q(Pl'" .,Pn)
o
5
p.
1
5 1, 1 5
i
n, and p +... +p
5
1
0, then Q must be identically zero.
P~ is complete, let g(x l ,.· .,xn ) be a symmetric
function such that E g(X , ... ,X ) = 0, all PEP. Consider any n
l
pn
n
x
To prove that
in
points
dP(x )
n
X and any probability measure
is a homogeneous polynomial of degree
n
P
such that
in
which
satisfies the assumptions of Lemma 1.1 and hence vanishes identically.
"
BY symmetry, t h e coe ff 1C1ent
0
f
Pl'"
Pn
.
1S
I ( 0
n.g
xl'" .,xn0) ' an d t h e
resul t follows.
With completeness of
every function
T
p
x
established, Halmos was able to show that
6(P) which admits an unhiased estimator of finite
7
n
A
8 (Xl" .. ,X n )
variance (for all PeeP) has a unique UMVUE
and that
A
e (Xl' ... , Xn )
is
symmetric in its arguments.
Further, each statistic
8(X , ... ,X ) synunetric in its arguments (and with finite variance under
l
n
each PEP)
is the UMVUE of its expectation.
(Since
T(X l ,··· ,X ) can
n
be shown fairly readily to be a sufficient statistic, these results
also follow from Theorem 1.2.)
Fraser (1954, 1957) gave the analogues of these results for some
other non-parametric families.
~
a
S
Let
a-finite measure on (X,S), and
be a a-field of sets of
P
x
ability measures on (X,S) dominated by
probability measures
P
(~)
~
the family of all prob-
(so that
P
x
(~)
S).
~
For the case
contains all
dP/d~
whose Radon-Nikodym derivatives
simple functions of sets in
X,
are
= Lebesgue measure on
X = R, Fraser showed that the order statistic based on a random sample
from (X,S,P), some PEP
x
(~),
is a complete sufficient statistic, and
he obtained the results corresponding to those of Halmos.
Again, the
proof of completeness is instructive because it reduces the problem to
the one considered by Halmos, and we will have cause to refer to and
use this reduction device later.
E
Suppose, then, that
g
pn
We give the proof for general a-finite
g(X , ... ,X )
I
n
is symmetric in its arguments.
respective (finite non-zero)
Hence
n
n
1
for all
n
a ,· .. ,a .
n
l
1
n
=
1
where
8
G(A.
n
1
, ... ,f\.. )
1
]n
PEP
x
(~)
be sets in
If p. 20,
1
0 , or
n
I···
I 1p 1. . . . p.1
l'
. =
1
=
0
AI, ... ,A
~-measures
E g(Xl, ... ,X)
pn
Let
=
o
, where
S with
l:s; i :s; n , and
The problem
1S
=
G(A l ,.·. ,An)' and we conclude that G(A l , ... ,An )
replaced by
AI"" ,An
now reduced to the one treated by Halmos, with g(x ,·.· ,x )
l
n
in S.
By a standard argument,
0
for all
for all
AES(n) ,
A
(n)
whence g = 0 a.e.
J gd~n =
0
[Px(~)].
The analogues of Halmos' results for general probability measure
spaces were given by Bell, Blackwell and Breiman (1960), using some
methods due to Fraser (1954).
r2 (X)
Define
the class of all probability measures on some measurable
o
space (X,S)
the class of all non-degenerate probability measures on
(X,S)
r2 2 (X) = the class of all non-atomic probability measures on (X,S)
r2 (A)
{PEr2 (X): P«A}
for some o-finite non-atomic measure
0
3
A on S
r2(T,A)
{AAIAET~} , where
=
T~
AA(C) = A(AC)jA(A)
=
{AETlo
for all
< A(A) < oo}
and
CES
Bell, Blackwell and Breiman stated the following result:
Let
Xl" ",Xn
be a random sample from (X,S,P), where (X,S) is
an arbitrary measurable space and
the order statistic
Further, if
T(X , ... ,Xn )
l
or r2 (X).
2
sufficient statistic if
X
jf
Then
is a complete sufficient statistic.
A is a non-atomic o-finite measure on
semi-algebra which generates
algebra on
PEr2 (X), r2 (X)
0
1
S, then
T(X , ... ,X )
n
l
PEr2(A,A), r2(S,A)
or
r2 (A).
3
S, and
A is a
is a complete
[A is a semi-
XEA; A is closed under fjnite intersections; and
9
A,BEA
c
with
A
c
B ~~>
Am = B and
{Ao,AI, ... ,A }
m
3:
Ai - A _ l E A for
i
The completeness property of
c:
A such that
lsi
A
A cAe
o
1
m].
<
T(X , ... ,X ) for the families of
l
n
probability measures considered by Halmos and Fraser (and for P (u))
x
is lost as soon as certain types of restrictions are imposed on the
families.
Hoeffding (1975) considered the sub-family
comprised of those
PEP
Px,k of Px
which satisfied
x
1 :s; i :s; k ,
(1.3.1)
where
are
S -measurable, and
c 1 ' ... , c k E R' .
x
These "gener-
alized moment" conditions are enough to destroy the completeness of
T(X l ,··· ,Xn ), for, if
hI" .. ,h
S(n-l)_measurable
are symmetric
k
n-l
functions which are P
-integrable for each
x
PEP
x,
k ' then the
symmetric function
k
n
I I [u.(x.)-c.]h.(xl,···,x.
l'X' l'''''x)
i=l j=l
J
JJ+
n
(1. 3.2)
1
1
1
is an unbiased estimator of zero for all PEP
x, k'
However, Hoeffding
showed that (1.3.2) characterized the symmetric unbiased estimators of
zero.
THEOREM 1.4
Let
P
(Hoeffding (1975)).
be a convex family of probability measures on (X,S )
x
which satisfy (1.3.1), and let
P
S(n)-measurable function such that
x
exist
k
symmetric
~
P k'
x,
E
pn
g =
If
0
g
is a symmetric
for all
PEP, then there
S~n-l)-measurable functions hI"" ,h k , which are
n-l
P
-integrable for each
PEP, such that (1.3.2) is satisfied for all
10
Hoeffding showed further that although
T
was not complete,
it was boundedly complete if every non-trivial linear combination
of u l ' ... ,u
k
was unbounded.
To prove Theorem 1.4, assume without loss of generality, that
c l =·· .=c k = 0, choose N>n, and let
{PI'" .,PN} be a probability
{xl' ... ,xN}
measure concentrated on
such that
O,l:s:i:S:k.
(1. 3. 3)
Then
N
N
I· .. I
(1.3.4)
il=l in=l
o.
p.
g(x. , ... ,x. )p.
11
1n
11
1
n
The system of equations (1.3.3) can be used to express
Pn+l'" .,PN
in terms of PI'" ·,Pn; substituting these expressions for
Pn+l,···,PN
into (1.3.4) yields a homogeneous polynomial equation of degree
in
n
which can be shown to hold for a set of values of
(Pl""'Pd forming a non-degenerate interval of
of Lemma 1.1
R~.
By an extension
(see Lemma 2.6) the homogeneous polynomial on the left
hand side of (1.3.4) (after substituting for
identically zero.
P + ,.·· ,PN) must be
n l
The equations obtained by setting coefficients of
distinct power products to zero may be manipulated to yield the desired
representation (1.3.2).
(P is assumed convex to prove the h.'s integrable.)
1
There are results corresponding to Theorem 1.4 and the bounded
completeness of
Let
Px
T
p
for dominated families of probability measures.
, k(~) be the sub-family of probability measures in
satisfy (1.3.1) , and let
P(~)
p
x
(~)
which
be a convex family of probability meas-
ures satisfying (1.3.1) and such that
Then a precise
analogue of Theorem 1.4 has been obtained by Hoeffding, with the slight
11
change that (1.3.2) was shown to hold a.c. [P(v)]. Similarly, hounded
completeness was shown to hold if every
of
were
P(v)-unhounded.
non~trivial
linear comhination
The analogue of Theorem 1.4
for dominated families is proved using Fraser's device.
An obvious question suggested by Hoeffding's work is the following:
if (1.3.1) is replaced by
= c
(1.3.5)
for some symmetric S(k)-measurable function
x
do corresponding results hold?
u
and some real
c ,
The main part of this dissertation
(Chapters 2 and 3) is devoted to showing that the answer is "yes,"
at least for k = 2, and provided
u(x l ,x 2 ) satisfies certain "admis-
sibility" conditions (to be defined later).
The methods used to
obtain the analogue of Theorem 1.4 are peculiar to k = 2, because they
rely on the solution of a quadratic equation.
The techniques used to
prove bounded completeness may, however, be applicable for general
Finally, we remark that, by taking
k.
\,k'
u(x l ,x 2 ) = Li=lu i (xl)u i (x 2 ) ,
the conditions (1.3.1) with general k' are seen to be a special case
of (1.3.5) with k = 2.
1.4 Characterization of Minimum Variance Unbiased Estimators in Some
Non-parametric Families.
Since the family
T
p
of probability measures considered by Hoeffding
is not complete, the question arises as to which functions
e(p), PEP,
admit UMVUE's, and which statistics based on a random sample Xl"" ,X
are UMVlJE's of their expectations.
application of Theorem 1.3.
We resolve this question hy an
(Again, take c =... =c k = 0 in (1.3.1).)
l
12
n
THEOREM 1.5
Suppose that the functions
integrable.
u ' ... ,uk in (1.3.1) are P-square
l
In order that a symmetric
S(n)-measurable function
x
be the (unique) UMVUE of its expectation, it is neces-
8(X , ... ,X )
n
l
sary and sufficient that, for all x ' ... ,XnEX,
2
o ,
(1.4.1)
Proof:
Let
8(X , ... ,Xn )
l
for all PEP
and let
variance, for all
where
8
PEP
hI"" ,h k
5
k , all PEP.
be a symmetric statistic with finite variance
g
be an unbiased estimator of zero with finite
Then
are each
1 ~ i
g
has the representation (1.3.2),
(
P n-l) -square integrable.
By Theorem 1.3,
is the UMVUE of its expectation if and only if
k
(1.4.2)
r
n
r
L L J... J
i:::l j:::l
8(x l ,···,xn )u.(x.)h.(xl,···,x.
l'x.J+ l""'x)
1
J 1
Jn
::: 0
for all
PEP
and all such functions
hI'" .,hk ,
By symmetry, (1.4.2) reduces to
r
k
(1.4.3)
r
A
) J... J 8(x l , .. ·,xn )u.(x
2 ,···,xn )dP(xl)···dP(x)
1
l )h.(x
1
n
1:::1
::: 0
for all PEP
, all hI"" ,h k
as specified above.
It is clear that
(1.4.1) is sufficient for equation (1.4.3) to be satisfied, so we concentrate on showing that (1.4.1) is necessary.
Since (1.4.3) holds for all symmetric
functions
P
(n-l)
hI'" .,h , it holds in particular for
k
leaving
13
-square integrable
h 2::: ... :::h k ::: 0,
(1.4.4)
all
PEP
A _ be an arllitrary set in
n l
Let
J... J { f
A _
s(n-l).
x
We have
S(x l ,· .. ,xn)u l (Xl)dP(Xl)}dP(X z )" .dP(xn )
n
l
==
O/(n-l)!)
I
rr z , ... , n
r··J {J
A
n-l
8(x l ,x i ""'X i )Ul(Xl)dP(X l )}
Z
n
n-l
x dP(x. ) ... dP(x. )
1
ln
Z
==
O/(n-l)!)9,IJ···J {f
(1.4.5)
S(Xl"",Xn)Ul(Xl)dP(Xl)}dP(Xz)···dP(Xn)
B9,
for some integer m ~ 1, and some S~n-l)-measurable sets
are invariant under permutations of their coordinates.
successively
hl(xZ'''''xn )
I[(x '" .,x )
Z
n
E
Bm].
==
which
B ,· .. ,B
m
l
In (1.4.4) choose
I[(xZ'''''xn ) E Bl], ... ,hl(xZ'''''xn )
==
It follows that each integral in (1.4.5) is zero,
whence
J... J {
f S(x l ,··· ,xn)u l (Xl)dP(Xl)}dP(X z )"
.dP(xn )
a
A _
n l
for all
AlE Sen-I) (and all
nx
and similarly for
u z , ... ,uk '
If we consider the family
be that (1.4.1) will hold a.e.
PEP).
Thus
The proof is thus complete.
P(~)
[P(~)].
instead, the only change will
Analogous results will also
obtain if (1.3.1) is replaced by (1.3.5).
Condition (1.4.1) is quite stringent; however, non-trivial examples
can be found in which it may he applied.
14
Thus, if
Px
is restricted
by the single condition
J xIllxl
for some
aER
1
+
>
o
a]dP
A
, then any symmetric statistic
zero outside {lxI'
:s;
8e x l ,··· ,xn )
which is
a, ... , IXnl ~ a} will be the UMVUE of its expect-
ation.
15
CHAPTER 2
CANONICAL FORMS FOR SYMMETRIC UNBIASED ESTIMATORS OF ZERO
2.1
Introduction
In this chapter we shall be concerned with deriving results corres-
ponding to Theorem 1.4 (and its analogue for dominated families) for the
sub-family Po of P
x
(PO(~)'
J J u(x,y)dP(x)dP(y)
(2.1.1)
where
its arguments.
Px(~))
the simple functions in
u
satisfying
= 0 ,
is an S -measurable function symmetric in
x
(Note that the right hand side of (2.1.1) could be taken
to be any fixed real number).
For certain types of functions
u, this problem can be reduced to
that solved by Theorem 1.4; Section 2.2 is devoted to a discussion of
this issue, and to consideration of conditions on
Xl'" .,x in X, there always exist N
n
>
u
under which. given
n, xn + l "" ,xN in X, and a dis-
crete probability measure {PI'" .,PN} concentrated on {xl"" ,xN},
such that Pl>O, ... ,PN>O and
N
I
where
N
I u .. p.p. = 0 ,
i=l j=l 1J 1 J
(2.1.2)
u ..
1J
= u(x.1 ,x.).
J
(Functions which have this property will be
termed "admissible.")
Section 2.3 contains the representation theorems which show that
if g(x , ... ,x ) is a symmetric S~n)-measurable unbiased estimator of
n
l
zero for all PEP
O
(or
PO(~))'
then g must have the form
I
u(x. ,x. )h(x. , ... ,x. )
11
12
13
In
ITl, ... ,n
2,n-2
(2.1.3)
(a.e. [PO(~)]) for some symmetric S~n-2)-measurable function
h
(pro-
vided that u is not one of the special cases discussed in Section 2.2).
Throughout this chapter and the next, the (unsubscripted) symbol
T will denote the order statistic based on a random sample drawn from
T
a population with probability measure belonging to P, say; p will then
denote the induced family of probability measures of T.
2.2
Some functional forms for u(x,y).
Unless otherwise specified, we shall assume that u(x,y) is admis-
sible.
Suppose that u(x,x)
u(xl,x l ) > O.
~
0, and, without loss of generality, take
Solving (2.1.2) for PI and selecting the larger root
(since PI > 0) yields
N
(2.2.1)
= -(
L ul·p·)
i=2
1
~
+
1
where
It is crucial to the proof in the next section that
function of P2" ··,PN when
x2""'~
not be a rational
are arbitrary, so we now examine
some exceptional cases in which it is.
17
~
(a)
If
~
=0
for all P2'" .,PN' then the coefficients of each power
product in ~2 must be zero.
Thus b .. = 0, that is,
l 1J
or, writing vl(x)
U(X,y) = vl(x)vl(y)
for all X,yEX.
In this case, (2.1,1) is equivalent to
J vldP
= 0
,
all PEP 0
,
and Theorem 1. 4 is applicable.
(b)
If, for any p 2 ' ... ,P N
then a comparison of coefficients of like power products on each side
yields
(2.2.3)
and
ciC j = uliu lj - ullu ij .
2 2
Since (c. c.) 2 _ c. c. , we have
1
J
1
J
(UliU lj - UllU ij )
2
2
= (u li
- u
2
U
ll ii
) (U
2
lj
2
u ij = UliUlj/Ull ± [(U li - u ll U ii ) (U lj
or,
(2.2.4)
U(X,y)
VI (x)v (y) ± v 2 (x)v 2 (y)
l
2
for all X,yEX, where
[(u (xIx) - u(xl,xl)u(x,x))/u(xI,x l )]
18
,~
It
2
is easily seen that a "+" sign in (2.2.4) implies that u i - u u < 0
l
ll ii
which contradicts (2.2.3) (unless ali
= 0,
which brings us back to (a)).
Hence
(2.2.5)
X,~X.
for all
We shall now show that, for u of this form, an unbiased estimator
of zero must have the form (2.1.3).
From (2.1.1),
whence
Let
P~ and P~ be the subsets of Po for which
(2.2.6)
(vI +v 2 )dP
=
o ,
J (v l -v 2 )dP
=
o ,
J
and
(2.2.7)
+
-
Then Po
U
Po = PO'
Write wI
=
vI + v 2 ' w2
=
vI - v 2 ' and let
g
be
a symmetric S~n)-measurable unbiased estimator of zero for all PEP '
O
+
For PEP O'
Theorem 1.4 implies that, for all (xl"
n
t·
,Xn)EX ,
n
(2.2.8)
g(xl,···,x n )
for some symmetric
=
L wl(x.)hl(xl,···,x.
l'x,1+ l""'x)
1
1n
. 1
1=
S~n-l)-measurable function hI' and for PEP~ , Theorem
n
1.4 implies that, for all (xl" .. ,Xn)EX
,
(2.2.9)
(n-l)
for some symmetric Sx
-measurable function h 2 ,
19
We may assume that wl(x) is not a constant mUltiple of w (x) for
2
all x (for otherwise (2.2.6) and (2.2.7) would imply JVldP = JV dP = 0
2
for all PEP, and Theorem 1.4 would be directly applicable).
The follow-
ing result (the proof of which is postponed to the end of this section)
will enable us to establish the representation (2.1.3) for g.
PROPOSITION 2.1.
Let g be a sYmmetric function which admits the
two representations (2.2.8) and (2.2.9) for all xl" .. ,xn ' where
hI and h 2 are sYmmetric, and wI and w2 satisfy
(2.2.10)
for at least one pair of points xi, x 2.
Then there exists a symmetric
function h(x , ... ,x _2 ) such that
l
n
(2.2.11)
I
wl(x. )w 2 (x. )h(x. , ... ,x. )
111
1
rrl, ... ,n
1
2
3
n
1,1,n-2
for all xl"" ,xn '
Using this result, the derivation of (2.1.3) is simple.
(2.2.11) can
also be written as
(2.1.12)
I
w2 (x. )wl(x. )h(x. , ... ,x. ).
111
1
rrl, ... ,n
1
2
3
n
1,1,n-·2
Since
wI (x. )w 2 (x. )
11
1
2
+
wI (x. )w 2 (x. )
1
1
1
2
== vI
(x. )v (x. ) - v 2 (x. )v 2 (x. ),
l 1
1
1
1
1
2
2
1
it follows that the average of (2.2.11) and (2.2.12) is just
I
u(x. ,x. )h(x. , ... ,x. )
1
1
1
1
n
2
3
1
rrl, ... ,n
2,n-2
as required.
(c)
k
Suppose u(x,y) has the form I~:lvi (x)v i (y), k
(2.1.1) is equivalent to JV~dP
= Q,
20
1 ~ ~ ~ k,
>
1.
Then
the case considered in
Theorem 1.4.
It is easy to see that, for such functions u, unbiased
estimators of
z'~ro
need not have the form (2.1.3) (choose k = 2, n = 1;
consider g(I) = vI (x)
+
v 2 (x)).
The proof in the next section will not
work for such functions because 6
2
can be shown to be non-positive for
all
- ullu 1)
.. )p.p.
1)
N
N
ull.L
L u .. p.p.
1=2 j =2 1) 1 )
~
0
by Schwarz' inequality, with equality if and only if
Vt(Xl)/L~=2V2(Xi)Pi =
In the present case, L~=lV2(xi)Pi = 0,2 = l, ... ,k.
the condition for equality in 6 2 ~ 0 is satisfied.
p, 1
~ 2 ~ k.
Hence
The discussion in (c) is an illustration of the fact that equation
(2.1.2) cannot be solved in the way desired by (2.2.1) if, for each
N
N = 2,3, ... , and (xl, ... ,xN)EX ,
N
(2.2.13)
N
L L z(x.;x.,x.)q.q.
i =2 j =2
1
1
)
1)
~ 0
where
(2.2.14)
z(w;x,y) = u(w,x)u(w,y) - u(w,w)u(x,y), all w,x,y E X
The question arises as to which conditions u has to satisfy in order to
be admissible, and in order that (2.1.2) can be solved as desired.
The
following sufficient conditions are satisfied by a wide class of functions:
21
(2.2.15)
(i)
there exist two points
Xl' x
in X such that u(xl,x )
l
2
and u(x 2 ,x 2) have opposite signs
or
(2.2.15) (ii)
u(x,x):;;: 0,
Fe 0 [:0; 0, Fe 0],
and there exist xl,x
such that u(xl,x ) < - (u(x l ,x )u(x 2 ,x Z)) ~
l
Z
Z
E
X
~
[u(xl'x Z) > (u(xl,xl)u(xZ'x Z)) ].
Observe that ullPl in (Z.Z.l) is positive if either
(2.2.16)
(i)
u 12P Z +... +ulNPN
<
0,
~Z
> 0
N
N
or
(2.2.16) (ii)
u 12 P 2 +... +ulNPN > 0,
L L u .. p.p.
i=2 j=2 1J 1 J
To derive (Z.Z.15)(i), suppose ull > 0, u
zz
< 0;
< 0 .
then a
12
> O.
If u lZ < 0, (Z.Z.16)(i) can be satisfied by choosing P3'" .,PN arbitrarily (but positive) and then choosing Pz sufficiently large and positive
with respect to P3'" ·,PN'
(2.Z.l6) (ii).
If
u lZ > 0, the same argument applies to
(Z.Z.15)(ii) is proved using (2.Z.l6)(i).
As an example, suppose X is an interval of the real line, and consider the function
2
u(x,y) = (x_y)Z - c , which restricts the family of
probability measures on (X,S ) to those with variance ~c2.
x
u(x,x)
Ix-yl >
= _c 2
clI ,
Here,
2
< 0, so (2.2.l5)(ii) requires that u(x,y) > 2c , that is,
for at least one pair (x,y) in X.
Thus in this example at
least, (Z.2.l5) (ii) is also necessary if the family is to be of interest,
for any random variable whose values lie (almost surely) in an inter.al
[a - c//2, a + c//2] must have variance less than or equal to 1~c2 , with
22
equality if and only if the distribution assigns mass 1/2 to each endpoint a - c/l:2, a
+
c/l:2.
A similar example is the function
X an interval of the real line.
u(x,y)
7
Ix-y! - c- , again with
=
(2.2.15) (ii) is just the condition that
2
there exist two points x,y in X for which Ix-yl > 2c .
Professor
Hoeffding (personal communication) has shown that this condition is also
necessary, since if X and Yare independent and identically distributed
with common probability measure P concentrated on the bounded interval
2
2
E 21X-Y1 ~ 2c , with equality being attained if
2
[a - c , a + c ], then
P
and only if P assigns mass 1/2 to each point a
~
2
c , a + c
2
The conditions (2.2.15) are, however. not necessary.
u(x,y) is a function such that u(x,x)
a 12
= U(X l ,x 2)2
~
0, all XEX, and u(x 'x ) <
l
- u(x ,x )u(x ,x ) < 0.
l l
2 2
°
b
and
2
- a
123
2
° =>
For N = 3, it can be shown from
(2.2.1) that there is a positive solution for PI' P2' P3 if
u 13 < 0, b 123 >
Suppose that
a
> 0.
12 13
u 12 < 0,
Corresponding results for
larger values of N seem to be rather more complicated.
We observe that a function
u(x,y) may be admissible, yet we may
(2.l,~
not be able to find the desired representation (either
or (1.3.2))
of an unbiased estimator of zero using either the methods developed in
this dissertation or using Theorem 1.4 of Hoeffding.
arise when the function
z
(2.2.13).
u(x,y) = I~1= IV.1 (x)v.1 (y), where {v.1 (x)} is square
Suppose that
summable for each XEX.
k =
00,
showing that 6
2
associated with
This situation may
u (see (2.2.14)) satisfies
Then the analysis in (c) above is valid with
<
° in
section are not applicable.
general, so that the methods of the next
Further, (2.1.1) is equivalent to
J vi (x)dP(x) =
so Theorem 1.4 is not applicable.
23
°
1
~
i <
00
(d)
We now turn to the case in which u(x,x)
= O.
Let
N distinct points in X, and consider the equation
be
I u .. p.p. = 0
lsi<jSN 1J 1 J
(2.2.17)
we seek conditions on
u
under which there exists a solution (Pl""'P )
N
of (2.2.17) for which
PI
>
0, .. "P
N
>
O.
Assume, without loss of generality, that
j
E
{2, ... ,N}.
(2.2.18)
u(xl,x )
j
i 0 for some
Then
PI - -
I u .. p.p./ I u .. p.
2Si<jSN 1J 1 J 2sjSN 1J J
We distinguish two situations, according as the denominator divides, or
does not divide, the numerator.
I.
lA.
I
2sjSN
ul·p·{
I u .. p.p.
J J 2si<jSN 1J 1 J
Suppose there is a point (say x 2 ) for which there exist two
points (xl and x 3 say) satisfying
(2.2.19)
By taking P2 sufficiently large, we can make the denominator of (2.2.18)
positive.
The numerator
may be written (incorporating the factor (-1))
as
(2.2.20)
Each bracketed expression in (2.2.20) is continuous in P4'" .,PN and
converges to zero as (P4"" ,PN) converges to (0, ... ,0); hence (2.2.20)
24
can be made positive for positive P2'" .,PN'
lB.
Suppose that the assumption of IA is not valid; then for any
given x o' either
Assume that
XEX.
that u(x ,x )
3 4
0
<
~
u(xO,x)
XEX, or u(xO,x) ~ 0 for all
for all
0
u(x ,x ) > 0 (so that u(xl,o)
l 2
~
0, u(x ,o)
(so that u(x ,o)
3
~
0).
0, u(x ,o)
~
4
0), and
~
2
Then
the denominator of (2.2.18) is always strictly positive, and the numerator
can be made so by choosing P3 and P4 large and arguing as in lA.
We assume that it is not possible to solve (2.2.17) along the lines
described in Part I for any N
Suppose then that
L(P2'" .,PN)
u 12 P 2
=
PI
uINP N and
Q has the form L(P2'" .,P N) °
= L~1= 2ul·c.P?
1 1 1
Q(P 2 ,···,PN)
2
Since all coefficients of
xl'" .,xN.
-Q(P2"" ,PN)/L(P2'" .,P N), where
=
+ ..• +
Then
1 and any
>
+
L2
'4'
~l~J~
NUl·c.P.P.·
1 J 1 J
must be zero, assume without loss of
Pi
generality that, for some M, 2
~
M ~ N,
(2.2.21)
Then
c.
(2.2.22)
some i,
=
J
2
~
i
~
0
c.
J
+
1
~
j
~
N)
follows from (2.2.22) that
~
j'
~
lJ
l
1
M.
If, for some j (M
M+ 1
u .. /u ·,
u..
lJ
=
0
for any i, I
~
i
~
M, it
u.· = 0, all i, 1 ~ i ~ M. Since u .. , = 0,
lJ
JJ
N, and since it is assumed that LIQ for all choices of N
25
and xl"" ,x N' it follows that
points
X
o of X for which u(xO,o)
u (x. ,0) == O.
Denote by X the subset of
o
J
O.
==
For the time heing, we shall assume
that xl'" .,XNEX-X O; thus cm+l~ 0, ... ,c N ~ 0, u
~ 0, l~isM, M+lsjsN.
ij
If 2 s ~ ~ M and M + 1 ~ k ~ N, (2.2.22) implies that u .. /u n • =
1J
x,J
(2.2.23)
and equation (2.2.18) becomes
M
I
(2.2.24)
I
i=2 j=M+l
where {u .. }
1J
(*)
N
M
u .. p.p./
1 J
1J
I
i=l
ul·p·
1 1
satisfy (2.2.21), (2.2.22) and (2.2.23),
l~i~M<j~N
It may be seen from (2.2.24) that for any subset
U
ik
=
u
jk
=
X.,
X.,
J
1
x
k
of
0, or precisely one of the
u's is zero.
We now seek conditions on u under which there exist PI
satisfying (2.2.17).
sign or zero), let
Under the conditions of IB
xl
be a point for which
u(x ,x)
2
~
0, ... ,u(xM,x)
~
0, ... ,PN
(u(x,o) of constant
u(xl,x)
~
0, all x, with
u(x ,x ) > O, ... ,u(xl,x ) > 0, as specified in (2.2.24).
l 2
M
have
>
Then we must
0, all x, so the numerator in (2.2.24)
(without the "-") is always positive.
Hence
PI
~
O.
It follows that
if u is of the type B, no univariate expectation involving u is zero,
Under the conditions of A, it is assumed that there exist three
(2.2.25)
26
>
0
The following proposition may be deduced from the analysis in II.
PROPOSITION 2.2.
For equation (2.2.17) to admit only solutions
of the type (2.2.24), for any N > 1, and any xl' ... ,xN in X - X ' it
O
is necessary and sufficient that there exist a partition of X - X '
O
Xl u X2 say, such that
(i)
(ii)
x. E Xl'
1
X. ,X.
x. E X => u
0
2
ij f
J
E Xl => u
i = 0
il
Z
1
Z
X. ,x. E X =0> u . .
J l J2
Z
J IJ Z 0
1
1
(iii)
Proof:
The conditions of the proposition are easily seen to be suffi-
cient, since (2.Z.17) reduces to (Z.Z.Z4).
(For convenience, assume
that x Z, ... ,~ E Xl' xI'~+I,··"xN EX 2 )·
Suppose then that every solu-
tion of (Z.2.17) (that is, for any N > 1, any xl" .. ,xN E X-X ) is necesO
sarily of the type (Z.2.24).
We have assumed u
lZ
>
0 under the conditions·A.
Write
Xl = {XEX-X : u(x,xl) f O} , and X = {XEX-X : u(x,x Z) = O} .
z
O
O
Then XluX is a partition of X-X . For suppose x~Xl: then u(x,x l )
O
Z
But
O.
=> u(x,x ) f 0, and since u(xl'x ) f
0,
u(xl,x ) f 0, hence by (*), u(x,x ) f 0, so XEX '
Z
Z
Z
xIEX Z' XZEX ·)
l
X-X O'
=
Similarly, XEX-X O and
Also, XlnX
Z
=
0 , for xEX l
x~X2 =>
XEX l ·
(Note that
Thus
XluX Z =
l
it follows from (*) that u(x,x ) = 0, so
Z
Z
x~XZ.
Thus
XluX Z is a parti-
tion of X-X '
O
We now show that (i), (ii) and (iii) are satisfied by Xl' X2 '
(i)
xi E Xl'
whence
x j E Xz
u .. f
1J
~=>
u il f 0,
o.
27
uIj = 0
~
(ii)
=
(iii)
a
follows from (2.2.23).
Finally, to show that the partition does not depend on Xl or x '
2
X = {x: u(x p x ) t a}. Since x EX , u
= O.
3
l 2
l3
3
We wish to show that X3 = Xl' If xEX I , u(xl'x ) t 0; since u(x I ,x ) = 0,
3
I
(*) implies u(x ,x ) t 0, so XEX . Similarly XEX => xEX , whence
I 3
3
3
I
let X EX , and write
3 2
X3 = Xl'
The proof of Proposition 2.2 is complete.
Now let
X
2
YIEX I , Y2EX2
with
u(YI'Y2)
>
O.
Then for any
xIEX I ,
EX ,
2
whence
say, where
!,;
WI (Xl) = u(x l ,Y 2 )/(U(Y I ,y 2 ))
2
!,;
w2 (X 2 ) = u(x ,y1)/(U(Y I 'Y 2 ))
2
2
•
We may extend the definitions of WI and w2 to all XEX by setting
a
a
Then we can write
for all x ,x in X.
I 2
From (2.2.24),
M
M
PI
N
L UI 1po)
(L Cop
L ul·p· - - (°2
1 o MI]]
i=2 1 1
0
1=
28
0
]= +
)
•
o ,
that is,
or
where
1
~
P
i
~
is the probability measure assigning mass Pi to the point xi'
N.
This is precisely the situation considered earlier in this section,
where it was shown that an unbiased estimator of zero had the representation (2.1.3) when the condition !udp
condition fWldP • !w dP = 0, all PEP '
2
O
2
= 0, all PEP ' reduced to the
O
(wI is clearly not a multiple
of w in this case.)
2
The results obtained in this section are summarized in the following
theorem.
THEOREM
2.1.
Let
P
be a convex family of probability measures on
(X,SX) which satisfy (2.1.1), such that PJP O' and let
symmetric S(n)-measurable function such that !gdp
x
If
u(x,y) has the form
there exist
k
I~=lVi (x)vi(y)
symmetric
n
= 0
g(xl, ... ,x ) be a
n
for all
PEP.
(some positive integer k), then
S~n-l)-measurable functions hI"" ,h k , which
n-l
are P
-integrable for each PEP, such that (1.3.2) is satisfied for all
for some Sx-measurable P-integrable functions
29
vI' v ' then there exists
2
(n-2)
a symmetric Sx
-measurable function
n 2
h(x l ,· .. ,x _ ) which is P - n 2
integrable for each PEP, such that (2.1.3) is satisfied for all
Proof:
If
u
has the form
assertion of Theorem 1.4.
I~=lVi(x)vi(Y) then this is precisely the
If
u
has the form
vl(x)vl(y) - v 2 (x)v 2 (y),
the only details not proved in this section are the P-integrability of
h and the uniqueness of h.
These points are dealt with in the proof of
Theorem 2.2 (in the next section).
Finally, we give the proof of Proposition 2.1, used earlier in this
section.
Let
xl and x 2
be points for which
We shall prove that h (x , ... ,x ) can be expressed in the form
2 3
n
I~ 3wI(x.)h(x 3 ,·· .,X. l'x, 1'" .,x).
J=
J
JJ+
n
From (2.2.8) and (2.2.9), we have
I
rr3, ... ,n
wl(x.1 )hl(x.1 , ... ,x.1 )+
4
3
n
l,n-3
(2.2.26)
I
w2 (x. )h 2 (x. , ... ,x. ) = 0
13
14
1
, ... ,n
n
l,n-3
rr 3
which expresses wI (x l )h l (x ,··· ,xn ) - w2 (x l )h 2 (x 3 ,· .. ,xn ) as a linear
3
combination of terms
hI (xl,o) and
h 2 (x l ,o).
Our first task is to
substitute for the terms hI (xI,o); suppose that wi(x j ) f 0, i,j = 1,2.
Denote the left hand side of (2.2.26) by
TO(x 3 , ... ,xn ), and, for
r = 1,2, ... ,n-3, write Tr (x 3 , .. ·,xn-r ) = TO(x 3 ,···,xn-r ,xl"" ,Xl)'
30
Then setting T (x 3 ' .. · ,xn _ l ) to zero yields
l
L
2wl(Xl)hl(X3"",xn_l,Xl) +
rr3, ... ,n-l
wl(x i )hl(x j ,,,,,x i
,xl,x l ) +
3
4
n-2
1,n-4
a ,
=
and this can be used to eliminate the terms like hI (x 3 ,· .. ,xn_l,x l ) from
(2.2.26).
However, terms like hI (x 3 ,. ",xn_Z,xl,x l ) will then be intro-
duced, and will have to be eliminated by substituting from equations
like T (x 3 ,. ",x _ ) = O.
2
n Z
Proceeding in this fashion, we are led to the
conclusion that the equation
(2.2.27)
=
a
gives a representation of h l (x , ... ,X ) in terms of the function h , wI
3
n
2
and w .
2
Explicitly, equation (2.2.27) gives
wI (x l )h l (x 3 ,···,xn ) - w2 (x l )h Z (x 3 ", .,xn )
L
rr3, ... ,n
h 2 (x i , ... ,x i
'Xl) [w 2 (x )wl(x l )
i
3
n-l
n
2 (X l )W l (xi )]/w l (xl)
W
1,n-3
(2.2.28)
+
n-2
\'
r-l
L. (-1)
rr3, ... ,n
r,n-r-2
r=Z
x ,-
\'
L.
L w2 (x.
-8 r
J n- r+ 1
rw 2 (x l )w l (x.
h 2 (x
)w l (x.
I n - r +l
i
Jn-r
,,,,,x
3
i
n-r
,xl' ... ,x l ) x
) .. ,w (x. )w (xl) +
l Jn l
) ... wl (x. )l/rw l (xl)r
I _
n
31
n
•
say, where
S
rr i n-r+ 1'" .,i n
1, r-l
r
Corresponding to equation (2.2.28) we have another equation.
(2.2.29)
based on the variable x 2 .
We now solve the two equations (2.2.28) and
D.hl(x3'·· .,xn )
Z(X 3 ,··· ,xn ; xl) j)
[ D.h (x ,·· .,x ) ] = [
2 3
n
Z(x 3 ,· .. ,xn ; x 2)
Thus
(2.2.30)
D. h 2 (x 3 ,·· . ,x )
n
-wI (x 2 )Z(x 3 ,· .. ,xn;x l ) + wI (x l )Z(x 3 ,···,xn ;x 2)
L
rr3, ... ,n
l,n-3
+
h 2 (x i , .. "x i
,x I )W[w 2 (x i )w I (xI)~w2(xI)wI (xi )] +
3
n.l
n
n
\
h (x. , ... ,X.
,x 2 )W
rr 3 , ... ,n 2 1 3
1 n _1
l,n-3
L
n-2
+
L
r=2
(_l)r
L
rr3, ... ,n
r,n-r-2
h 2 (x
i3
"",x
-1
i
[w 2 (x. )wI(x2)-w2(x2)wI(x, )]+
1n
1n
,xl, ... ,xl)W
x
n-r
n-2
\ (_l)r
\ h 2 ( xi "",x
L
L
,x 2 '· ... x 2 ) W-1 x
i
r=2
rr3, ... ,n
3
n-r
r,n-r-2
32
•
Now only the first two members of the right hand side of (2.2.30)
contain terms to be eliminated, since, for r = 2" .. ,n-2. each h 2 (o)
is coupled with a factor
wI (x.Q)'
.Q,
> 2, and is thus acceptable.
(In
the sequel, we shall denote such "acceptable" terms by one of the letters
Al ,A 2 , . .. .)
Indeed,
in which a typical term in the second member is [h 2 (x 3 ,··· ,xn_l,x l ) -1
W h 2 (x 3 ,·· "xn_l,x2)]w2(xn)'
that
If we set
x
n
=
xl in (2.2.30), we find
-1
D.[h2(x3"",xn_l,xl) - W h 2 (x 3 , ... ,xn _l ,x 2 )]
- 2W
-1
h (x.1 , ... ,x.1 _ ,x ,x )
2
l 2
3
n 2
In general, we shall show that, for s = 1, ... ,n-3,
33
.
1S
represented as
5
D.
x
I
r=O
h2(x3"",xn_s,xI"",xl' x2"",x2)(~)(-W)-r
..
•
,,,
•
s-r
r
5+1
I h 2 (x i , .. "x i
'Xl" .. ,x I ,x 2 '· .. ,x 2 )
r=O
3
n-s-l
j
5+ 1)
-r
( r (-W) w2 (X
i
)/(5+1) + A
n-s
3
x
.
(2.2.31)
For given 5, 1
for 1
$
r
~
$
5
$
n-3, set
x =.. '=X _ + = Xl in (2,2.30). and,
n s l
n
5+1, consider the general term in (2.2.30) which contains
)h (x ,
,xl'" .,X l )· We seek the coefficient of w2 (x i
2 i
n-s
3
n-r
""xi
,XI,···,X I )· [Note that the coefficient of h 2 (x.1 , ... ,x.1 _
n s
3
n-s-l
xl" .. ,x I ) is zero.] For fixed r, this is
h 2 (X i "",x i
3
5)/ rW ()r-l
( _l)r"W.w ()r-l
.w I (xl) ( r-l
+ A4 '
I xl
I xl
and so, summing over appropriate r, the coefficient is
) W.
wI ( xl'
5+1
\'L. (_l)r( r-l
s ) /r + A5
r=l
(2.2.32)
Now, for given 5 and X =... =x
1 = 1, consider the general term
n
n-s+
in (2.2.30) which contains h (x ,. ",x
2 i
i
3
n-r
,x 2 ' ... ,x 2).
We seek the
coefficient of the terms
(2.2.33)
and
w2 (xi
n-s
)h 2 (xi , ... ,xi
,xl" .. ,xl' x 2 '··· ,x 2 )
3
n-s-l
•
s-r+l
r
j
' - .
h 2 (x 3 ,·· .,Xn_S,X I '·· .,x I ' x 2 '··· ,x 2 ) .
'---w----'"" ~
s-r
r
For given r, 1 $ r $ s+l, the coefficient of (2.2.33) is
(2.2.34)
34
./
(-W)
-r
•
s+l
WI (x 2 ) ( r )/(s+l) + A6
this is just the desired coefficient of a general term on the right hand
side of (2.2.31), for 1
~
r
~
s+l, the coefficient of (2.2.34) is
~
s+l, and the coefficient for r = 0 is
supplied by (2.2.32).
Finally, for 1
~
r
r -1 s
r-l
r-l
- (-1) W (r)r.wl(x l )
[w 2 (x l )w l (x2)-wl(xl)w2(x2)]/rwl(x2)
(_l)rW-r(s) .0.
=
r
Combining this coefficient with (2.2.34), and summing over r, yields
s
I
-0
r=l
h 2 (x 3 ,· "'Xn_s,X l ,· .. ,X l '
'-.
'
s-r
and this may be transferred to the left hand side of (2.2.30) (evaluated
yield the left hand side of (2.2.31), establishing the desired relation.
\fuen s = n-3, (2.2.31) gives
'-~----'
n-2-r
and we wish to show that the right hand side of (2.2.35) is zero.
Con-
sider (2.2.30) when s = n-2, that is, x =... =x 3=x l .
n
All terms
h 2 (X
For fixed r, the
term
i
, .. "x
3
,xl'" .,x ) have coefficients zero.
l
n-r
h 2 (x l ,· .. ,x l ' x2 ' ... ,X 2) has coefficient
i
n-2-r
r
35
Thus (2.2.30) evaluated at x =",=x = xl reduces to
n
3
-0
n-2
\'
L (-W)
-r
r=l
h (x ,···,x ,
2 l
l
~
n-2-r
whence
n~2 n-2
L
r=O
(
r )(-W)
-r
r
h2(X~'
.... ,x , x 2 '· .. ,x?) = 0 .
n-
r
~
r
This completes the proof of the proposition if wI (xl)
w2 (x )
l
t
0, w2 (x 2 )
t
t
0, wI (x )
2
t
0,
O.
The proof is unaffected if either w (x ) = 0 or w (x ) = 0 (but
2 l
2 2
not both, as 0
to).
If one of wI (xl) or wl (x 2 ) is zero, we may inter-
change the roles of the variables xl and x 2 ' the functions wI and w2 '
and the functions hI and h 2 .
The final case to consider is wI (x ) =
2
w2 (x l ) = 0 (or equivalently, wl(x ) = w2 (x 2 ) = 0).
l
Here the conclusion
follows directly from (2.2.28) as it provides a representation of
I IT 3 , ... ,nw2 (x.
1
1,n-3
2.3
3
)h(x. , ... ,x. ).
1
1
4
r
Canonical forms for symmetric unbiased estimators of zero.
For convenience, we restate one of the special forms for u(x,y)
discussed in the previous section.
F: z(w;x,y)
N
= u(w,x)u(w,y)
s::
0 for all N = 2,3, ... , xl' ... , XNEX
J=
THEOREM
(X,Sx)
satisfies
N
L L z (xl; x.1 , x.)
q. q.
. 2 . 2
J 1 J
1=
- u(w,w)u(x,y)
2.2.
Let P be a convex family of probability measures on
which satisfy (2.1.1), such that
P~PO'
is admissible and does not satisfy condition F.
36
and suppose that u(x,y)
If g(x ,·· .,x ) is a
l
m
symmetric S(n)-measurab1e function such that !gdp
x
n
0 for all PEP,
=
then there exists a symmetric S(n-2)-rneasurab1e function h(x ,· .. ,x 2)
1
x
n2
n
which is p - -intcgrab1e for each PEP, such that (2.1.3) is satisfied
r+l
Lemma
2.1.
- (r+l)
tr~ (t~l) ~
r
\'r
t r
Lt=O(-l) (t) (k+t-l) (s) (k-t) -
Lemma 2.2.
1 s s < r-l
0
_I (_l)r-lrl
s
(r) [k(k-l) (s-r+l)
l
s = r-l
+ (r-k)k(s_r+l)] (-1)
for 0 s r s k-l.
\'r
Lt=O (-1)
r-l
(r-l)!,
r-l
<
s = k
(r-k)k!
Proof:
1,2, . .. .
t
r
(t) (k+t-l) (s) (k-t) =
from which the result follows.
37
s
<
k
•
Lemma 2.3.
Let R(z) be a rational function of the complex variable
z, analytic in a domain D containing the open interval (a,b) of the real
line, and let I(z) be an irrational function of z, analytic in a domain
D' which also contains (a,b).
(2.3.1)
R(z)
If
= I(z),
z E (a,b) ,
then R and I must each be identically zero throughout the complex plane
t.
[For the purposes of this statement, the function which is identi-
cally zero in t may be considered either rational or irrational.]
If P(z) is the denominator of R when R is expressed as the
Proof:
(simplest) ratio of two polynomials, we may multiply both sides of
(2.3.1) without affecting the statement of the proposition in any more
than a trivial way.
Hence, assume that R is a polynomial.
analytic in the open plane Izi <
since R(z) = I(z) for zE(a,b)
c
00
and has only a pole at
Then
00
R is
Therefore,
DnD', we can extend this equality by
analytic continuation to all z EDuD', that is, to Izi <
00.
Now suppose that R is of precise degree m, and let c be a point
outside (a,b).
Then R(z)/(z-c)
m
=
I(z)/(z-c)
m
for zE(a,b), and we may
extend this equality by analytic continuation, to all finite z except
z = c.
Further, R(z)/(z_c)m is analytic at Izi =
m
I(z)/(z-c).
Izi <
00
,
00,
hence so is
It follows that I(z) has no essential singularity in
and that I(z)/(z_c)m has no essential singularity at Izi =
00
,
that is, I(z) has no essential singularity in t, and it is singlevalued
in t.
Hence, by a theorem in Phillips (1957), I(z) is a rational function
in t.
Thus I(z) -
a
in t.
38
Lemma
u(y,y)
t
Let xl •...• xn, y be any n+l distinct points in X, with
2.4.
0, and define
(2.3.2) (a)
(2.3.2)(b)
g
(r)
(xl'"
l:S;r:S;n ,
.,X
n-r )
(2.3.2)(c)
(2.3.2)(d)
h
(r)
(xl' ... , X n- 2 -r ) - h
If g(x , ... ,x ) admits
l
n
(r-l)
(xl.···,xn- 2 -r .y),
a representation of the form (2.1.3), then
n [n/2]
{
(r)
D(r,s;n)g
(x.
, ... ,x.
) x
r=2 s=l 5
1 r _l
1n~2
I
(2.3.3)
s-l
r-2
[rr u(x.
,x. )][ rr
u(x. ,y)][u(y,y)]-(r-s)}
t=l
1 2t _ l 1 2t
t=2s-l
1t
x
h
5 = {C 1' " .. ,I. _)
were
1
n 2
L I
E
l , ... ,n-2
rr n-r.2s-2,r-2s
1j an d t h e quan t"1t1es D( r.s;n )
do not depend on xl" .. ,xn _2 or y.
Proof:
From (2.1.3) and (2.3.2) (a) - (d) we obtain the following reeur-
renee relation:
(2.3.4)
h
(r)
{g
(xl,···,xn- 2 -r )
I
- r
5
-
5
(r+2)
1, r
(xl,···,xn- 2 -r ) +
(r+l)
u(x. ,y)h
(x., ...• x.
) +
1
12 1n-2-r
1
u(x.1 ,x.1 )h (r+2) (x.1 , ... ,x.1 _ _ ) }
1
2
3
n 2 r
2,r
t
L
{(~)u(y,y)} ,
where 51
5
2,r =
,r
{C'
= {(il,···,i
1 1 ""
.
,I n _2 _r
)
2 )
n- -r
E
E
rrll" ·3·,n-2-r}
,n- -r
rr l , ... ,n-2-r}
(2,n-4-r)
39
and
The proof may now be accomplished by using (2.3.4) to trace a path to
any particular term in braces, in (2.3.3).
Alternatively, (2.3.3) may
be proved using (2.3.4) and induction: this is straightforward but
laborious, and is omitted.
Lemma 2.5.
= 0,
Suppose that u(x,x)
and let
xl" .. ,xn 'Yl'Y2
be any n+2 distinct points in X, with u(Y l 'Y2) f O.
Define
(2.3.5)(a)
g
g
(r-l,s)
(xl,···,x
n-r-s
,y l ),
(r,s)()
xl"",xn _r _s -
2:O;r:O;n,
l:O;s:O;n,
g
(r,s.l) (
(2.3.5) (b)
)
xl ' ... , xn-r-s , Y2 '
l:o;r:O;n,
2:o;s:O;n,
(2.3.5)(c)
h
h
(r, s)
(xl""'xn- 2 -r-s ) -
(xl'···,xn- 2 .r-s ,Y l )
2:O;r:O;n~2,
J
l
h
(2.3.5)(d)
(r-l,s)
(r, s-l)
l:o;s:o;n-2,
(xl,···,x n ...r.s
2
,Y 2 )
l:O;r:O;n.2, 2:O;s:O;n-2.
If g(x , ... ,xn ) admits a representation of the form (2.1.3), then
l
\L \L [
\ [m/2]
\ g(r,m-r) (x.
L
L
, ••. ,x.
)
m=2 S £1+£2=3 r=l
1m_l
1n_2
£1'£2>0
r-l
s-l
x I H(r,m-r,s;n){[ IT u(x.
,x.)] x
s=o
t=l
1 2t - l 1 2t
n
=
(2.3.6)
r+s-l
x [
where S = {(i
1"'"
x
m-2
s -m+ 11 )
u(x i ,Y£)][ IT u(x i ,Y£ )] x [u(Y l 'Y2)]
J
t=2s-l
t
1
t=r+s
t
2
i)
n-2
IT
E
nl , ... ,n-2
n-m,2s-2,r-s-l
40
} and the quantities
H(r ,m-r, s; n) do -not depend on Xl"" ,xn _2 ' or y1 or y 2'
Proof:
From (2.1.3) and (2.3.5) (a) - (d) we obtain the following recur-
rence relation:
h
(r,s)
_
(xl' ... , Xn- 2 -r-s ) (r+l,s+l)
{g
=
(Xl'" .,xn- 2 -r-s ) +
\
(r+l,s)
(x.1 , ... ,X.1 _ _ _ ) +
S r L u(x.1 ,yl)h
2
1
n 2 r s
l,r,s
+
I
u(x. 'Y2)h
s l,r,s 1 1
s
\
L u (x.
-
S
where S
S
, X.
11
2,r,s
)
(r,s+l)
(x.1 , ... ,x.1 _ _ _ ) +
2
n 2 r s
h (r+l,s+l) (x.
12
{C 1' 1 , ... ,1. n- 2 -r-s )
E
)
1n- 2-r
13
)
{ ( 1. , ... ,1. 2
1
l,r,s =
n- -r-s
2 ,r,s =
, ... , X.
-s
rr 1l , ...3 ,n-2 -r-s 1J
E
}
and
,n- -r-s
rr 2l , ...4 ,n-2-r-s}
The rest follows by
,n- -r-s
analogy with the proof of Lemma 2.4.
Lemma 2.6.
(Hoeffding (1975)).
A homogeneous polynomial on RN
which is zero on a non-degenerate N-dimensional interval is identically
N
zero on R .
Proof of Theorem 2.2.
The proof is divided into two parts, according as
u(x,x) is not, and is, identically zero.
Part I:
Let
u(x,x) 1 0;
assume
u(xl,x l ) >
°
Xl'" "xn +l be n+l distinct points in X, and let xn +2 "" ,x N
be N-n-l other distinct points in X such that there exists a probability
measure {PI"" ,PN} concentrated on {xl···· ,x N} and belonging to P,O'
41
Then from (2.1.1),
N
I
(2.3.7)
N
I
u .. p.p. = 0 ,
i= 1 j =1 1J 1 J
and from (2.1.3),
(2.3.8)
= 0 .
The argument will be conducted through the following stages.
I (i).
Solve (2.3.7) for PI and substitute into (2.3.8), yielding an
equation of the form
(2.3.9)
P ,
L
where
~
2
, and P
are homogeneous polynomials of respective degrees
R
n, 2, and n-l, in P2, ... ,PN'
Lemmas 2.3 and 2.6 imply that PL and PR
are identically zero, hence the coefficients of individual power products
in each are zero.
The equations obtained by setting these coefficients
to zero will yield a representation of the form (2.1.3) for g(x 2 ,· .. ,xn + ).
l
I (ii).
The function
h
determined implicitly in lei) will contain a
dummy variable, Xl; we show that h is independent of the choice of the
dummy variable.
I (iii).
Equation (2.3.3) in Lemma 2.4 implies that h is P-integrable
for each PEP ; we show that h is P-integrable for each PEP,
O
I (i).
Solving (2.3.7) for PI and selecting the larger root yields
+ ~
42
,
where
A2
0
~L
=
. 2
1=
al·p·2
1 1
2
+
\
L
b · .p.p.
2"
<N l 1J 1 J
~l<J-
The cases ('.,2 = 0 and
('.,2 a perfect square have been dealt with in Section 2.2, and ('.,2 < 0
has been excluded from consideration.
So we assume that ('.,2 > O.
We rewrite (2.3.8) to isolate the terms involving p
I
--
i
n
PI
l: -.,
1.
r
i=l
L
+
n-i,N-l
+
(2.3.10)
n
--multiply both sides of (2.3.10) by u ll ' and substitute for uIIP from
l
(2.3.8):
then
n
n-i
un
i=l1!
l:
+
n
(2.3.11)
(n
+ U ll
r
L lr
n,N-l
')
r2
rN
Jg*O
P
,,·P
N
.,rN
r 2 ·· .rN 2
2 ,··
Using the identity
i
n
.
L c. I d .. ('.,J
. 1 1 . 0 1J
1=
J=
n
-
. n
l: ('.,J I d .. c.
. 0 . . 1J 1
J=
l=J
with
n-i
un
c. = I (i <0)
.I
1
1.
and
_
d .. -
1J
.
1
J
(.)(-1)
.. ( N
1-J
') i-j
l£=2\ ul~,P.R,J
L
(2.3.n) becomes
43
'
=
0 .
n-i
. n
U
N
,i-j
1
1-)
ll
I 6) I {I(i>O) -'-,(.)(-1)
I
ul£P£J
x
j=O i=j
1.
)
£=2
.
n
"l
x
= 0 ,
whence
n-i
"n
U
. "IN
,i-j
1.
1-)
ll
I E 6) I {I(i>O) -.-,(.) (-1)
I ul£P£J
n
j=O
i=j
1.
)
£=2
x
(2.3.12)
1 I
0 j l
= -6
{I(i>O)
6 j=l
i=j
n-i
u~~
1.
x
which has the form (2.3.9).
Equation (2.3.9) is valid for a set of
values of (P , ... ,P ) forming a non-degenerate open interval
2
N
~_lcR~-1
.
,,2.
S1nce u 1S not the square of a polynomial in P2"" ,P , at least for
N
(P 2 , ... ,P N)
E
N-l
R+
N-2
- R+
' and is not identically zero in this set, it
follows from Lemma 2.3 that PL (P 2 '···.P N) and PR(P2'" .• PN) are each
identically zero for (P2""'P N) E AN_I' and hence. by Lemma 2.6,
N-I
identically zero on R
. Therefore the coefficients of all power
r
products
r
P2 2 . "P N appearing in PL and PR must be zero.
N
44
We shall proceed according to the following schema to derive the
desired representation of g23 ... n+l·
(A).
Setting the coefficient of P2" 'Pn in PR to zero gives an equation
of the form
o
= u
n-l
11
n-l
+
g
123 ... n
+
t-l
gl ... I'1 ... 1, [K(n,t)u ll u l 1· 1 .,.u l 1' +
2
t=l S 1
t
t+
n
L
\
L
(2.3.13)
+
where Sl
and k,
,
J 3 · .. I n - t
is symmetric in its arguments.
Similar representations
may be obtained for g12
... n -1 ,n+ 1,···,g134 ... n+ l'
(B) .
Setting the coefficient of P2 ... Pn+l in PL to zero gives an equa-
tion of the form
o=
u
n
g
+
11 23 ... n+l
n
+
L L gl . . . I'1
t=l S3
+
2
t-l
. [K (n,t)u ll u l ·
1
.,.1
t
t+ 1
.. ,u l ·
1
n+ 1
+
(2.3.14)
, ... ,n+l}
' ) E II 2t-l,n+l-t
S
{C'J ,· ··,I. + - ) E
S3 =, {C 1' ", .,In+l
h
w.ere
' 4 =
2
l
n l t
lt +1,···,1 n+ 1 1
d k+
.
. . .
II 2
j ' an
,
,
lS symmetrlc ln ltS arguments,
1
,n- -t
J2" ·I n + l _t
(C) .
Using (2.3.13), substitute for the terms g123 .,.n ,··"g134 ... n+ 1
in (2.3.14); it will be found that the resulting equation will have the
t-l
same form as (2.3.14) except that all terms like u ll Ul,t+l" .ul,n+l
45
will have cancelled out.
The desired result will have been obtained.
Returning to (A), and recalling that g123 ... n
= gi ... 10 ... 0'
we
obtain g123 ... n from PR [see right hand side of (2.3.12)] by setting
i = rZ= ... =r
n
= 1, rn+l= ... =r
N
= 0; since j
~
i, j = 1, also.
So
g 123 ... n is contained in the coefficient of P2" .Pn' which we now
determine.
In order not to obtain the power of any p. exceeding 1, it is
J
~
necessary that 0
rs
~
1 for 2
~
s
~
n
(r + =... =r = 0).
n l
N
the only repeated x-value, (if any), in the argument of
be xl'
To find the coefficient of a typical term g
Thus
g~
1r 2 " .rN
1. .. 123 ... t
will
in the
coefficient of P2 ... Pn' choose
Then
L
r n-l,N-l
contributes
(2.3.15)
it remains to find the coefficient of p t+ l"'Pn in
n-i
.
"
N
..
u l1
1
1-J \'
1-J
L 6 J - L I(i>O) -.-,- (.)
(-1)
(L uup~)
.
j=l
i=j
1.
J
£=2
nO
(2.3.16)
. In
This last expression contains several terms irrelevant to the computai
is fixed, so (2.3.16) may be reduced to
t-l
u
ll
(n-t+l)! • I(n-t+l > 0) x
(2.3.l7)(a)
n-t+l
(2.3.17)(b) Xj~~
+ 2(u
{[2(u l ,t+l u l,t+2- u ll u t +1 ,t+2)Pt+lPt+2+'"
u -u u
)p
p] (j-l)/2 x
l,n-l In 11 n-l,n n-l n
46
(2.3.17)(c)
x (n - ~ + 1) ( -1) n - t + 1 - j
J
L
r
x
n-t+l-j ,n-t
r
r r
r
(n-t+l-j)!
t+l
n t+l
n}
x
,
rt+l ....
r n ,u
. l ,t+ 1" . u l n Pt +1 ... p n
If, for example, u ll u t +l ,t+2 is chosen from (ul,t+lul,t+2
u ll u + ,t+2)r) in (2.3.17)(b), an overall term of the form
t l
u t + 1 , t +2k l , t +3 , t +4 , ... ,n will be obtained from (2.3.17) (and such terms
have then been determined sufficiently well for our purposes).
So we
consider only the first member of an expression like (*) in order to
calculate K(n,t).
For fixed j, we may extract the factor
t-l
u
I(n_t+l>O)2(j-l)/2(n-~+1) (_l)n-t+l-j(n_t+l_j)! =
(n-t+l)!
J
(2.3.18)
In (2.3.17)(c) the number of distinct products of the form u · ... u .
l1
l1
1
n-t+ 1"
-J
is
(
(2.3.19)
n-t)
~n-t+l-j
=
(n-t)
j -1 '
where (i , ... ,i _ ) is a permutation of (t+l, ... ,n).
l
n t
For a fixed product
of this type, we need the multiplicity of the complementary product
u ·
l
1
n-t+ 2-J'
... u ·
l1
in (2.3.l7)(b).
n-t
'-1
There are (J
2
) different pairs ulpu
lq
of interest in (2.3.l7)(b),
giving (j;l) possibilities for the first choice.
Once a pair ulpu
lq
has
been chosen, no other pair containing either u
or u
is eligible for
lp
lq
. 3
selection; hence there are only (J; ) pairs available for the second
choice, and so on.
The multiplicity of u ·
l1
47
n-t+ 2"
-J
... u "
is thus
l1
n-t
(2.3.20)
Finally, summing over appropriate j, and taking into account (2.3.15),
(2.3.18), (2.3.19) and (2.3.20), the coefficient K(n,t) of
ut-lu
u g
in (2.3.13) is found to be
11 l,t+l'" In 1. .. 123 ... t
(2.3.21)
I
(n-t+l>O) (-1)
n-t+l
\'0
n-t+l
L
j=l
Returning to (2.3.12), we carry out step (8) analogously to the
method used for step (A).
g
occurs in the second member of
23 ... n+l
PL as the coefficient of u~ln!P2"'Pn+l' The rest of the overall
coefficient of P2' ··Pn+l in PL occurs in the first member of PL and
is linear in the functions gIl
...
1"
1
1
2 3
1"
", t
To find the coefficient of
the typical term gl ... 123 ... t' set r 2 =···=rt = 1, rt+l=.··=r N = 0,
i = n-t+l, with 0
~
j
~
n-t+l, j even.
of the first member of P
L
u
(2.3.22) (a)
A suitably reduced version
is
t-l
ll
(n-t+l)! • I (n-t+l>O) x
n-t+l
(2.3.22)(b)
x
j~~
+ 2(u
(2.3.22) (c)
{[2(u l ,t+l u l,t+2
u
l n l ,n+
1 - u
'/2
)p P
]J
x
11 n,n+l n n+l
u
x (n - ~ + 1) (-1) n - t + 1- j
J
I
r
x
n-t+l-j,n-t+l
r
r
r
r
(n-t+l-j)!
t+l
n+l
t+l
n+ll
x rt+l! ... r + ! u l ,t+ l' .. u l ,n+ lPt +1 ... Pn+ 1 J
n l
•
An argument similar to that following (2.3.17) enables us to con48
centrate on finding K+(n,t).
For fixed j, we extract from (2.3.22)
~
the factor
t-l
u 11
I (n _t +1>0) 2j / 2 (n - ~ +1) (-1) n - t +1- j (n _t +1_j )! =
(n-t+l)!
J
(2.3.23)
The number of distinct products u ' ... u '
l1
l1
1
n-t+ l-J'
in (2.3.22)(c) is
( n-t+1 ) _ (n-~+l)
n-t+1-j J
(2.3.24)
and, for a fixed product, the multiplicity of the complementary product
u '
11
... u '
.1
1
n-t+ 2 -J
in (2.3.22)(b) is
n-t+ 1
'I/2 j / 2 .
J,
(2.3.25)
Summing over appropriate j, and taking into account (2.3.23) - (2.3.25),
the coefficient K+(n,t) of
u1~lUl,t+l" .ul,n+l in (2.3.14) is found to
be
n-t+1
(2.3.26)
I(n-t+l>O) (_1)n-t+1
IE
(n-~+l)
j=O
J
To accomplish part (C), observe that upon substitution for
g123 ... n ,···,g134 , ... n+ 1 in (2.3.14), precisely n-t+l of these functions
involve the function gIl ... 123 ... t· So the overall coefficient of
t-l
u 11 u l,t+l'" u l,n+l g 11. .. 123 .. ,t after substitution will be (using
(2.3.21) and (2.3.26))
n-t-t:l
I(n-t+l>O) (_l)n-t+l LE (n-~+l) +
j =0
-(n-t+l)I(n-t+l>O)(-l)
J
n-t+l
n-t+ 1 ,,0
L
j=l
49
which is identically zero, by Lemma Z.l.
This completes the proof of
I (i) .
I(ii).
In lei), the function h (determined implicitly) involves
a dummy variable, xl'
To show that the value of h does not depend on
the choice of dummy variable, let xl" ",xn be n arbitrary points, and
hI(Xl,·.·,X n _Z)' hZ(x I " ",xn _Z) the functions obtained in (Z.1.3)
when Yl' yz respectively, are used as dummy variables (u(Yl'Yl) f 0,
u(yZ'YZ) f 0).
Subtracting the two representations of g(x l ,· .. ,xn ),
and writing h+(x I ,· ",x _Z) for the difference hI (xI' ... ,x _Z) n
n
hZ(x l " ",xn _Z), we have
(Z.3.27)
L
0 =
IT 1 , ••• ,n
u . . h:
1112 1
.
3 ... 1 n
Z,n-Z
Suppose that u(xl,x l ) f 0 (for example, choose xl = Yl):
+
that h (xZ""'xn _l )
+
= O.
+
From (Z.3.Z7), ullh ll ... 1
we shall prove
= 0, whence
+
h il ... 1 = 0, and Cn_I,Iulihii ... 1 + Cn_I,ZullhII, .. Ii = 0. whence
h+ • . . l'1 = 0. for 2 s i s n-I.
II
+
hI
...
l'
1
1
"
.
.1
=
0 for all j-subsets of Z, ... ,n-l. then, since
j
il,····i j +I
Sz = IT 2 ,j_l
), it follows that
(where Sl
h
+
1. ..
l'
l
n-Z. h+
.
l
In general, if we have shown that
... 1 +
j l
Z... n-l
= 0
= 0,
for all (j+I)-subsets of 2, ... ,n-l.
as required.
50
When j + 1
I(iii).
To prove that h is P-integrable for all PEP, consider (2.3.3)
in Lemma 2.4 in which a typical term in the representation of
(2.3.28)
(up to a multiple D(r,s;n)[u(y,y)]-(r-s)).
If A = {y}, and P
y
y
is a
probability measure in Po concentrated on y and N-l other points, then
(2.3.28) may be rewritten as
M(y).
J... J
A
Y
x
g(x r- 1" .. ,xn- 2'Yl'" .,yr )dP y (Y l ) ... dP y (y r )
x
A
Y
l'-
r-2
,- s-l
IT u(x2t_l,X2t) I
IT
- t=l
- - t=2s-I
where M(y) is a power of P (A ).
y
J
A
Y
The P-integrability of h for any PEP
Y
may now be deduced as in Hoeffding (1975), using the convexity property
of P.
This completes the proof of Part T.
Part II:
Let x , ... ,x + be n+2 distinct points in X, and let x + , ... ,xN
l
n 3
n 2
be N-n-2 other points in X such that there exists a probability measure
{Pl""'P N} concentrated on {xl'" .,xN} and belonging to PO'
from (2.1.1),
(2.3.29)
L L u .. p.p.
l~i<j~N 1J 1 J
and from (2.1.3),
51
= 0
Then
(2.3.30)
=0
(Note that u(x,x)
=> g(x, ... ,x) - 0.)
As in Part I, the proof will
be in three stages.
II(i).
Solve (2.3.29) for PI and substitute into (2.3.30), yielding
a homogeneous polynomial equation in P2, ... ,PN'
The equation holds for
N-l
(P2""'P N) belonging to a non-degenerate subset of R+ ; hence by
Lemma 2.6 the coefficients of all power products of the p's are zero.
The equations obtained by setting these coefficients to zero, used in
conjunction with those obtained by solving (2.3.29) for P2 instead of
PI and substituting for P2 in (2.3.30), will yield a representation of
the form (2.1.3) for g(x 3 ,· .. ,x + 2).
n
II(ii).
The function h determined implicitly in II(i) will contain
two dummy variables, xl and x ; we show that h is independent of the
2
choice of dummy variables.
II(iii).
We show that h is P-integrable for each PEP.
II(i). From (2.3.29),
(2.3.31)
PI = -(
N
I I
u .. p.p.)/l
2~i<j~N 1J 1 J
I
~=2
U
1n Pn) •
N
N
Again, we exclude the possibility that \N
u P I
L~=2 l~ ~
as this is dealt with in Section 2.2.
\\
u P P
LL2~i<j~N ij i j
From (2.3.30). we have
i
n-I PI
(2.3.32)
I 1..
+
I
i=l
+
r
L lr
n-l,N-l
n
) g*O r
2 ,·· .• r N
52
r
P
2
2 ·· .rN
2
r
... P
N
= 0 .
N
Multiply throughout (2.3.32)
(2.3.31) for
(L~=2u12P2)Pl
and substitute from
then
(2.3.33)
x
~
r.
L
n-l,N-l
+ (
~L
n!
r2
r N}
r'
r f g~
P2 ... p
N
2···· N' lr 2 ·· .rN
p )n-l
In n
2=2 N N
r
~
u
L
n-l,N-l
+
(
n)
r2
rN
gO*
P2
..
·PN
r 2 ,·· .• r N r 2 ,··rN
= 0 .
Equation (2.3.33) is a polynomial equation homogeneous in (P2, ...• PN)'
which is valid for a set of values of (P3" .. ,PN) forming a non-degenerate
.
l 'ln RN- l
open lnterva
+
Thus the coefficient of each distinct power
product must be zero.
In (B) of Part I(i), we were concerned with determining the functions
in equation (2.3.14) which had to be eliminated from the representation
of g 23 ... n+l·
Correspondingly, our first task here is to find a repre-
sentation for g34 ... n+2 somewhat analogous to (2.3.14) for g23 ... n+l;
the end result is (2.3.46), which comprises the terms to be eliminated
from the representation of g34 ... n+ 2'
Now g34 ... n+2 occurs only in the second member of (2.3.33). where
n-l
it is part of the coefficient of P2 P3" .Pn+2'
It will transpire that
n-l
knowledge of the coefficients of power products like P2 P3" .Pn+2-t
(with P2 the only p appearing to a power higher than 1) will be sufficient to establish the representation (2.1.3) of g.
So we shall
rewrite the left hand side of (2.3.33), omitting all terms involving
53
Pn +3 "",P N and all terms involving powers of P3"",P n +2 higher than
the first.
Note first that
(2.3.34)
( I I
u·kP.Pk)i =
Isj<ksN J J
N
(P 2 I U 2k Pk +
k=3
'I~
= 1. ~
£-0
£
P2
Ifl \'
1
u 2j ... u 2 j Pj ... Pj
x
-rr3, ... ,n+2
I
£ I
£L
£
and
(2.3.35)
n-l
=
I
m=O
x {
[(n-i-I)!/m!]u~2P~
I
rr3, ... ,n+2
n-m-i-l
UI £
I
x
... U I £
. P£ ... P£
.} +
n-m-l-l 1
n-m.l-l
2
n-i-l
n.i-l
)
+ (P 3 ,·" 'P n + 2 ,P n + 3 ,· .. 'P N
Using (2.3.34) and (2.3.35), the left hand side of (2.3.33) may be
reduced to
54
n-1
(2 . 3 . 36 ) (a)
.
(I (-1) 1 x
i=l
(2.3.36)(b)
f
x
i
-
l ~ P~ I
£-0
l
L
u 2j · .. u 2j Pj ... p j I
- rr3, ... ,n+2
1
£ 1
£ -
x
£
x ,-
L
rr3, ... ,n+2
2i-2£
J}
uk k ... uk .
k.
Pk' .. Pk.
1 2
21-2£~1 21-2£
1
21-2£
x
n-i-1
(2 . 3. 36) (c)
x {
L [(n-i-1)!/m!]u~2P~
x
m=O
x
1-
L
- rr3, ... ,n+2
n-m-i-1
(2.3.36)(d)
n-1
(2.3.36)(e)
+ ({
x
1-
L [(n-1)!/m!]u~2P~
x
m=O
L
- rr3, ... ,n+2
n-m-1
(2 . 3 . 36) (f)
x
n-1+t
We shall now find the coefficient of P 2
P3 · .. pn+ 2 -t '
a
~
t
~
n-1.
For t = 0, the coefficient will involve g34 ... n+ 2' and other terms like
gij1 ... 10 ... 0 in which x k appears in the argument at most once,
3 ~ k ~ n+2.
In particular, terms like gOj1 ... 10 ... 0 which appear in
n
the coefficient of p 2 - 1P 3' .. pn+ 2 with g34 ... n+ 2 will themselves be detern-1+j
mined from the coefficient of P2
P · .. p 2 ..
3
n+ ~J
ss
From (2.3.36)(e) select a representative term
(2.3.37)
[(n-l)!/m!]u~2P~
I
rr3, ... ,n+2
n-m-l
Ul~ "'Ul~
1
n-m-l
p~ ... p~
1
n-m-l
and from (2.3.36)(f) the complementary term
(2.3.38)
n-l+t-m
[n!/(n-l+t-m)!]p2
p~
... p~
g(x 2 , ...
n-t
n-m
,x2'x~
, ...
n-m
,x~
)
n.t
where {~l""'£
I} and {~
, ... ,£ t} are disjoint subsets of
n-mn-m
n{3, ... ,n+2-t}.
Multiplying (2.3.37) by (2.3.38) and summing over appro-
n-l+t
priate m gives the coefficient of P2
P3 ... Pn+2-t from (2.3.36)(e)x(f)
as
(2.3.39)
n-l
\ (n-l)!
n!
m
\
L
m!
(n-l+t-m)! u 12
L
m=t
rr3, ... ,n+2
n-m-l,m-t+l
x g(x 2 , ...
,x2'x~
, ... ,x£
n-m
).
n-t
Turning now to (2.3.36)(a)x(b)x(c)x(d), (2.3.36)(b) and (2.3.36)(c)
can contribute at most n-l P2's and (2.3.36)(d) at most n-i P2's.
there are at most 2n-i-l P2's, so 2n-i-l
~
n-l+t, i
~
n-t.
But i
Thus
~
n-l
also, hence
~
i
(2.3.40)
min(n-t, n-l) .
For such i, select P~ from (2.3.36)(d).
~
at most n-l P2's, n-l+t-j
(2.3.41)
t
Since (2.3.36)(b)x(c) contributes
n-l, that is, j
~
j
~
~
t.
Hence
n-l .
j
Suppose that P2 P i+j-t+3" .Pn-t+2 is selected from (2.3.36) (d). It
n-l+t-j
remains to find the coefficient of P2
P3" .p 1+). . t + 2 in (2.3.36)(b)x(c).
56
m
Choose P2 from (2.3.36)(c):
if n-i-l
m
~
n-l+t-j.
~
n-l+t-j-i.
(2.3.42)
Since £
m may achieve its upper limit only
i in (2.3.36)(b), we shall also require
~
Hence
m-l+t-j-i
~
m
~
min(n-l-i, n-l+t-j) .
For m in this range, (2.3.36)(c) provides terms like
(2 .3. 43)
'
. 1)'. / m..
P2mP j-t-n+m+2i+4··· P i+j-t+2 u l,j-t-n+m+2i+4· .. u l ,i+j-t+2 (n-l-
n-l+t-j-m
This leaves P2
P3" 'Pj-t-n+m+2i+3 to be obtained from (2.3.36)(b),
so choose £ = n-l+t-j-m, giving the term
(2.3.44)
n-l+t
From (2.3.39) - (2.3.44), the coefficient of P2
P3" 'P n +2 - is
t
,
obtained as
(2.3.45)
n-l
\'
n!(n-l)!
\' m
m! (n-l+t-m)! L u 12 u a ... u l £
g(x 2 ,··· ,x 2 ,x£ , ... ,x£
)+
m=t
S
1
n-m-l
n-m
n-t
L
1
+
min(n-l,n-t).
n-i min(n-l-i,n-l+t-j)
I
(_l)ln! (n-i-l)! I
I
x
i=l
j=t
m=n+t-j-i
say, where Sl = rr 3 , ... ,n+2
n-m-l,m-t+l
and
57
S
2
= rr 3 , ... ,n+2-t
£,2i-2£,£+j-t-i,n+2_i-j
Observe that, had we solved (2.3.29) for P2 and substituted in
(2.3.30) for P2 instead of PI' an
expression V2 (t) precisely analogous
to VI (t), but with the roles of xl and x 2 interchanged, would have been
obtained.
We shall term such an expression "the complement" of (2.3.45).
Setting VI (t) to zero yields, for t = 0, a representation of
g3 , ... ,n+ 2 as a linear combination of other g's with coefficients which
are sums of products of u's.
Some of these products have to be elimi-
nated, namely those in which no u
r,s
with both r,s > 2 appears.
is suggested by (2.3.6) of Lemma 2.5).
(This
Such "nuisance" products also
arise in VI (t) for general t (and, of course, in V2 (t)), so we will be
concerned with finding an appropriate linear combination of VI (0),
VI (1), ... ,V 2 (0), V2 (1), ... which eradicates these products.
Accordingly, we write down the subset of terms in VI(t) and V2 (t)
to be eliminated.
(Note that the terms from VI(t) have been multiplied
t
t
by uI,n~t+3·· .u I ,n+2/ u I2 ' and those from V2 (t) by u2,n~t+3· .. u 2 ,n+2/ u I2·
These multipliers will comprise the non-constant components of the coefficients in the desired linear combination.)
m
= n-l+t-j,
Setting
2i~~
= ~,
that is,
in the second member of (2.3.45) and in its complement
yields
(2.3.46)(a)
n-l
\'
(n-l) In!
\' { m-t
(2.3.46)(b) x L (n-l+t~m)!m! SL u l2 u2~1·· .u2~
IU 2 ,n-t+3·· .u 2 ,n+2
m=t
n-mI
x g(x l , ...
,xI'x~
n-m
58
, ...
,x~
n-t
)} +
x
min(n~l,n-t)
(Z.3.46)(c)
L
+
i=l
.n-i
(n~i-1) In!
n~l~i-j
(-1)~ I
L { (n-1+t-i.j)!j!
u lZ
x
j=t S3
x u 1k . "u 1k .
u Zk
u . . t U z,n _t +3"
. ] Zk
1
J
J+
1+J-
min(n-1,n~t)
I
i=l
x
U
Zk ..
J
.n-i
(n-i-1)!n!
(-1)~ I
L { (n-1+t-i-j)!j!
j=t S3
'U
1
'-
1
1
z, . •, , , x -'z,
'-
J
t
i
J
- G(t: 3,4" ..
S
1
n-1-i-j
U 1Z
x
U
Zk .
1k
, 1 " ' U 1k . . t U 1 ,n~ t + 3' "u 1 ,n+ Z x
J
J+
~+J-
x g(x , ... ,x , x
say, where
+Z x
l
~
+
Lo,n
g(:J""'X 1 ' xZ,· .. ,x Z' x k . .
,,,,,x k
)J +
~
'---y----'
~+J -t+1
n-t
x
(Z.3.46) (d)
.u~
= rr 3 , ... ,n+Z-t
n-m-1,m-t+1
)}
x ..
' , .. , x
k
k
~+J ~t+1
n ..t
,n+Z~t)
and S3 = rr~,:",n+~-t
J,~.t,n~J
We seek a linear combination
n-1
I
L
A.G(j: i 1 , .. ·,i n~J.)
j=O rr 3 ,: .. ,n+2 J
n~J
which is identically zero.
(2.3.47)
Now
A.
J
gOj1 ... 10 ... 0
Choose A = 1,
O
= (-l)j (n+j -2).
(J -1)
(n-j -1)
= g(x 2 ,·· .,x 2 ,x 3 '· .. ,xn+2 _j )
appears in (2.3.46)(a)
for 0 ~ t ~ j, and for each t there are (j~t) expressions G(t;i 1 ,· .. ,i n _ )
t
n-1+J'
which contain g*
The coefficient of g*
U
x
Ojl. .. 10 ... 0'
Ojl. .. 10 .. ,0 12
u
3 .... u
2 in G(0;3, ... ,n+2) is
1 ,n+ -J
l ,n+
(2.3.48)
59
say, and in G(t;3, .•• ,n+2~t) it is
(2.3.49)
say, 1 s t s j.
Thus the overall coefficient of g*
in L
Ojl".lO ... O
is, from (2.3.47), (2.3.48) and (2.3.49),
= {(n-l) (j) +
~.
l.
t=l
t
(i) (-1) (n+t-2) (t -I) (n-t-l) (n-I) (j -t) +
+ (-l)jjl}nl/jl
~.
t
.
= { l. (iH-l) (n+t-2)('_1) (n-t-l) + ( .. l)Jjl}nl/jl
t=O
J
(2.3.50)
In Lemma 2.2, put r = j, k = n-l, s = j-l; the term in braces in (2.3.50)
is seen to be zero, identically in j, 1 s j s n-l.
Similarly all terms involving giOl ... lO ... O vanish in L, 1 s i s n-l.
It remains to show that all terms un-l-i-j u
l r · .. u l r u 2sl ... u 2si x
12
j
l
g*"l
10
0
have
coefficient
0
in
L l , where rl, ... ,rj,sl'" .,si is a
1 J '"
•••
permutation of n+3-i-j, ... ,n+2.
n-l-i-j
g*"1
u 12
u l , n+ 3-1-J
· · · · .u l ,n+ 2 -1.u 2 ,n+ 3'"
-1 .u2 ,n+ 2 1J
... 10 . .. 0
(2.3.51)
For 1 s
Again, fix a typical term
t
s j,
this term will occur in (2.3.46)(c), and there are (.jt)
J-
terms G(t;i 3 .... ,i + _ ) like G(t;3, ... ,n+2-t) which contain (2.3.51).
n 2 t
Similarly, (2.3.51) will occur in (2.3.46)(d) for 1 s i s t, and there
i
are (i-t) terms G(t;i 3 ,· .. ,i n +2 _t ) like G(t;3, ... ,n+2-t) which contain
(2.3.51).
Therefore the overall coefficient of (2.3.51) in L(x 3 ,· .. ,xn +2 )
is found to be
60
(2.3.52)(a)
~
j n! (n-j-l)!
i
t
t~l(-l) i!(n-l+t-i-j)! (t)(-l) (n+t-2)(t_l)(n-t-l) +
(2.3.52)(b)
l
i n! (n-i-l)!
j
t
+ t~l(-l) j!(n-l+t-i-j)! (t)(-l) (n+t-2)(t_l)(n-t-l) +
I
(2.3.52) (c)
+ (_l)J,
, 1) I
n.I ( n-J.
i!(n-l-i-j)! +
(2.3.S2)(d)
+ (_1)1.
. 1)'
n.I ( n-1.
j 1 (n-l-i-j) 1
Now (2.3.S2)(a) =
=
(-l)jnl i
.
il
t~l (-l)t(~)(n-j-l)... (n-j-i+t) (n+t-2) (t-l) (n-t-l)
j
(-l)jnl
i
t i
= . I (-1)
L
(-1) (t) (n-t+2) (. '-1) (n-t-l)
1. n
(j)t=O
l+J
(-1) nl (n-l) (.l+J')
i! (n-l) (j)
the second member of which cancels with (2.3.52)(d), and the first member
of which reduces, with the aid of Lemma 2.2, to
(-l)jnl
(i+j-l).
.
i-I,
,
[(n-l-J)(n-l)(.) - (n-l-1)(n-l)(,)](-1)
(1-1)!
1. n
(j)
J
J
J
'I( -1)
= (-l)(i+j-l)(i+j_l)!(i_j)/(ilj!) ,
and this will cancel with the term resulting from (2.3.52)(b) + (2.3.52)(d).
Hence L(x , ... ,x +2 ) is identically zero, as required.
n
3
II(ii).
Write Z(x l ,x 2 ; x 3 , ... ,x + ) =
n 2
n-l
I
t=O
At(V1(t)
+
Vz(t)).
Then Z = 0 is an equation which expresses g(x , ... ,x + ) as a function
3
n 2
of the form of the right hand side of (2.1.3), where the function h
contains xl and x 2 .
We have to show that the choice of dummy variables
61
is immaterial (except that u(x ,x ) cannot be zero).
l 2
As in Part I,
consider two representations of g(x l , ... ,xn ) (derived from the two different pairs of dummy variables (Yl'Y2) and (Y3'Y4)' where (x1,· .. ,xn )
is now the given arbitrary argument of g), subtract one from the other,
and write h+ for the difference between the two respective h-functions.
+
we shall show that h (x 3 ""'xn ) = O.
The proof is quite similar to that in Part I, the difference being
that for each j
1,
=
...
,n- 2 we h ave to
5 h OW t
·
h at f or 1,
+
(2.3.53)
h (xl"'" xl' x 2 ' ... 'x 2 ' x.1 , ... ,x.1 )
1
~
n-2-j-i
0,
.
. . . ,1.E
J
rr 3. , ... ,n ,
J
i = 0, ... ,n-j-3.
j
i
For the purposes of this proof only, we shall write the left side of
(2.3.53) as h
+(n-2-j-i i)
' (x. , ... ,x. ).
1
1
1
j
From (2,3.27), recalling that u(x,x) - 0, we have
u
similarly
u
Further,
h+(n-2,0)
=0
h+(0,n-2)
=0
h+(n-3,1)
=0
12
12
whence h+(n-2,0) = 0;
whence h+(n-3,1) = 0 ,
and in general
u
(2.3.54)
for 0
~
i
~
12
h
+(n-2-i,i)
0, whence h
=
Now introduce x.
n-2.
1
Cn _2 ,lu12 h
+(n-3,0)(
xiI
)
1
+(n-2-i i)
'
= 0 ,
(with n-2 xl's, one x 2 ):
C
h+(n-3,1)
h+(n-2,0)
+ n-2,lu li1
+ u 2il
= 0 .
From (2.3.54), only the first term on the left hand side is non-zero,
hence h+(n-3,0)(x. ) = O.
1
1
62
Suppose we have shown that
h+(n - 2- 9, , 9,-1) (x. )
(2.3.55)
1
0 , 9,
=
=
1, ... , i-I .
1
•
Then in the equation
+(n-2-i,i-l)() + C
.
u. h+(n-2-i,i) +
Cn- 2 -1+,
. 1 1C1,
. lU 12 h
x.1
1 1 11
n- 2-1+,
1
1
+
c.
u. h+(n-2-i+l,1) = 0
1,1 21
1
only the first member of the left hand side is not zero, hence
h
+(n-2-i i-I)
'
(x.) = 0, and so (2.3.55) is established by induction for
9,
1
1
i; hence for all 9,
=
1, ... ,n-3.
=
In general, suppose that (2.3.53) has been established for
{I
$
i
$
n-2-j, 1
all j -subsets
j
$
J}
$
x. , ... ,x.
1
1
1
il,···,i J
Sl = III , J-l
'
S2
=
j
U
{I
$
i
$
I-l,j = J} , I - I
of x ' ... ,xn ' in each set.
3
n-2-J, and
<
From (2.3.27), with
il,···,i J
II 2 ,J-2
+ C
\
+(n-J-I-l I)
n-J-I,l L u l 9, h
' (x9, , ... ,x9, ) +
Sl
1
2
J
\
u9, t h
S2 1 2
+ L
+(n-J-I I)
' (x9,' ... , x9, ) =
3
J
o,
and by the induction hypothesis, all terms are zero except the first term
on the left hand side; hence h+(n-J-I-l,I-l)(x. , ... ,x. )
1
1
1J
=
o.
A similar induction can be performed on j when (2.3.53) has been
established for {I
$
i
$
n-2-j, 1
63
$
j
$
J-l}.
It follows that
II(iii).
The proof of the P-integrability of h for all PEP follows
the proof of I(iii), with the modification that we choose two fixed points
Yl'Y2 (u(Yl'Y2)
~
0) and a distribution Py on Yl'Y2 and some other appro-
priate points.
The proof of the theorem is completed.
Remark:
Suppose that
X contains only a finite number of points (xl"",xn
O.
~
say) for which u(x,x)
The question arises as to how we may obtain
the representation of g(xl, ... ,x ), there being no other point y, with
n
u(y,y)
~
0, available as a dummy point.
of Part II of the proof may be used.
The answer is that the methods
The complementary problem (in which
only a finite number of x's in X have u(x,x) = 0) is treated analogously.
Now let
~
be a a-finite measure on (X,S), and write U(A,B) =
J ud~2 for all A,BES. We shall call U ~-admissible if, given AI" .. ,An
AxB
in S, there always exist N > n, An+l""'~ in S, and strictly positive
finite numbers al, ... ,a
N such that
N
L a.~(A.)
.11
1=1
=
1
and
N
N
L L a.a.U(A.,A.)
i=l j=l
1
J
= O.
J
1
Corresponding to condition F at the beginning of this section, we
introduce condition F
~
F
~
Z(A;B,C) - U(A,B)U(A,C) - U(A,A)U(B,C)
N
I
N
I
i=2 j=2
satisfies
Z(AI;A. ,A.)b.b. ~ 0
1
64
J
1
J
for all N
=
2,3, ... , AI' ... , AN
Let
THEOREM 2. 3.
P(~)
S, and all (b 2 , ... , b N)
E
E
N-l
R+ .
be a convex family of probability measures
which are absolutely continuous with respect to a a-finite measure
on (S,X) and which satisfy (2.1.1) (for u
ing
F~),
P(~) ~ PO(~)'
and suppose that
S(n)-measurable function such that fgdp
~-admissible
~
and not satisfy-
If g(xl, ... ,xn ) is a symmetric
n
= 0 for all PEP(~), then there
exists a symmetric S(n-2)-measurable function h(x , ... ,x _ ) which is
l
n 2
n-2
P
-integrable for each PEP(~), such that (2.1.3) is satisfied a.e.
(n)
[p(~) ] .
Proof:
As in the proof of Theorem IB in Hoeffding (1975), the proof of
this theorem may be accomplished by structuring the assumptions in such
a way that the methods of the proof of the discrete case (Theorem 2.1)
are applicable.
There is a dichotomy of the proof (corresponding to
that in the proof of Theorem 2.1) according as there does or does not
exist a set AES
of
~-finite
o
(non-zero) measure for which
I J ud~21
<
<
00
•
AxA
Part I:
0 < fUd~2 <
ANxAN
00
for some ANES, ~(AN) <
00
Let A be the class of all sets A in S such that
Y
~
for all pairs A
Yl
posi tive so that
2(A
,A
Y x AY )
1
2
Y2
in A.
j
+
Let
I
. 1
a.~(A.) = 1
1
<
00
Al, ... ,ANEA, and choose al, ... ,a
N
1=
luld~2
A A
Yl Y2
1
65
N
and
N
I
N
I
a.a.U..
i=l j =1
where U..
1J
1
J
0,
1J
= U(A.,A.)
= J ud~2
1
J
Then
= L\~1 = la.I[xEA.]
is a prob1
1
p(x)
A.xA.
1
J
ability density with respect to
~
of a distribution in
PO(~):
hence
= 0
f
where Gl ... n - G(A l ,·· "An) =
gd~n,
and
Al x ... xA
n
is the analogue of g*
in the proof of Theorem 2.2. We may now
r l •• .r N
use the argument in that proof (or the argument of (b) in Section 2.2
if U(A,B) has the form VI (A)V l (B) - V2 (A)V 2 (B)), with Pj' X, x j ' u ij
and g.
1
l'
1
, .. n
replaced by a., A, A., U.. and G.
J
J
1J
1
.
1
, .. 1
respectively, to
n
n-2
conclude that there exists a sYmmetric real-valued function H on A
such that,for any (A , ... ,A _ )EA
l
n 2
n-2
,
I
U. . H.
ITl, .•• , n
2,n-2
1
1
l 2
1
.
3 · · · 1n
Lemma 2.4 implies a representation for H(A , ... ,A _ ) in terms of G and
l
n 2
U; further, we may write
H(A l ,···,An _2) =
J
J
h(xl"",xn_2)d~(xn_2)···d~(xl)
AI' .• A _ 2
n
where h is given by (2.3.3) in Lemma 2.4 with the adjustments
g(k)(Xl,···,X n _k ) =
f· ··f
g(x l ,
~ ~
66
""xn_k'Yl'···'Yk)d~(Yl)···d~(Yk)
and u(x. ,y) replaced by Ju(x.,y)d~(y).
1
A
1
Now write w(x l '" .,X ) =
n
n
g(xl, ... ,x ) - z: 1
u(x. ,x. )h(x. , ... ,X. ).
n
IT , ••• ,n
11
1
1
In
2
3
2, n-2
far, that
F (C) =
for all sets C
Ic
wd~
n
n
Alx ... xAnEA.
=
= 0
Following the proof of Theorem lB in
Hoeffding (1975), let B be a set in A.
hence w(xl, ... ,x )
n
=
Then F(EnBn )
=0
for all EES(n),
n
0 a.e. [~n] on B .
PEPO(~)'
Finally, let
We have showI), thus
let p be a version of
dP/d~
(hence, a simple
function of the form z:kla.I[XEA.]for
some integer k, positive numbers
1
1
and let AEA.
(2.3.56)
We want to show that B£EA, that is,
J J(l +
lu(x,y)l)d~(x)d~(y)
<
00
B A
£
for all AEA.
Since B£ is just the union of those sets A among Al, ... ,Ak
j
for which p. > £, the left hand side of (2.3.56) is just the sum of
J
integrals over sets Aj x A, all of which are finite by definition. Thus
00
n
n
B EA for all £ > O. Therefore w(x l , ... ,xn ) = 0 a.e. [~ ] on U B
'
£
m=l l/m
n
a set of P -measure one. It follows that
~
u(x. ,x. )h(x. ,
11
12
13
II 1 , ... ,n
... ,x. )
In
2,n-2
The fact that h does not depend on the arbitrary set AN is proved
analogously to the result for P, by working with HI (A , ... ,A _ ) and
l
n 2
H2 (A l ,· .. ,An _ 2) based on two different choices for AN'
67
The integrability
of h for all
PEP(~)
is also proved analogously to the corresponding
result for P.
Part II:
The proof proceeds in similar fashion to that in Part II
of the proof of Theorem 2.2 (or to that in (d) of Section 2.2, if U(A,B)
has the form Vl(A)V l (B) - V2 (A)V 2 (B)) with modifications as in the proof
of Part I of the proof of this theorem.
If U(A,B) has the form
Theorem 2.3 is thus established.
Ik
V. (A)V. (B), the situation is reduced to
i=l 1
1
that considered by Hoeffding in the analogue of Theorem 1.4 for dominated
families.
The analogues of (2.2.1S)(i) and (2.2.1S)(ii) provide suf-
ficient conditions on u for
~-admissibility.
68
CHAPTER 3
BOUNDED COMPLETENESS OF SOME NON-PARAMETRIC
FAMILIES OF PROBABILITY MEASURES
T
Although the families p and pT(~) of probability measures, corresponding to the families P and
P(~)
of Theorems 2.2 and 2.3, are
not complete, they will be boundedly complete provided that u satisfies
certain conditions.
THEOREM
3.1.
in Theorem 2.2.
Let P he the family of probability measures defined
If u is unbounded, then pT is boundedly complete.
Proof:
Suppose that g(xl, ... ,x ) is a sYmmetric bounded S(n)-measurable
n
x
n
function satisfying !gdp = 0, all PEP. We wish to show that g(xl, ... ,x )
n
= ° if
u(x I ,x 2) is unbounded.
The proof is in two parts, according as
there does or does not exist a sequence {x k } for which \u(xk,x k )
1+
00.
Generally speaking, within each part we shall prove first that h(xl, ... ,x )
l
=
0, then that h(x , ... ,x ,x ) = 0, and so on, until we obtain
l
l 2
There are, however, two cases in which we can prove directly that
h(x , ... ,x _ ) = 0. If \u(xk,x )\ + 00 as k + 00, and we replace y by x
k
k
n 2
l
in (2.3.3) of Lemma 2.4, we see that the right hand side of (2.3.3) will
converge to zero if u(xi,xk)/u(xk,x k ) is bounded for 1
~
i
~
n-2,
because of the preponderance of u-factors in the denominator compared
with the numerator of each summand.
A similar argument obtains when
u(X,X) is bounded, [u(xk,y k )
I
~
00
as k
~
00
,
U(Xi,Yk)/U(Xk,y k ) are both bounded for 1 s i
~
n-2, and Yl and Y
2
in equation (2.3.6) of Lemma 2.5 are replaced by x
k
and Y .
k
Thus
the main part of the proof of this theorem is devoted to cases in
which u(x,x k ) and u(x'Yk) are not so well-behaved.
Part I:
u(x,x) unbounded.
Let {x } be a sequence of points in X for which Iu(xk,x ) I ~
k
k
as k
~ 00,
and let x "",x _ be n-2 distinct points in X.
l
n 2
00
We prove
first that
h*n-2,0 ... 0 =
°
(and so h(x, ... ,x)
- 0)
and
*
h n - 2 - i ,0 ... Oi
1 :::; i :::; n-2 ,
=
where
- h(x·l"······,,
..x 1 , ••• ,x,n- 2""'x1)- 2' ~"",xk)
~
~
11
{n-2
i n _l
for (il,···,i
n-
1)
E
r n- 2 ,n- l'
(Correspondingly, the last subscript of
g* will be associated with x ).
k
Consider, then, the following system of
equations, derived from (2.1.3):
(3.1.1)2
(3.1.1).
1
g*
n-l,O ... Ol
=
C
u h*
+ C
U h*
n-l,2 11 n-3,0 ... 01
n-l,l lk n-2,0 ... 0
g*
n-2,0 ... 02
=
C
u h*
n-2,2 11 n-4,0 ... 02
°
g*n-l,
. ... 0'1
+
C
C
u h*
+
n-2,1 2,1 lk n-3,0 ... 01
+
u
= Cn-1,
. 2u 11 hn-1* . 2 ,0 ... 0·+
1
x h* .
.
n-l-1,0 ... 0,1-1
70
+
h*
kk n-2, 0 ...
°
Cn _"1, lC,1, 1 u 1k x
C.1,~~ukkh*n-l,
: a ... O'
,1- 2
(3 .1. 1) n-3
g*30 ... O,n-3
C3,1 u 11 h*10 ... 0,n-3
x
(3.1.1) n- 2
(3.1.1)
I (i) .
n-
1
g*
10 ... 0,n-1
u
h*
11 0 ... 0,n-2
+
C
u. h*
n-3,2 Kk 30 ... 0.n~5
+
C C
u h*
2,1 n-2,1 1k 10 ... 0,n-3
+
C
u h*
n-2,2 kk ZO ... O,n-4
= Cn-1,1 u 1k h*0 ... 0,n-2
+
=> h*
n-
0 ... O,n-Z
C
u h*
n-l,2 kk 10 ... 0,n-3
=
-1
o(ukk
)
-1
1 => h*10 ... 0,n-3 = O(~k)
(3.1.1)2
=> h*
n-Z,O ... 0
=
-1
O(ukk )
1
. 0 . , . O'1
whence h*_Z
n
, 0 ... 0 = 0, and h*
n- Z-1,
II(ii).
Assume
u
1k
/u
We may assume that
k
~
00,
+
Assume ulk/u
bounded.
kk
(3.1.1)n
(3.1.1)
g*
20 ... 0,n-2
h*
.
20 ... 0,n-4
C3,lCn_3,lu1k x
+
and
~k
kk
00
as k ~
00,
:0;
n-2.
so that lulkl ~
00
as
In the following sequence of implications,
D , ... ,D _ are non-zero and not dependent on x k .
n 2
1
From (3.1.1),
h*n- ZOO
# 0 => u 11 # 0, h* n-3, 0 ... 0 1 ~ Dl u 1k '
, '"
(3·1.1)Z
=> h*
(3.1.1)n_Z
~~c>
~
n-4,0 ... OZ
contradicting (3.1.1) n- l'
i
unbounded.
Iu 1k /ukk I ~
= o(u lk ).
:0;
h*
0 ... 0,n-2
~
D u2
2 1k
D un-Z
n-Z lk
Hence h~_2,0 ... 0 = o.
71
It remains to show that
h*
(3.1.2)
n-2-i,0 ... Oi
=
O(u
-1
lk
),
1
~
i
'r"
The following method wi 11 be termed "method
n-3.
~
for reference purposes,
as it will be used several times in later sections.
Then
Suppose, contrary to (3.1.2), that ulkhio ... 0,n-3 is unbounded.
(3.l.l)n_2
=> u
kk h*20 ... 0,n-4 unbounded;
h*
= o(h*
)
10 ... 0,n-3
2 ... ,n-4
(3.l.l)n_3
=> u
kk h*30 ... O,n-S unbounded;
= o(h*30 ... O,n-S )
h*
20 ... 0,n-4
h*
unbounded;
kk n-3,0 ... 01
=> U
h*
= 6(h*
)
n-4,0 ... 02
. n-3,0 ... 01
(3.1.3)
(3·1.1)2
=> h*
n-4,0 ... 02
constant
+
(if u
ll
to),
x
ulkh~_3,0 ... 01 +
o(h*
)
n-3,0 ... 01
contradicting the implication in (3.1.3).
Hence
ulkhiO ... 0,n-3 must be bounded, and we may proceed similarly to show
that (3.1.2) is true for all appropriate i.
If u
ll
=
0, the contra-
diction will arise in (3.1.1)2 because the right hand side is bounded
and the left hand side is not.
We now introduce x 2 into the argument of g.
proof we disposed of the case where uik/u
so in I(iii) below we shall assume that u
definiteness, that ulk/u Zk is bounded).
72
kk
2k
In the preamble to the
was bounded for 1
/u
kk
~
i
~
n-2,
is unbounded (and, for
In the preamble, mention was
made of the sequential nature of the proof, in particular that we would
show that h(x , ... ,x ,x ) = O.
l 2
l
This will be demonstrated (in a round-
about way) below, by showing that
(3.1.4)(a)
h~1,n- 2'
-1, 0 ... 0 = 0
and
-1
h*
= O(u)
i,n-2-i-j,0 ... OJ
2k'
(3.1. 4) (b)
for i
0(u
-1
2k
= 1,2, ... ,n-3.
) for 1
I(iii).
$
Assume
i
$
1
$
j
$
n-2-i ,
(Note that I(ii) implies that h*
=
0,n-2-i,0 ... Oi
n-3).
u2k/~k
unbounded, u
lk
/u 2k bounded.
Consider the following system of equations:
(3.1.5)1
(3.1.5)2
g*
= C
u h*
+
1,n-2,0 ... 01
n-2,2 22 1,n-4,0 ... 01
+
C
u h*
n-2,1 12 0,n-3,0 ... 01
+
Cn-2,1 u 2kh*1,n-3,0 ... 0
g*
= C
u h*
+
1,n-3,0 ... 02
n-3,2 22 l,n-S,O ... 02
+
+
+
(3.1.5)3
C
u h*
+
n-3,1 12 0,n-4,0 ... 02
C
C u h*
+
n-3,1 2,1 2k 1,n-4,0 ... 01
C u . h*
. + u h*
2,1 lk 0,n-3,0 ... 01
kk 1,n-3,0 ... 0
g*
= C
u h*
+
1,n-4,0 ... 03
n-4,2 22 1,n-6,0 ... 03
+
C
u h*
n-4,1 12 O,n-S,O ... 03
+
C
C u h*
n-4,1 3,1 2k l,n-S,O ... 02
+
C u h*
3,1 lk 0,n-4,0 ... 02
+
C u h*
3,2 kk 1,n-4,0 ... 01
73
+
+
+
(3.1.5),
gi,n-i-l,O ... Oi = Cn-i-l,2U22hi,n-i-3,0 .. ,Oi
1
(3.1.S)n_3
(3.1. 5)n_2
+
Cn-1, 1 , lU 12 h O
* ,n-1. 2 , 0 '" 0'1
+
Cn _i _l lC i ,lU 2k h i,n_i_2,0 ... O,i-l
+
Ci,lUlkhO,n_i_l,O
O,i-l +
+
Ci ,2 Ukk h i,n-i-l,0
0,i-2
+
+
g*
= u h*
+ C
u h*
+
120 ... 0,n-3
22 10 ... 0,n-3
2,1 12 010 ... 0,n-3
+
C C
u h*
2,1 n-3,1 2k 110 ... 0,n-4
+
C
u h*
+
n-3,1 lk 020 ... 0,n-4
+
C
u h*
n-3,2 kk 120 ... O,n-S
g*
= C
u h*
110 ... 0,n-2
n-2,1 2k 10 ... 0,n-3
+
+
C
u h*
n-2,1 lk 010 ... 0,n-3
u. h*
n-2,2 kk 110 ... 0,n-4
+ C
We prove first that h(x l ,x 2 , ... ,x 2 ) = 0, and then use the reduced
equations in conjunction with method + to prove that u
cannot be unbounded,
°
~
i
~
n-4.
° (which implies, using
Suppose then that hi,n-3,0 ... 0 f
that u
22
f 0).
h*
2k liO ... 0,n-3-i
(3.1.5)1'
In the following sequence of implications, E , ... ,E _
l
n 3
are non-zero multiples of hi,n-3,0 ... 0 which do not depend on xk .
(Note that terms like ulkh *O,n -'-I
1
,
(3.1.5)1
=> h*
(3.1.5)2
=> h*
°
...
1,n-4,0 ... 01
~
0 ,1'-I are bounded.)
E u
l 2k
2
E u
2 2k
l,n-S,O ... 02
#
I-
(3.1.6)
(3.l.5)n_4
~~>
h*
110 ... 0,n-4
~
E un-4
n-4 2k
e
74
(3.1.7)
(3.1.5)n_3
=> h*
1l0 ... 0 ,n-4
E _ un-6 u
n 3 2k kk
~
so that (3.1.6) and (3.1.7) are mutually contradictory unless E
n-
3 and
En-4 are zero, that is, unless h*l ,n -3 , 0 ... 0 = O.
After removing from (3.1.5) all terms involving h*
, we
1,n-3,0 ... 0
may use method t to derive a contradiction if u h*
is un2k 10 ... 0,n-3
unbounded,
bounded. Then (as in I(ii)), consider in turn u h*
2k 1l0 ... a ,n-4
u h*
unbounded, and so on, deriving a contradiction at each
2k 120 ... 0,n-5
stage. In this way, (3.1.4) is proved for i = 1.
Proceeding by induction, suppose that (3.1.4) has been established
for i
= 1, ... ,~-1.
We shall prove that (3.1.4) holds for i
=~ «
The equations are:
(3.1.8)1
(3.1.8)2
g*~,n-~-l,O ... Ol = Cn-~-1,2 u 22 h*~,n-~-3,0 ... 01 +
+
C u
+
C
C
u
+
C
u
h*
~2
11
h*
~-2,n-~-1,0
n-~-l,l ~,1
n-~-l,l
2k
12
... 01
+
h*
~-1,n-~-2,0
~,n-~-2,0
... 01
+
... O
g*~,n-~-2,O ... 02 = Cn-i-2,2 u 22 h*~,n-~-4,0 ... 02 +
+
C
+
C
r.
+
C
C u h*
2,1 2k ~,n-~-3,O ... 01
+
C
~,2
u
11
h*
~-2,n-~-2,0
n-~-2,1 ~,1
n-~-2,1
~,l
+ u
kk
75
u
12
h*
... 02
+
~-1,n-~-3,O
... 02
C u h*
2,1 1k ~-1,n-~-2,0 ... 01
h*
~,n-~-2,O
... 0
+
+
n-3).
(3.1.8)r
C
g~
0
N,n-N-r,O
... 0r
n-~-r,2
+
C
+
C
+
C
~,2
u
u
22
11
h*
~,n-~-r-2,0
h*
~-2,n-~-r,0
C
n-~-r,1 ~,l
u
12
+
... 02
+
... Or
h*
~-l,n-~-r-l,O
... Or
_
... O;r~l
C u h*
r,l 2k ~,n-~-r-l,O
C
C
u h*
~,l r,l lk ~-l,n-~-r,O... O,r-l
C u h*
r,2 kk ~,n-~-r,O ... 0,r-2
n-~-r,l
+
+
+
+
+
(3.l.8)n_~_2
g*
~20
... 0,n-~-2
= u
22
h*
~O
+ C
2 ,1
C
~,l
u
12
C
+
... 0,n-~-2
~,2
h*
~-1,lO
u
11
h*
~-2,20
... O,n-~-2
+
... 0,n-~-2
+
+
C
C
u h*
+
2,1 n-~-2,1 2k ~lO ... O,n-~-3
+
C
+
C
C
~,l n-~-2,1
n-~-2,2
u
kk
u
lk
h*
~-1,20 ... 0,n-~-3
h*
~20 ... O,n-~-4
(3.l.8)n_~_1
g*
~lO
... O,n-~-l
=
C
~,2
u
+
C
+
C
+
C
+
C
11
~,l
h*
~-2,10
u
12
... 0,n-~-1
h*
n-~-l,l
~-l,O
u
2k
C
h*
n-~-1,2
u
kk
... 0,n-~-1
~O
~,l n-~-l,l
u
+
+
... O,n-~-2
lk
+
h*
h*
~,lO
~-l,lO ... O,n-~-2
+
... O,n-~-3
By the induction hypothesis, all terms containing h*~~l,. or h*~-2,.
are bounded and may be transferred to the left hand sides of their
respective equations; the resulting system is then analogous to (3.1.8),
and a similar argument shows that (3.1.4) is true for i =
limiting case
(~ =
n-3),
(3.l.8)n_~_1
76
reduces to
~.
In the
(3.1.9)
g*
= C
u h*
n-3,10 ... 02
n-3,2 11 n-S,10 ... 02
+
C
u h*
n-31 12 n-4,0 ... 02
+
C2,1 u 2k h*n-3,0 ... 01 + Cn_3,IC2,lulk x
x
h*
+ u h*
n-4,10 ... 01
kk n-3,10 ... 0
+
and (3.1. 8) 1 to
(3.1.10)
g*
= u h*
+ C
u h*
+
n-3,20 ... 01
22 n-3,0 ... 01
n-3,2 11 n-S,20 ... 01
C
u h*
2 ,1 n-3,1 12 n-4,lO ... Ol
+ C
+
As before,
(3.1.10)
h~-3,10 ... 0 ~
C u h*
2 1 2k n-3,10 ... 0
0 => u 22
~
0 (from (3.1.10)), and then we have
=> h*
~
constant
x
=> h*
~
constant
x
n-3,O ... 01
u
h*
,
2k n-3,10 ... 0
and
u
(3.1.9)
n-3,0 ... 01
10 ... 0 = O.
a contradiction unless h*_3
n ,
u
u
kk h*
n-3,10 ... 0
2k
Finally, (3.1.9) implies that
h*
is bounded.
2k n-3,O ... Ol
Thus far, we have shown that, for any x,y in {xl" .. ,x _2 }
n
h(x, ... ,x) - 0,
h(x, ... ,x,y, ... ,y)
. Iu(x,x k )
h(x, ... ,x,y, ... ,y,xk,.·.,x k ) = O[mln(
==
,-I ,
0 ,
Iu(y,xk ) 1-1 )].
(Note that the proof in I(iii) has not used the fact that u(xk,x k ) is
unbounded.)
The proof of Part I may now be completed by induction, as
follows in I(iv).
77
I(iv).
Assume u~k/ukk unbounded, uik/u~k bounded, 1 ~ i s £-1.
The induction hypothesis will assume that for i = 1, ...
,~-1,
and for all i-subsets
(3. 1. ll)(a)
h(y
,···,y,,···,y·,···,y·)
'- 1
1~1·
·'1 = 0,
~
1
J1
(3.1.11)(b)
for (j1,···,j·)Er
2'
1 n-,l
Ji
h(Y1""
.... 'YI""'Y'"
...1... ,y.J ,xk""'X~)
~
j1
ji
ji+1
for (j 1 ' ... , j.1+ 1) E r n- 2 ,1+
. l'
and we posit that (3.1.11) is true for i =
~.
Because of the complexity of the notation, we shall restrict attention to the case j1="
.=j~-l
case is straightforward.
= 1;
the complete generalization from this
For such values of j1" .. ,jt-1' consider the
following system of equations:
(3.1.12)1
U nh~
len
n-N, 1 SN
l
+ S
1
+
IS
u
l
. _ ,n-Nn 1 ,... 01
t 1
···l
h~.
+
st 11···1~_1,n-t,0... 01
2
+ Cn_~,lU£khj1" .jt_1,n-t-1,0 ... 0
(3.1.12)r
+
IS
1
+
C
u h~
n-t-r+1,1 st 1 1 "
2 u st h~1
S
2
1
"
78
.
.
.1~_1,n-t-r,O
.1~_1,n-t-r+1,O ... Or
+
... Or
+
+
+
I cr,l u sk h~1
S
1
+
C
n-~-r+1,1
1"
.
.1~_1,n-~-r+1,0 ... 0,r-1
+
C
u h~
.
r,l ~k J1 ... J~_1,n-~-r,0... O,r-1
+
+ Cr,2U~~hj1" .j~_1,n-~-r+1,0 ... O,r-2
(3.1.12)n_n
N
g~
.
lOOn
J1 ... J~_1' ... ,n-N
=
IS
u nh~
.
+ C
u
0
SN 11···1~_1'
...
0
n +
,n-N
1
n-~,l ~k
h~
.
J ... J~_l,O ... O,n-~-l
1
where 51 = {i 1=···=i _ = i s+l=" .=i~_l = 1, i = 0 ,
s 1
s
+
1
~
s~~-I},
and 52 = {i =·· .=i _ = i s+l=" .=i _ = it+1=···=i~_1= 1, i = it = 0,
1
s 1
t 1
s
1 ~ s
<
t ~ ~-1}.
I
By the induction hypothesis, all terms involving
are bounded and so may be removed to the left hand sides of their
respective equations.
The system (3.1.12) is then of the same form as
(3.1.8) and we may proceed in corresponding fashion to conclude that
= 0 , and
(3.1.13)
, r= 1 , ... ,n - ~ - 2 ,
for j1="
·=j~-l
= 1.
(3.1.13) can be established for general j1'"
79
·,j~-l
When £ = n-l, (3.1,13)
by induction on jl' then on j2' and so on.
furnishes the desired result that hi ... 10 = O.
This completes the
proof of Part I.
Part II:
u(x,x) are bounded.
To accommodate the extra variable Yk we shall add an extra subscript
to g* and to h*, and for emphasis the last three subscripts of each will
be written explicitly.
The methods of proof for this part are determined by the boundedness or otherwise of u(x,x ) and u(x,y ).
k
k
If either is unbounded, the
techniques employed in Part I suffice; if both are bounded, further
argument is required.
II(i).
We suppose first that u(x,x ) is unbounded.
k
Assume lu(x,x ) I ~
k
as k~
00
00.
Consider the system of equations (3.1.1), except with g* . 0
O'
n-1, ... 1
replaced by g*_.
and h*_'_l
1
n 1 , 0 ... 0 ,1'-I replaced by
n 1, 0 ... 0'0'
h*n-1. 1 , 0 ... O'
,1- 1 , 0 for 0
~
i
~
n-l.
If u ll = 0, (3.1.1)1 implies immed-
iately that h*
= 0, so assume that u
1 O.
ll
n-2,0 ... 000
In the following
sequence of implications, F , ... ,F _2 are non-zero multiples of
l
n
h~-2,0 ... 000 and independent of x k and Yk'
(3.1.1)1
=> h*
F u
1 lk
(3·1.1)2
=> h*
2
F u
2 1k
(3.1.1)n_2
=> h*
F
u
0 ... 0,n-2,0 - n-2 1k
n-3,0 ... 010
n-4, O... 020
n-2
80
which contradicts the implication of (3.l.l)n_l that h*0 ... 0,n-2,0
n-2
o(u lk ) unless h~-2,0 ... 000 = O.
If u
kk
~
0 for infinitely many k, method t may be used to show that
h~-i-2,0... OiO
If u
kk
=
=
-1
=
O(u lk )
0 for all k > kO' this follows readily from (3.1.1).
The general result hi ... 100
=0
may now be obtained by following the
methods in I(iii) and I(iv).
II(ii).
Assume u(x,xk ), u(x'Yk) bounded, u(xl,x l )
~
O.
Again returning to (3.1.1),
=> h*
n-3,0 ... 010
(3.l.l)n_2
=
0(1) ,
=> h *
O. .. 0 ,n -2 , 0 = 0(1) .
We introduce x 2 into the argument of g, yielding the following system
of equations:
(3.1.14)1
g*
n-2,10 ... 010
= Cn-2,2 u 11 h*n-4,10 ... 010
C
u h*
+
n-2,1 12 n-3,0 ... 010
C
u h*
n-2,1 lk n-3,lO ... OOO
+
+
(3.1.14).
1
+
g~-i-l, 10 ... 0iO
+
Cn-l-,
. 1 lU 12 h*n-l. 2 , 0 ... 0'0
1
+
Cn_i_l,lCi,lulkh~_i_2,10 ... 0,i-l,0
+
C.1,1 u 2k h*.
1 0 .. , O'
n-1-,
,1- 1 , 0
81
+
+
+
cn-2,1 u 1k h*010 ... 0,n-3,0
g*
(3.1.14)n_2
no ... 0,n-2,0
+
C
u h*
n-2,1 2k 10 ... 0,n-3,0
+
C
u h*
n-2,2 kk 110 ... 0,n-4,0
=> h*
Then (3.1.14) 1
n-4,10 ... 010
=> h*
(3.1.14)n_3
010 ... 0,n-3,0
= 0 (1)
=
+
,
0(1) .
In general, having introduced x 2 ,···,x r ' we find that h*_._
n1r -1 , 1
is bounded, 1
~
i
n-r-1.
~
10
It follows that all functions h containing
either x or Y (but not both) in their arguments are bounded.
k
k
It remains
to show that functions h which contain both xk and Yk in their arguments
are O(u(xk,y k )
be sketched.
-1
).
The proof of this is straightforward, and will only
There are two cases to consider.
(a)
Either u(xk,x k ) or u(Yk'Y k ) non-zero for infinitely many k
(b)
u(xk,x k )
=
u(Yk'Y k )
0, k
=
For (a), suppose that u(xk,x k )
~
O.
>
k O.
In order to distinguish Yk from x k
in the following system of equations, we shall subscript Y by
the understanding that
(3.1.15)1
~
g*
O... O,n-l,l
<->
=
1
gO* ... 0 ,n-1,1
..
=
k.
C
u h*
n-l,2 kk o... O,n-3,1
+
(3.1.15) .
0·0
1
+
C
u h*
n-l,l k£ 0 ... 0,n-2,0
Cn-1,
. 2ukk h O* ... 0 ,n- 2
'· +
-1,1
+ Cn-1,
. 1C1,
. 1u kNnh O* • • • 0 ,n- 1-1,1·· 1
+
C.1 , 2U £~h*O , ... , 0
'·
2
, n -1
, 1 -
82
+
~,
with
(3.l.l5)n_l
g*O... Ol,n-l = uk~h*O... 0,n-2,0
+
Cn-l,2 u ~~h*0 ... 00,n-2
In the following implications, Gl , ... ,G _2 are non-zero constants
n
independent of xk and
y~.
uk~hO ... 0,n_2,0
Suppose that
is unbounded.
(3.1.15)1
(3.1.1S)n_2
=> h*
0 ... 00,n-2
~
G
n-2
(u
k~
/~)
n-2
Kk
h*
0 ... 0,n-2,0
which contradicts the implication of (3.l.lS)n_l·
Hence
uk~hO ... 0,n_2,0
must be bounded, and by repeated use of this procedure, we conclude that
-1
h*
= O(u ko ),
N
0 ... 0,n-2-i,i
~
0
n-2 .
~
i
We may now introduce xl' write out the system of equations for
g*lO . .. 0 ,n -2l,···,g*10
,
. .. 0 , 1 ,n -2' and re-apply the above argument to show
that
hio ... 0,n-3-i,i
~
i
~
n-4 .
we find that for any xl" .. ,xn _4 '
Introduce x 2 ,x 3 ... sequentially:
h(x l , ""xn_4,xk'y~)
1
-1
= O(~~).
In case (b), the results of (a) follow trivially from the same sets
of equations.
The proof in this section is now essentially complete.
We have
(3.1.16)
+
L
nl, ... ,n-2
1,n-3
+
I
nl, ... ,n-2
1,n-3
'Yk) +
u(xk,x i )h(x i , ... ,x i
1
2
n-2
u(Yk'x, )h(x. , ... ,x.
,x k )
111
1
2
n-2
83
+
+
I
u(x. ,x. )h(x. , ... ,x.
,xk,Y )
k
11 1 2
13
1n _2
rr 1 , ... ,n- 2
2,n-4
and each term in (3.1.16) is bounded except for u(xk,y ).
k
h(x , ... ,x _ ).
1
n 2
~
This proof has required that u(x,x)
one xE{x , ... ,x _ }.
1
n 2
•
Hence
0 for at least
In the last section (II(iii) below) we consider
= u(xn- 2'xn- 2) = O.
II(iii).
Assume u(x,x k ), u(x'Yk) bounded, u ll =·· .=un - 2 ,n-2 = O.
We begin with the observation that u(x,x) may be assumed to be
identically zero.
For, suppose that u(x
I
n-
l'x
n-
1) ~ 0, and write
hex. , ... ,x.
1
rrl, ... ,n-l
n-2
1
1
)
n-2
and
I
rrl, ... ,n+l
2,n-l
u(x. ,x. )H(x. , ... ,x.1
L
rrl, ... ,n+l
1
1
1
2
1
3
)
n-l
g(x. , ... ,x. )
1
1
1
n
n
Then G is bounded, and so the arguments of II(ii) imply that H(x ,·· .,x _ )
l
n l
= 0, whence
LIT 1 , .•. ,n- lh(x.1
, ... ,x.
1
1
_
n 2
) = O.
Since hex. , ... ,x.
1
1
1
_
n 2
) = 0
n-2
ldx. , ... ,x.
}, all terms in H vanish except h(x l , ... ,x _2 ), so
n1
1 _
n
n 2
1
it too must be zero.
if x
Thus we are left with the case where u(x,x)
and u(x,y ) is bounded.
k
= 0,
u(x,x k ) is bounded,
As remarked in the preamble, the result follows
from Lemma 2.5.
84
THEOREM
3.2.
Let
defined in Theorem 2.3.
P(~)
If
be the family of probability measures
u
is P(~)-unbounded, then pT(~) is
boundedly complete.
Proof:
The proof of Theorem 3.1 applies here to the functions
G123 ... n , U12 , and H123 ... n-2 as defined in Theorem 2.3.
85
REFERENCES
C.B. BELL, D. BLACKWELL AND L. BREIMAN
"On the completeness of order statistics,"
Ann. Math. Statist. 31 (1960): 794-797.
D.A.S. FRASER
"Completeness of order statistics,"
Canad. J. Math. 6 (1954): 42-45.
D.A.S. FRASER
Non-parametric Methods in Statistics
New York: John Wiley and Sons (1957).
P.R. HALMOS
"The theory of unbiased estimation,"
Ann. Math. Statist. 17 (1946): 34-43.
W. HOEFFDING
"Some incomplete and boundedly complete families of distributions,"
Institute of Statistics Mimeo Series #984 (1975) .
...
E.L. LEHMANN AND H. SCHEFFE
"Completeness, similar regions, and unbiased estimation - Part I,"
SankhyCf (AJ 10 (1950): 305-340.
E.G. PHILLIPS
Functions of a Complex Variable
London: Oliver and Boyd Ltd. (1957).
..
86