PROBABILITY AND EXPECTATION INEQUALITIES
by
Stamatis Cambanis* and Gordon Simons**
University of North Carolina
Abstract
This paper introduces a mathematical framework within which a wide
variety of known and new inequalities can be viewed from a common
perspective.
Probability and expectation inequalities of the following
types are considered:
(b) Ek(Z)
~
(a) P(ZEA)
~
P(Z'EA) for some class of sets A,
Ek(Z') for some class of functions k, and (c) Et(Z)
some class of pairs of functions t and k.
~
Ek(Z') for
It is shown, sometimes using
explicit constructions of Z and Z', that, in several cases, (a) <=> (b) <=> (c);
included here are cases of normal and elliptically contoured distributions.
A case where (a)
~>
(b)
<~>
(c) is studied and is expressed in terms of
"n-monotone" functions for (some of) which integral representations are
obtained.
Also, necessary and sufficient conditions for (c) are given.
Key Words and Phrases:
probability inequalities, expectation inequalities,
normal distribution, elliptically contoured
distributions, stochastic orderings, quasi-monotone
functions, n-monotone functions
*Research supported by the Air Force Office of Scientific Research under
Grant AFOSR-75-2796.
** Research supported by the National Science Foundation under Grant
MCS78-0l240.
2
1.
Introduction
There is an extensive literature dealing with probability inequalities of
the form
A E
P (ZEA) ;:: P (Z ' E A) ,
A,
(1.1)
and expectation inequalities of the form
Ek(Z) ;:: Ek(Z ') ,
(1. 2)
where Z and Z' are random vectors (or more general random elements) with common
range space R, A is a class of (Borel) subsets of R, and F is a class of real
l
(measurable) functions on R.
Here we also focus attention on expectation
inequalities of the form
E9-(Z) ;:: Ek(Z') ,
(1. 3)
where F is a class of pairs of (measurable) functions on R.
2
When the classes
A,F ,F are progressively richer then conditions (1.1),(1.2),(1.3) are
l 2
progressively stronger.
Specifically if lA E Fl , i.e. if lA E F for all A E A,
l
then (1.1) <= (1.2); and if F
l
c
{k: (k,k)
E
F } =: F then (1.2) <= (1.3).
2l
2
The more interesting conclusions are therefore those which lead from (I.l) to
(1. 2) to (1. 3) .
The first question of interest is of course to describe conditions on the
distributions of Z and Z' which guarantee (1.1) for specific classes A of sets,
and there is a vast literature on this.
The second question is, given a class
of sets A, to describe a class of functions F , depending of course on A, for
l
which (1.1) => (1.2); if such a class Fl contains lA then in fact (1.1) <=> (1.2).
The third question is, given a class of functions F , to describe a class F? of
l
3
pairs of functions for which (1.2) => (1.3); and if furthermore F
l
then (1.2) <=> (1.3).
C
F2l
Clearly this equivalence holds for the class F
2
defined by what may be called the "separation approach":
F2 = {(2,k): 2
~
m
~
k for some m E Fl ,
and the expectations in (1.3) are defined}
(1.4)
(This approach is most useful when there is a simple direct description of
F2 , one that does not predicate the existence of quantities with certain
properties.)
When positive answers to the second and third questions are
feasible, one of the following relationships will follow:
i.
(1.1) => (1. 2) => (1.3)
ii.
(1.1) _._> (1. 2) <=> (1.3)
iii.
(1.1) <=> (1.2) <--> (1. 3)
(1.S)
An interesting example of (l.S.iii) is described by Kemperman [5].
Suppose that R is a partially ordered space and that (1.1) holds for the class
A of all measurable increasing sets A (i.e. a
the partial ordering, imply b
E
A).
E
A and a
~
b, in the sense of
Then, by consjdering simple function
approximations, one obtainS (1.1) <=> (1.2) where F is the class of all
I
-.
measurable increasing functions k (in the sense of the partial ordering) for
which the expectations in (1.2) are defined.
Using the separation approach one
also obtains (1.2) <=> (1.3), where F is defined by (1.4) and has the
2
alternative direct description as the class of all pairs of functions (2,k)
satisfying
k(x) ~ 2(y) ,
x
~
y ,
(1. 6)
--
4
and for which the expectations in (1.3) are defined, provided the "separating"
increasing function m defined for instance by
m(y)
=
sup{k(x),xsy}
is measurable, as is the case when R is the real line.
(We infer from a
comment made by Kemperman [S] that measurability of m may fail even in
m2 .)
New examples of (l.S.iii) are described in Section 3 for bivariate random
variables with normal distributions (Theorem 3.1) and with certain elliptically
contoured distributions (Theorem 3.2).
An interesting example of (l.S.ii) is described in Section 2 when the
range of Z
=
(X,Y) and Z'
=
(X' ,V') is the real plane.
If
A is the class of
all closed symmetric rectangles then (1.1) implies that
(1.7)
for all nonincreasing, convex functions h on the positive half line; i.e.
(1.1) => (1.2) where F is the class of all functions k(x,y) of the form
l
2
h(x +y2) with h as above. It should be pointed out that (1.1) no longer
implies (1.2) if h is either not nonincreasing or not convex; convexity would
2
be unnecessary if (1.1) implied that X2+y is stochastically larger than
X,2+ y ,2, which is not true in general.
In order to use the separation
approach, we note that functions f and g on the positive real line can be
separated by a convex function h,
f
s h s g ,
(1. 8)
if and only if
f[~s
+
(l-~)t]
s
~g(s)
+
(l-~)g(t)
,
sst ,
os
~
s 1 ,
(1. 9)
5
and then the convex separating function h can be defined (not necessarily
uniquely) by
t-u
u-s
h(u) = inf{---t
-s. g(s) + --t--s get),
(1.10)
Also, this choice of h, or some simple modification of it, is nonincreasing if
and only if
f(t)
~
g(s) ,
s
~
t
(l.11)
.
Consequently (1.1) implies
(1.12)
for all functions f and g satisfying (1.9) and (1.11) and such that the
expectations in (1.12) are defined; i.e. (1.1) => (1.2) <=> (1.3) where F is
2
2 2
2 2
the class of all pairs of functions £(x,y)=g(x +y ),k(x,y)=f(x +y ) where f and
g are described above.
Again, in this case, the class F , originally
2
introduced via the separation approach, has a direct description.
Section 2
includes additional implications of the form (1.1) => (1.2), which are
described for n-dimensional vectors Z and Z' (n
~
2), and also integral
representations for certain "n-monotone" functions which may be of
independent interest.
Kemperman [5] also describes an alternative approach based on a theorem of
Strassen [10] which guarantees that when R is a partially ordered complete
separable metric space and
A is the class of all measurable increasing sets,
then (1.1) is equivalent to the following:
There exist two random variables ZO,Zo with the same marginal
distributions as Z,Z' and such that Zo
~
Zo a.s.
(1.13)
6
It is then immediate that (1.13) => (1.3) => (1.2) => (1.1) where F ,F2 are
l
defined in the paragraph describing Kemperman's example, and thus
(1.13) <=> (1.1) <=> (1.2) <=> (1.3).
It turns out that this use of surrogate
random variables with certain specified a.s. properties (cf. (1.13)), the
"surrogate approach," provides a necessary and sufficient condition for
expectation inequalities of the type (1.3) in cases where no other approach
seems to work (including the separation approach) and even in cases where no
useful necessary and sufficient condition for (1.3) of the type (1.1) can be
found.
We illustrate the usage of this surrogate approach in a case treated in
Section 3.
Let Z
= (X,Y)
and Z'
= (X',Y')
be two-dimensional random vectors and A the
2
class of all principal lower and upper ideals in R , i.e. all rectangles of
the forms (-oo,x]x(-oo,y] and [x,oo)x[y,oo).
Then (1.1) is equivalent to saying
that Z and Z' have common marginal distributions and that (1.1) holds for all
principal lower ideals (_oo,x]x(_oo,y].
It is sho\Vll in [2] and [12] that
(1.1) <=> (1.2) where F is the class of all quasi-monotone functions k, i.e.
l
functions k which satisfy the inequalities
(1.14)
for which the expectations in (1.2) are defined and which satisfy some minor
regularity conditions.
It is not completely clear how the quasi-monotonicity
condition (1.14) should be modified in order to derive inequalities of type
(1.3).
The separation approach would yield (l.S.iii) with F defined by (1.4)
2
as the class of all pairs of functions 2,k which are separated by a
quasi-monotone function.
It then follows that
(l.b)
7
but we have been unable to find a direct description of the class F defined
z
through the separation approach.
Condition (1.15) is necessary but not
sufficient for £,k to be separated by a quasi-monotone function, as shown by an
example in Section 3.
Another example shows that (1.1) does not imply (1.3)
for all functions k and £ satisfying (1.15) (and for which the expectations in
FZ to satisfy
additional inequalities which are in the same spirit as (1.15). For instance,
(1.3) are defined).
One could conceivably require
(~,k) E
if the additional inequalities
do not hold, it is possible to construct examples where (1.1) holds but (1.3)
fails.
But even these inequalities are insufficient and we have failed to
obtain usable conditions describing the class
approach.
Fz by continuing with this
An alternative approach is to assume that k and £ satisfy no more
than (1.15), i.e. to define F as the class of all pairs of functions k and £
Z
which satisfy (1.15), and for which the expectations in (1.3) are defined, and
to impose additional assumptions on Z and Z', i.e. to strengthen condition
(1.1).
This can be best achieved through a variation of the surrogate approach.
To this end consider the following condition:
(CO)
There exists a four-dimensional random vector (Xl,XZ,Yl,Y Z) whose
values are in the set F = {(xl,xZ'Yl'Y Z) = (xZ-xl)(YZ-Y l ) ~ O}
and whose bivariate marginals Fll,FlZ,FZl,FZ2 for
(Xl'Yl),(Xl'YZ),(XZ'Yl)'(XZ'Yz) respectively satisfy
Fll + FZZ = 2H and FIZ + F = ZH', where Hand H' are the
Zl
distribution functions of Z and Z' respectively.
When (1.15) holds, condition (CO) implies
8
(1.16)
which, upon taking expectations (assuming they are defined), yields (1.3).
Hence (Cm => (1.3).
It is shown in Section 3 that when 2 and 2' are
normally distributed with common means and variances then
p
~
p' <=> (ca) <=> (1.1) <=> (1.2)
~
(1.3) ,
where pep') denotes the correlation between the components of 2(2').
In
Section 3, a similar result is obtained when Z and Z' have elliptically
contoured distributions; and also a generalization from two to higher
dimensions is described.
For the example of the preceding paragraph, as was mentioned, (1.1) does
not imply (1.3) in general.
(CO) <=>
(1.1)'
<=>
It is shown in Section 4 that
(1.3), where A2 is the class of all pairs of Borel sets A
and A' in the plane which are such that the functions l
A
and lA' satisfy (1.15)
~d
P(ZEA)
~
P(Z'EA') ,
(A,A')
E
A2 .
(1.1)'
Using a generalization of a theorem by Strassen [lOJ we obtain in Section 4
several further results of the type (CO) <=> (1.1)' <=> (1.3), where (1.1)' is
stronger than (1.1).
Among these the following is related to the inequalities
of Section 2 described earlier.
When R
=
2
R ,
A is the class of all closed
symmetric rectangles; F is the class of all functions k which satisfy
I
9
and for which the expectations in (1.2) are defined; and F is the class of all
2
pairs of functions
Q,
and k which satisfy
and for which the expectations in (1.3) are defined; it is shown in Section 4
that (I. 1) <=> (1.2) < > (1.3).
k(x,y)
=
It should be noted that the functions
2
f(x +y2), with fnonincreasing and convex, considered in Section 2, do
not belong to the class Fl'
Finally Section 4 derives several new inequalities
of the type (1.1) and (1.3) for normal and elliptically contoured distributions.
2.
n-monotone functions
In this section we develop inequalities for expectations of n-monotone
functions (to be defined below) of the squares of the moduli of n-dimensional
random vectors.
We begin with the case n
the general case n
~
2 (Theorem 2.2).
=
2 (Theorem 2.1) and then proceed to
In the process of establishing
Theorem 2.2 we develop an integral representation for certain n-monotone
functions (Lemma 2.3) which may be of independent interest.
Theorem 2.1.
Suppose Z
=
(X,Y) and Z'
=
(X' ,V') are bivariate random vectors
for which
P(IXI::;;a,IY!::;;b)
2
p(IX'I::;;a,IY'I::;b)
J
a
2
0
J
b 20.
(2.1)
Then
(2.2)
for every nonincreasing convex function f on [0,00).
10
·e
2
2
Condition (2.1) is equivaient to saying that aX' v bY' is
2
2
0 b ? -0, were
h
u v v d eno t es th e
stochastically larger than aX v by for a ~,
Proof.
maximum of u and v.
Thus for any bounded nonincreasing function h on [0,(0),
a
0 ,
~
b
~
0 ,
(2.3)
and, consequently,
E
J! [x~
h
cos
e
Y-;::~;--e] sin e cos
v -.
Sln
e de
~
E
i!
h[
X'~ e v _y-,2--::2;--e]
sin e cos
sin
e de .
cos
(2.4)
2
2
2
y
v
Now with the substitution of (x +y2)u for
x
. 2
.
cos 2 e
Sln
7T
2
x
Y
. 2 8] sin 8 cos 8 d8 simplifies to
Jo h·cos 2 e v Sln
[2
2
e
, the integral
2 2
foo h((x +y )u)
2 1
2
du .
1:.
u
Thus (2.2) holds for functions f of the form f(s)
=
OO
f1
h(su)
2 du, s
~
O.
But,
u
according to the lemma below, the class of such functions coincides with the
class of bounded nonincreasing convex functions.
boundedness is easily removed by truncation:
The unwanted restriction of
If f is any nonincreasing convex
function, then f v (-n) is a bounded nonincreasing convex function whose limit,
as n
+
00, is f.
Then (2.2) follows by means of the monotone convergence
o
theorem.
Lemma 2.1.
The cZass of bounded nonincreasing convex functions on [0,(0) and
the cZass of
•
funct~ons
f of the form f(s)
=
foo h(su)
1 .2
u
noninoreasing and bounded on
[O,oo)~
coincide.
du~
s
~ O~
with h
11
Proof.
[0,(0) .
Suppose f(s)
OO
= J1
5
~ 0, with h nonincreasing and bounded on
5
u
Quite obviously f is nonincreasing and bounded.
= 5 Joo5
convex, observe that f(s)
°<
h(su)
2 du,
To see that f is
h(v) dv for s > 0 and f(O)
2
= h(O).
Thus for
v
< t,
s+t)
f(s) + f(t) - 2f (--2--
OO
= 5 J5
h(v) -h
v
(T)
dv
2
oo h(v) -h [s+t)
2
+ t
Jt
s+t
v
--2--
sJ;t
=
s
h(v) -h
(T)
2-
5
dv +
=
° is similar.
2
dv
dv
h[S;tJ-h(V)
2
dv
s+t
v
t
t
v
The argument when 5
2
h(V)_h(S;t)
(s+t) r~
-
v
~
°.
--2--
Consequently, f is convex on [0,00).
Conversely, suppose f is a bounded nonincreasing convex function on [0,00).
Define a nonincreasing function hon [0,00) as follows:
h(s)
= f(s)
=
- sfO(s)
f(O)
for
5
>
°
for
5
-
0
where fOes) denotes, for definiteness, the smallest slope among all tangent
lines to f at s.
(Thus, when f is differentiable at
~
Then for fixed n
1 and
5
5
n
----·-s~/n~---- ~
due to the convexity of f.
> 0, fOes)
> 0,
1
f(s (1+1:.)) -f(s)
f(s) -
5
Thus
h(s)
~
f(s) -
5
f(s)-f(s(l--))
n
sIn
= f'(s).)
12
(n+l) f(s)
-
1
nf(s(l+-)) :s; h(s) :s; nf(s (l_l))
n
n
-
(n-l) f(s)
.
But
J~
(n+l) f(su) -nf(su(l+.!.))
n
u
du
2
=
1
1+(n+l) J n f(su)
2· du
1
u
-+
,£(s) as n
-+ 00 ,
since f is continuous at every point s > 0, and likewise
oo
J1
nf(su(l-~))-(n-l)f(su)
2
du
-+
f (s) as n
-+ 00 ,
U
from which it follows that
f(s)
roo h(su) du ,
2
=
Jl
s
~
0 .
u
From this integral it is easily checked that h is bounded.
o
There are unbounded nonincreasing convex functions f which cannot be
expressed in the above integral form, with h nonincreasing and necessarily
unbounded, e.g., f(s)
= -s,
s
~
O.
An analogue of Lemma 2.1 can be established
for nonincreasing convex functions f defined on the open interval (0,00).
Boundedness is not essential on (0,1], but is on [1,(0).
See Lemma 2.3 below.
Likewise Theorem 2.1 can be modified to cover functions f defined on (0,00)
which are nonincreasing and convex.
Such functions can be approximated from
below by functions of the type described in Theorem 2.1; and through use of the
monotone convergence theorem, we can obtain:
~
Corollary 2.1.
If, in addition to (2.1), P(Z=(O,O)) = 0, then (P(Z'=(O,O)) = 0
and) (2.2) hoZds for each nonincreasing convex function f on (0,00) for which
the expectations contained therein exist.
13
It is apparent from the nature of assumption (2.1), appearing in Theorem
2.1, that inequality (2.2) can be extended to
."
B :?:
a :?: 0 ,
0 .
There are, of course, many nonincreasing convex functions f to which
Theorem 2.1 or its corollary is applicable.
As an example the assumptions of
Theorem 2.1 imply ERa ~ ER,a, 0 < a ~ 2, where R2 ~ X2 + y2 and R,2
= X,2
+ y,2,
while the assumptions of Corollary 2.1 permit the conclusion ERa:?: ER,a, a ~ O.
The value of Theorem 2.1 and its corollary depends, of course, upon the
reasonableness of assumption (2.1), an inequality of type (1.1).
Theorem 2.1
of Das Gupta et al. [3] states easily checked conditions under which this
inequality hOlds for pairs of related elliptically contoured distributions,
as well as conditions under which assumption (2.7) holds in Theorem 2.2 below
and in its corollary.
The requirements in Theorem 2.1 that f be nonincreasing and convex are
both necessary for the generality of the theorem:
If f is any function on
[0,00) which satisfies (2.2) whenever (2.1) hoZds and the expectations make
sense~
Proof.
then f must be nonincreasing and convex.
The need for f to be nonincreasing can be seen by considering
nonstochastic Z and Z' of the form (x,O) and (x',O), 0
~
suppose f is nonincreasing and satisfies (2.2) for all Z
Z'
(X',Y') satisfying (2.1).
=
and Z
= S~y[~
i)~,
For s > 0 and p
E
x
x' <
~
= (X,Y)
(0,1], let Z'
00.
Now
and
=
!<
s2y
where Y is uniformly distributed on the unit circle.
Since
Z and Z' are elliptically contoured vectors which satisfy (2.1) (cf. Theorem 2.1
of
[3]), inequality (2.2) holds, which translates into
f(s)
~ 1T
-1 J·l
2 -!<2 f(s [l+up]) du ,
.
(l-u)
-1
s > 0 ,
P
E
(0,1] .
(2.5)
14
Replacing s by s - lin and letting n
£(s-)
~ TI
-1 Jl
00 yields
+
2 -~ f(s[l+up])du,
(l-u)
s > 0 ,
P
E
(0,1] ,
-1
which, in turn, due to the monotonicity of f, yields
f(s-)
~ TI
-1 JO
2 -!<:2 f(s-sp)du + Jl0 (l-u)
2 -!<:2 f(s+)du
(l-u)
= 2"1
1 f(s+)
f(s-sp) + 2"
-1
for s > 0 and P
(0,1].
E
+ 0, we obtain f(s+)
Letting p
~
f(s-), s > 0, which
establishes the continuity of f on (0,00).
Now suppose f is not convex so that for some 0
~
a < b, we have f(a) > feb)
and
(2.6)
f(a) + feb) < 2f (a;b) .
Consider lines t
= ms + c, a
~
s
For large values of c, the line t
[a,b].
~
b, of negative slope m
= ms
+ c > f(s) over the entire interval
Let c decrease until the line first touches the graph of f at some
point in the interval [a,b], and let
with this line.
Due to (2.6),
p =
= (f(b)-f(a))/(b-a).
f- :oJ
A
So
So
be the smallest such point of contact
(Since f is continuous on (0,00), both c and
So
are well-defined.)
is in the open interval (a,b).
So
and
[~o -1
J,
so that 0
~
p
~
1 and a
~
Setting s
=
s(1+up) < b for -1 < u
~
1, we
obtain from inequality (2.5):
f(sO)
~ TI
-1 Jl
2 -~ f(sO [l+up]) du
(l-u)
-1
tit
~ TI
-1
Jl
-1
2 -~ (ms [l+up] +c) du
(l-u)
O
This can only happen if
-1
~
u
~
1 ,
15
which is impossible (for negative u) due to the way
So
is defined.
Thus f must
~
o
be convex.
.
2
')
We remark that the random variables Rand R'''- , associated with the random
vectors Z
= s~ V (~
2
i] ~ and
ZI
= s~ V
(used in this proof), are not stochastically
= ER, 2 = s.
Thus condition (2.1) can hold without R2 being
stochastically smaller than R, 2
It follows, of course, that condition (2.1)
ordered since ER
can hold without (2.2) holding for every nonincreasing function f.
Finally it should be pointed out that the argument used in establishing
Theorem 2.1 shows that inequality (1.2) holds for all functions k of the form
Tf
k(x,y) =
J-02
F
[~v
e cos e
-.JxL]
d1J(e)
sin e
I-'
where Fe(r) is jointly measurable in (e,r) and nonincreasing in r for each
fixed e, II is a measure on the open interval (O,~), and the indicated integral
exists and is finite.
= h(r)g(e)
By choosing for instance Fe(r)
and nonincreasing and g bounded and ~ 0 (e.g. gee)
=
with h bounded
(sin e)n (cos e)m) and II
Lebesgue measure, we can generate a large class of symmetric as well as
nonsymmetric functions k(x,y).
The choice gee) = sin e cos e gives Theorem 2.1
2 2
for bounded nonincreasing convex functions of x +y ; k(x,y)
= f(x 2+y 2).
Theorem 2.1 can be generalized to higher dimensional vectors, and this is
done in Theorem 2.2 where the following terminology is used.
For 2
~
function f defined on [0,00) or (0,00) is said to be n-monotone if its k
divided differences are of alternating signs for 1
for odd k and of nonnegative sign for even k.
k
~
th
order
n, of nonpositive sign
(Thus [xO,xl;f], defined by
(f(xO)-f(xl))/(xO-x ), is nonpositive for distinct
l
f;
~
n < 00, a
o and
X
xl in the domain of
[x O,x l ,x 2 ;f], defined by ([xO,xl;f]-[xl,x2;f])/(xO-x2)' is nonnegative for
distinct x o' xl and x ; etc.)
2
It follows from Theorem A, page 238, of Roberts
•
4IJ
16
and Varberg [7] that f is n-monotone iff (i) it is nonincreasing on its domain,
..
(ii) it is (n-2)-times continuously differentiable on (0,00) with
Ie
s > 0 ,
= I, ... , n- 2
and (iii) (_1)n-2 f (n-2) is nonincreasing and convex on (0,00).
reference, we note that (iii) is equivalent to:
,
For future
(iii') (_1}n-2 f (n-2) is
locally absolutely continuous with a nonpositive and nondecreasing
(Radon-Nikodym) derivative (_1)n-2 f (n-l).
A function f defined on [0,00) or
(0,00) is said to be oo-monotone if it is n-monotone for all n, i.e., if f is
nonincreasing on its domain, and f is infinitely differentiable on (0,00) with
k k
(-1) f (s)
~
0, s > 0, k
~
1.
In Lemma 2.3 an integral representation is
obtained for all bounded n-monotone functions defined on [0,00), 2
well-known related notion is that of complete monotonicity.
~
n
~
00
A
A function f
defined on [0,00) or (0,00) is called completely monotone if it is continuous
on its domain, and it is infinitely differentiable on (0,00) with
(_l)kf(k)(s) ~ 0, s > 0, k ~ O.
Thus a completely monotone function is
oo-monotone, and if f is oo-monotone on [0,00) or (0,00), then _f(l) is completely
monotone on (0,00).
Completely monotone functions on [0,00) are Laplace
transforms of finite measures on [0,00), and completely monotone functions on
(0,00) are Laplace transforms of (not necessarily finite) measures on [0,00)
for which the Laplace transform is finite on (0,00).
Theorem 2.2.
(See Widder [13].)
If the randOm veotors Z = (Zl"",Zn) and Z' =
(Zi""'Z~)~
n ~ 2~
satisfy
(2.7)
17
and if f is an n-monotone function on
[O,oo)~
then
•
(2.8)
Remarks.
1.
Theorem 2.2 can be viewed as an extension to higher dimensions of
a well-known result in one dimension, provided one interprets a I-monotone
function as a nonincreasing function.
2.
Some examples of oo-monotone functions on [0,00), to which
Theorem 2.2 is applicable are:
e- as (a > 0), _sa (0 < a ~ 1),
(s+a) a (a < 0, a > 0), -log(s+a) (a > 0).
3.
An example of an n-monotone function which is not (n+l)-monotone
is the function f defined by f(s)
=
((l-s) v 0) n-r, s~ 0 (n ;: : 2).
Before we prove Theorem 2.2, we must gather together a number of facts,
some of which we state in the form of lemmas.
Lemma 2.2.
If h is a bounded function and the random vector (VI'"
is uniformly distributed on the surface of the n-dimensional unit
"Vn)~
sphere~
n ;: :
2~
then
(2.9)
Proof.
For n
=
2, (2.9) is established in the proof of Theorem 2.1.
We now
assume (2.9) is true when n in (2.9) is replaced by n-l, and proceed to
establish its validity for n (n ;: : 3).
Using the facts that V has density
n
18
n-3
2
(1_v )-2-
r~
2
=
'IT
that, conditioned on V
n
!<2'
cn-1
2
= v,
2 -1z
(I-v)
-1 < v < ] ,
(V1, ... ,V _ ) is uniformly distributed on
n l
the (n-l)-dimensional unit sphere (see for instance Lemma 3 in [1]), and
therefore that (by the induction hypothesis) the conditional expectation of
h
[:~
v ...
v
:~] Iv
nI given Vn = v is
1 ... V
n-l
2 -2v(l-v )
r [n;l)
n-l
-2'IT
J
001
(u_l)n-3
n-l
u
(n-3)!
we obtain (after some minor simplifications)
4~]
n
[n
Ellvn IV.]. I
1
V~].
1
= Jl
-1
+[~ X~]~IV.I Vn
=V
1 V. 1
].
]fVn (v)
dv
].
(2.10)
r [~)
= n 2
J~
'lT / (n_3) !
(u_l)n-3
un-l
u(r 2 _x 2)
With the change of variable v
+
1
fo h
x2
y: ---~2~n- v ~
I-v
=
[(r 2-xn)u
2
I-v
2
v
i 2] v(l-v 2) n-2
x
n
2
r y, the inner integral in
v
(2.10) simplifies to
r
x 2( r 2y-x 2)n.-2
2
h(r
y) n
In
+ ( r 2-x 2)n-l} y ·n dy,
2(n-l)
n
2 2
2
nr
{
u(r -x )+x
u
n
n
u
n-1
r
and (2.10) becomes
2
dvdu .
19
2
2
r y-X n
~
n
2
-2(n-l) Joo her y)
r
1
n
(u-l) n-3
J1
y
2 (n-3)!
2
r -x n
2
x 2 (2
r y-x 2)
n u - n
n 1
{
+
2 2 n-l }
(r -Xn)
dudy .
IT
The inner integral equals
2 2 jn-2
-xn
xn (r y-x n )n_2 1- 2
2.
[ r y-x
2
2
1
r
+
jn-2
r 2 y-x 2
(r 2 _x 2)n-l _L
n_ l
n-2 [ 2 2
n
r -x
n
n
_ 1
2(n-l)( 1)n-2
- n-2 r
y,
and thus (2.10) becomes
r [E.2 1
---~---~n 2
lT / (n-2) !
J7
(1)n-2
2
y- n
her y)dy
y
o
By using the same argument used in proving Theorem 2.1, together with
Lemma 2.2, one can readily establish (2.8) for functions f defined on [0,00) of
the form
f(s)
=
Jool
(u-l)
u
n-2
n
where h is bounded and nonincreasing.
h(us)du
S ::0:
°,
(2.11)
The class of such functions is
characterized in Lemma 2.3, which follows.
Lemma 2.3.
The class Fn of functions f described by
and nonincreasing function on
n-monotone functions on [0,00).
[o,oo)~
(2.ll)~
with h any bounded
coincides with the class of bounded
The class
F~
of functions f of the form
20
=
f(s)
s
r: (v-s) n-2 h(v)dv "
s
s >
n
v
°"
(2.12)
1;.J1:th h any nonincreasing function on (0,00) and bounded on [1,(0)" coincides luith
the cZass of n-monotone functions on (0,00) which are bounded on [1,(0) (n?- 2).
An immediate consequence of this lemma is that the classes
Remark ..
Fn and F'n
If f E F (EF') for every n, then f-f(oo) is a
are nonincreasing in n.
n
n
completely monotone function on [0,(0) (on (0,00)), and, therefore, f must be of
the form
f(s)
=c
roo -su
+ )0 e
dV(u) ,
S
?- 0 (s > 0) ,
where c is any real number and V is a finite measure on [0,(0) (V is a measure
on [0,(0) for which the integral is finite for all s > 0).
Proof.
We shall prove the characterization of F'.
n
Since the right-hand side
of (2.12) is equal to the integral in (2.11) when s > 0, the stated
characterization of
Suppose f
E
F'.
n
Fn is easily inferred from that for F'.
n
It is clear from (2.11) that f is bounded on [1,(0), and
from (2.12) that f is (n-2)-times continuously differentiable on (0,00) with
f
(k)
(s)
= (_I)k-l
k 2
(n-2)! foos (V-S)nn- (n-k)!
[kv - (n-I)s]h(v)dv ,
v
s >
°,
(2.13)
1 ::; k ::; n - 2 .
Also f(n-2) is locally absolutely continuous with a (Radon-Nikodym) derivative
f(n-l) (s)
=
(_I)n-2 (n-I)! foo h(v) dv
s
v
n
+
(_l)n-l (n-2)! s-(n-l) h(s) a.e. on (0,"') .
21
which, after integrating by parts, simplifies to the version we will use:
f(n-l) (s) = (_l)n (n-2)!
foos
v-(n-l) dh(v),
s > 0 .
Clearly f(n-1) (00) (=lim f(n-l)(s)) exists and equals zero.
(2.14)
In fact,
s-?oo
s k-l f (k) (s)
+
0 as s
+
00 ,
1:-::;k:-::;n-1.
(2.15)
This is obvious from (2.14) for k = n - 1 and from (2.13) for 1 :-: ; k :-: ; n - 2.
.
.
Ob serve t h at (l)
- n-lf(n-l).1S non1ncreas1ng.
follows by (2.14) that (-1)
induction:
S·1nce h·1S non1ncreas1ng,
.
.
.
1t
n-l (n-1)
f
is nonnegative.
Proceeding by backwards
From the nonnegativity of (_1)n-1 f (n-1), we infer that (_1)n-2 f (n-2)
is nonincreasing.
Since f(n-2) (00) = 0 (implied by (2.15)), (_1)n-2 f (n-2) is
nonnegative, etc.
Thus (-l)jf(j) is nonnegative for j = 1, ... ,n-1, and
(_1)n-1 f (n-1) is nonincreasing, which together say that f is n-monotone.
Conversely, suppose f is n-monotone on (0,00) and bounded on [1,00).
Define
h on (0,00) by
n-l
h(v) = (n-1)
I
(-~~
k
vkf(k) (v)
v > 0 ,
(2.16)
k=O
which is nonincreasing because each term in the sum is nonincreasing (a
consequence of f being n-monotone).
Observe that h is of bounded local
variations and
(1)
( -1) n-1 n-l
dh(v) = (n-2)! v
df n- (v),
i.e. ,
(2.17)
22
(1)
( _l)n-l
n 1
hey) - h(s) = (n-2)! fV u - df n- (u),
s
o<
s
~
v <
00
•
(2.18)
Then for s > 0 (using (2.18)),
S
n-2
( v-s )
oo -=----"-(h (v) -h (s)) dv
fs
n
=
v
(_l)n-l s foos (v_s)n-2 fV un - 1 df(n-1) (u)dv
(n-2) !
n
s
v
n-1
roo { n-1
n-l} (n-l)
= en-I)!
1s u
- (u-s)
df
(u) :: Fn-l (s) ,
(-1)
say, where we have applied Fubini's theorem for nonnegative functions (without
knowing, as yet, that F
n-
l(s) is finite).
In what follows, we shall need to
use (2.15), which should be justified in the present context.
Lemma 2.4 below for the k
2
~
k
~
n - 1.
th
This is done in
derivative of an arbitrary n-monotone function,
The remainder of (2.15), for k
= 1,
is valid in the present
context since, by assumption, f is bounded on [1,00).
Now, using integration by parts and (2.15) for k
(-1)
n-1
F 1 (s) - - (n-1)!
n-
s
=n
- 1, we obtain
n-l (n-l)
f
(s) + Fn _2 (s) ,
where
(-1) n-2 roo { n-2
( 1)
u
- (u_s)n-2} f n- (u)du.
Fn _2 (s) =. (n-2)! 1
s
Proceeding by backwards induction, we are eventually led to
s
f:
(
)
v-S n
v
n-2
n-l
(h(v)-h(s))dv = f(s) -
\'
L
k=O
(-1)
k!
k
s
kf(k) ( )
s
23
which, in view of (2.16), establishes (2.12).
The boundedness of h on [1,00)
0
may be inferred from (2.12).
If f is n-monotone on [0,00) or (0,00), then (_1)k f (k) is nonnegative and
nonincreasing on (0,00) for k
o~
s(_l)k f (k)(s) ~
IS
= 1, ... ,n-l,
and, hence,
(_l)k f (k)(u)du
s/2
(2.19)
s > 0 ,
1
~
k
~
n - 1 .
These inequalities permit us to describe the behavior of the derivatives of f
as s
~oo
and s
Lemma 2.4.
+ 0:
If f is n-monotone on [0,00) or (0,00) for some n
s
Proof.
k-l (k)
f
(s)
~
0 as s
~oo
,
~
3, then
2~k::;n-l.
(2.20)
This follows by induction from (2.19), provided (corresponding to
k - 1 = 1)
£(1) (s) _ f(1) (s/2) ~ 0 .
But this is the case since f(l) is nonpositive and nondecreasing.
Lemma 2.5.
If f is n-monotone on [0,00), or n-monotone on (0,00) with f(O+)
finite, for some n
~
2, then
(2.21)
Proof.
In either case, f(O+) exists and is finite.
s + 0, and (2.21) follows from (2.19) by induction.
Thus f(s) - f(s/2)
~
0 as
o
24
Proof of Theorem 2.2.
In view of the remark preceding Lemma 2.3, we may take
(2.8) to be established for all bounded n-monotone functions f on [0,00).
The
proof for unbounded f requires the removal from f of its (possible) linear part
and then a truncation argument.
~
Suppose first that f(s) = -cs, s
0, where c > 0.
Inequality (2.8) can
be expressed as
n
n
I1 EZ~1
I
~
1
,2
EZ.
1
,
which must hold since (2.7) implies Z: is stochastically smaller than Z~,
1
1
i=l, ... ,n.
Now suppose f is any unbounded n-monotone function on [0,00), i.e. f(oo) = _00.
~
Since f(l) is nonpositive and nondecreasing, the finite nonpositive limit
f(l) (00) exists.
We shall assume, without loss of generality, that f(l) (00) = 0.
For otherwise, we may express f as the sum of two n-monotone functions,
f = f
l
+ f
2
where fi l ) (00) = 0 and f (s) = f(l) (00) • s, s ~ 0, and treat the
2
parts independently.
(2.16) and h(O) = (n-l)f(O).
f
x
Define h on (0,00) by
We shall truncate f as follows:
For x > 0, let h (v) = h(vAx) , v
x
~
0, and define
by (see (2.11))
f (s) = fool (u-l)
x
u
n-2
n
h (us)du
s
x
~
0 .
(2.22)
Since h is nonincreasing on (0,00) (see (2.17)) and
h(O+) = (n-l)f(O+)
~
(n-l)f(O) = h(O) (cf. Lemma 2.5), it follows that h is
nonincreasing on [0,00) and, consequently, h
function on [0,00) for every x > O.
x
is a bounded nonincreasing
This implies that (2.8) holds for each
25
function f , and it only remains to show, if possible, that f
X
X
+ f as x
that (2.8) follows for f itself by the monotone convergence theorem).
is nonincreasing in x, so is f
x
x
=
ro
(so
Since h
x
(apparent from (2.7.2)), and thus it is only
necessary to show the pointwise convergence of f
From (2.22) we have f (0)
+
(n-l)h(O)
=
to f.
x
f(O) for all x > O.
focus our attention exclusively on points s > O.
Thus we may
For such points, it is more
convenient to use the following variant of (2.22) (see (2.12)):
f (s)
x
=
s
foos
(v-s)
n
n-2
v
h (v)dv
x
s > 0 .
For x > s > 0, we have (using (2.18))
f (s)
x
t
= s s (v-s)n
v
= s f~
n-2
h(v) dv
+ S
Cv_s)n-2 {C_l)n-l
v
n
(n-2) !
+
= ,,;,~~_="""i~~~_1
(v-s)
n
v
ts un-l
s
e
dv • hex)
{I _(l_~ln-l}
x)
+
~(l_~]n":l
n-l
x
- [1- ~rllfCn-l) Cn)
~~~)
{I - (l- f r
_ _ ( _l)n-l fX (u_s)n- 1 df (1)
n- (u)
(n-l)!
n-2
df Cn-l) Cn) } dv
hex)
(n-l)·
f~ nn-l{(1- ~rl
+
!':sO
1
}+
~~~) (1_~rl
hex)
n-l
+--
By repeatedly integrating by parts (much as in the proof of Lemma 2.3), we
obtain
4It
26
f
x
(s)
=
x > S >
°.
Since, as x ~ 00, f(l) (x) ~ f(l) (00) = ° (assumed without loss of generality),
it follows from Lemma 2.4 that the sum converges to zero as x
f (s)
~
x
f(s) as x
~
~
Thus
00.
00, which completes the proof.
0
Theorem 2.2 can be extended to n-monotone functions on (0,00), to allow for
functions which are unbounded at zero as well as at infinity.
If, in addition to (2.7), P(Zl=o""'Zn=O) = 0, then
Corollary 2.2.
(P(Zi=o, ... ,z~=O) = ° and) (2.8) hoZds for each
function f on
for which the expectations in (2.8) are defined.
(0,00)
~
n-monoton~
Proof.
Let f be n-monotone on (0,00) with f(O+) = 00 and f(oo) = _00.
f with smaller ranges can be handled similarly or more easily.)
(Functions
Let So be the
zero of f, f(sO) = 0, and for each k > (2/s ) define fk(s) = f(s+f)' s ~ 0.
0
Then each f k is n-monotone on [0,00), and by Theorem 2.2, Ef (llzI1 2) ~ Efk(IIZ'
k
Also as k t 00, f t f on (0,00). More precisely, f~ t f+ and by monotone
k
2
2
convergence Ef~(1 Izi 1 ) t Ef+(1 Izi 1 ). Also for s ~ So - f' since ° ~ _f(l)+,
11
2
).
we have
1
s+-
= - Jk
f(1) (u) du
s
s
e
and thus ° ~ f (s) - f- (s) ~ fl f(l) ( 2°)
k
I,
s ~ O.
It
2
and Ef- ( II Z 11 ) are finite or infinite together and thus Ef
....
~------~-----------~~-~-_.-
k(\I Z\12)
follows that Ef
2
k(II Z11 )
2
+ Ef- ( II Z 11 )
27
(by dominated convergence if they are finite or trivially if they are infinite).
Since Ef(IIZI12) is defined by Ef+(llzI12)_Ef-(llzI12) iff at least one of the
n
two terms if finite, (2.8) follows.
Some examples of an oo-monotone function, to which Corollary 2.2 is
applicable, are:
sa (a < 0), -log s.
We have already shown that the 2-monotone functions provide the appropriate
class for the result of Theorem 2.1.
The following example shows that
3-monotone functions provide the appropriate class for the result of
Theorem 2.2 for n = 3 (by constructing, for a 2-monotone function which is not
3-monotone, 3-dimensional random vectors Z and Z' which satisfy (2.7) and for
which (2.8) fails), and we anticipate that similar examples would show the same
for n > 3.
Example.
a and b, a
Suppose f is 2-monotone but not 3-monotone on [0,00).
~
Then for some
0, b > 0, one has
f(a+3b) - 3f(a+2b) + 3f(a+b) - f(a) > 0 .
(2.23)
(Implicitly we are saying that functions f which are 2-monotone and satisfy the
converse of (2.23) for all a and bare 3-monotone, which can be verified.) Let
222
3a =a and 2a + B = a + b be used to define a and B (0 ~ a < B), and let Z
and Z' be three-dimensional random vectors whose distributions are described by
the following table:
28
z
P (Z~z)
P(Z'~z)
P (Z=z)
P(Z'=z)
11
9
11
9
BY
BY
8T
15
(a,a,a)
BY
(a, a, B)
BY
(a, B, a)
4
6
15
15
8T
15
BY
8f
8f
BY
CB,a,a)
6
8f
BY
(a, B, B)
8
8f
(B,a,B)
Sf
6
Sf
6
8T
(B, B, a)
8
6
8f
15
8I
27
8I
27
8f
27
8I
(B, B, B)
34
Sf
Sf
36
Sf
15
8I
27
8I
27
Sf
27
Sf
1
1
4
6
BY
4
8
BY
From the last two columns it is apparent that condition (2.7) holds.
R2
= zzt
and R,2
= z'z,t,
Now for
we have
2
11
12
24
34
Ef(R ) = 8f f(a) + 8f f(a+b) + 8f fCa+2b) + 8T f(a+3b) ,
2
Ef(R' )
9
= Sf
18
18
36
f(a) + 8f f(a+b) + 8f f(a+2b) + 8T f(a+3b) .
From (2.23) it follows that Ef(R,2) > Ef(R 2).
Consequently, the assumption of
3-monotonicity in Theorem 2.2 when Z and Z' are three-dimensional is essential;
o
it is impossible to consider a larger class of functions.
3.
~
Expectation inequalities for pairs of functions
In this section we consider random vectors Z and Z' (i.e. R =
mn )
which
satisfy (1.1) with A the class of all principal lower and upper ideals (-oo,z]
29
and [z,oo), z
E
n
R.
When n = 1, (1.1) says that Z and Z' have the same
distribution, which is not interesting.
When n = 2, (1.1) says that Z and Z'
have the same marginal distributions and that their bivariate distribution
functions H and H' satisfy H
~
H'.
Our attention will be focused on the
bivariate case; and only at the end of the section will we consider a higher
dimensional case.
It is shown in [2, 12] that (1.1) < > (1.2) with
FI the class of all
quasi-monotone functions (cf. (1.14)) for which the expectations in (1.2) are
defined and which satisfy certain minor regularity conditions.
Theorem I in [2].)
(See
The separation approach yields (1.5.iii) with F defined by
2
(1.4) as the class of all pairs of functions
quasi-monotone function m:
~=m+f,k=m-g
~,k
which can be separated by a
where f and g are nonnegative.
(Large
4IJ
classes of quasi-monotone functions are known or can be constructed; see for
instance [2].)
When
~
and k are separated by a quasi-monotone function then
they satisfy (1.15) (cf. (1.4) and (1.14)).
for
~
However (1.15) is not sufficient
and k to be separated by a quasi-monotone function:
functions
~
there exist
and k satisfying (1.15) which are sufficiently close that no
quasi-monotone function can exist between them.
This is easily demonstrated
with the aid of Figure I.
In Figure I, relevant values of k and
~
are indicated at various points
within an array of eight points possessing a particular geometric orientation
in the plane.
In order to obtain a contradiction, it is assumed that a
quasi-monotone function m satisfying
~ ~
m
~
k does exist.
Figure I indicates
two points in the array where it is impossible to define m simultaneously.
not given in Figure I.
In
30
(k=O)
0
0
I
I
I
I
I
I
(£=0)
(m=S)
(k=£=l)
0-------------0-------------0
(£=0)
0-------------0-------------0
I
I
I
I
I
I
I
I
I
I
I
I
I
(£=0)
(k=O)
(m=a)
FIGURE I
By referring to the four points in the lower left-hand corner of Figure I, one
can easily deduce from £
inequality a
+
~
m
~
S ~ 1 must hold.
contradicting inequalities a
~
k and the quasi-monotonicity of m that the
In the same way, one can obtain the
1 and
S
~
1 by examining the four points in the
lower right- and upper left-hand corners of Figure I, respectively.
quasi-monotone function m exists which satisfies £
~
m
~
k.
Thus no
This example
suggests the possibility that (1.1) can hold without (1.3) holding for all
functions k and £ which satisfy (1.15) (and for which the expectations in (1.3)
are defined).
In fact, an example based upon Figure I is easily constructed:
Let the distribution of Z assign mass 1/3 to each of the points in Figure I at
which £
= 0,
and let the distribution of Z' assign mass 1/3 to each of the
three points in Figure I at which k is explicitly defined.
It is easily
checked that (1.1) holds but E£(Z) < Ek(Z').
Thus if Z and Z' are bivariate random variables with equal marginal
distributions and with bivariate distribution functions satisfying H
~
H' (i.e.
if (1.1) holds), then in general this does not imply that (1.3) holds for all
functions k and £ satisfying (1.15).
We now show that this implication is true
in certain special cases, using the variation of the surrogate approach
involving condition (CO), which is described in the introduction.
with the normal case.
We begin
31
Suppose Z and ZI are bivariate normal random variables with
Theorem 3.1.
oommon means and varianoes and with oorrelation ooeffioients p and p'
saMsfying the inequality p
k and
Q,
~ p'.
Then (1. 3) holds for every pair of funotions
whioh satisfy (1.15) and for whioh the expeotations appearing therein
make sense.
Proof.
Let (S,T,U) be normally distributed with a zero mean vector and the
covariance matrix
=
E
(S,T,U)
1
p
(p'_p)
p
1
(pl_p)
(pl_p)
(pl_p)
2(p_pl)
Define
X =]1 +0' S , Y =]1 +0' T , X =]1 +0' (S+U) , Y =]1 +0' (T+U)
l x x
l y y
2 x x
2 y y
where
(]1 ,]1 )
x
y
of H and H'.
is the common mean vector, and
0'2
x
and
0'2
Y
,
are the common variances
It is easily checked that (Xl,Y ) and (X 2 ,Y ) have distribution
l
2
function H and that (X ,Y 2) and (X ,Y ) have distribution function H'.
l
2 l
Moreover (X 2-X l ) (Y 2-Y1)
= O'xO'y U2
~
O.
Thus condition (CO) is satisfied and
(1.3) follows from (1.15), via (1.16) as discussed in the introduction.
0
Thus when Z and Z' are as in Theorem 3.1, and F is the class of all
2
pairs of functions
Q,
and k satisfying (1.15) and for which the expectations in
(1.3) are defined, we have p
~
p' => (1.3).
On the other hand we clearly have
(1.3) => (1.2) and, as was already mentioned, (1.2) <=> (1.1) which in this
case is equivalent to H
~
H'.
Thus p
~
p' => H
~
H', which is a special case
of an n-dimensional result due to Slepian [9], and which implies p
~
p' <=> H
~
e
H;.
32
It then follows that when Z and ZI are bivariate normal variables with common
means and variances then
p
~
pI <=>
H
~
HI <=> (1.2) <=> (1.3) .
Theorem 3.1 can be extended to higher dimensions and from normal to
elliptically contoured distributions.
If Z is an n-dimensional random (row)
vector and, for some n (row) vector lJ and some nxn nonnegative definite matrix
E, the characteristic function ¢Z -]J (s) of Z-lJ is a function of the quadratic
t
form sEs , ¢Z
(s)
.
= ¢(sEs t ),
we say that Z has an elliptically contoured
-lJ
distribution with parameters lJ, E and ¢, and we write Z ~ EC (lJ,E,¢). When
n
¢(u)
=
exp(-u/2), EC n (lJ,E,¢) is the normal distribution Nn (lJ,E).
The location
and scale parameters lJ and E can be any n vector and any nxn nonnegative
definite matrix, while the class
rank k of E, r(E)
=
~k
of admissible functions ¢ depends on the
k, and consists of all functions of the form
¢eu)
=
J
2
~(r u) dF (r) ,
u
~
0 ,
[0,00)
2
k
for some distribution function F on [0,00), where ~(llsll ), s E :rn. , is the
characteristic function of the uniform distribution on the surface of the unit
k
sphere of R.
This follows from a theorem of Schoenberg [8] and is discussed
in [1] where the following useful stochastic representation is also introduced.
Let E
= At A be
a rank factorization of E, Le. A is kxn and r(E)
= k = r(A).
Then Z has the "canonical" stochastic representation
Z ~ lJ + RU (k) A
where the equality is in distribution, R is a nonnegative random variable (with
distribution F), u(k) is a k-dimensional random vector uniformly distributed on
33
the surface of the unit sphere in R k , and Rand U(k) are independent.
Suppose that Z
Theorem 3.2.
and p
~
~ EC2(~'~'¢)
~ EC2(~'~"¢)
Proof.
Then (1.3) hoZds fop evepy paip of functions k and
pl.
~
which satisfy
If p = 1 and p' = -1 we have
where A =(oI,02)' Ai=(oI,-02)'
I
E~(Z)
~
Since Rand U(I) are independent, in order to
Ek(Z') it suffices to show
r
Since k(o) and
r·~
whepe
and fop which the expectations appeaping in (1.3) ape defined.
(1.15)
show
and Z'
O.
~(.)
satisfy (1.15), so do k(Wro) and
~
0 .
~(wr·)
(3.1)
for every ]J and
Thus it suffices to show (3.1) when]J = 0 and r = 1, i.e. it suffices
to show E~(U(I)AI) ~ Ek(u(I)Ap, which is written as
and follows from (1.15).
Now assume that at least one of p,p' differs from 1 in absolute value.
Putting
tIJ
34
cos CI.'
cos CI.
A
= [::
:] and A'
sin CI.
' -TI,
h
were
CI. and CI.,
4 ~ CI.
~
CI.
~
sin CI.'
TI are d ef'lne d by
4'
have L: = At A and L: I = AIt A'.
p = sl'n 2CI. and p I = sl'n 2CI. ' , we
When -1 < p' < p < 1 then both L:, L: ' are full
rank and r (A) = 2 = r (A ') so that L:=At A , L: I=A ItA' are rank factori zations of
It then follows that
L:,L:'.
(3.2)
When one, but not both, of p,p' equals 1 in absolute value, say -1 < p' < P = 1,
then Z' g ]..I + RU(2)A' and EeiS(Z'-]..I)t = <j>(sL:l s t ) = J n (r 2SL:l s t )dF(r)
2
[0,00)
where F is a distribution of R.
Since
2
t·
J n2 (r sL:s ) dF (r)
[0,00)
it is easily checked that Z
g ]..I
+ RU(2)A.
Hence (3.2) holds provided at least
one of p,p' differs from 1 in absolute value.
Because of the independence of
R,U(2), arguing as before, it suffices to show that E1(U(2)A) ~ Ek(u(2)AI).
This will be done by defining a random vector (Xl ,X 2,Y l ,Y 2) which satisfies
condition (CO); (Xl'Y ) and (X ,Y ) will be distributed as U(2)A, (X ,Y ) and
2 2
l 2
l
(X ,Y ) will be distributed as U(2)A', and the product (X -X l )(Y 2-Y ) will be
2 l
2
l
nonnegative.
The random vector U(2) can be taken to be (sin 8,cos 8), where 8 is
uniformly distributed on any interval of length 2TI.
e
U(2)A
= (0
1
sin(8+CI.) ,
O
2
cos(8-Cl.)) , U(2)AI =
(0
1
Then
sin(8+CI.') ,
O
2
cos(8-Cl. ' )) .
(3.3)
35
Let
S=sin(8+a) , T=cos(8-a) , V=sin(a'-8) ,
and define
Since 2 sin(a-a')cos(a+a') = sin 2a - sin 2a' = p - p'
(X2-XI~(Y2-YI) ~
O.
~
0, it follows that
By further trigonometric manipulations, one obtains
S + 2 cos(a+a')V = sin((2a'-8)+a) , T+2 sin(a-a')V = cos((2a'-8)-a) .
Since 2a'-8 is uniformly distributed on an interval of length
well as (XI,Y ), is distributed as U(2)A.
I
and (X ,Y ) are distributed as u(2)A'.
2 I
Remark.
2~,
(X 2 ,Y 2), as
Similarly, one finds that (X ,Y )
I 2
Implicit in this proof is the use of a fact about any two ellipses
which are inscribed in the same rectangle.
Each point on one of the ellipses
is the vertex of a rectangle whose opposite vertex is on the same ellipse and
whose adjacent vertices are on the other ellipse.
rectangles.)
(In fact there are two such
Whether this is a known fact from projective geometry is unknown
to the authors.
(In the present context, the two ellipses are the ranges of
the random vectors U(2)A and U(2)A'.)
When k and
~
are functions of x ± y.
Suppose k and ~, which are defined for (x,y)
sum x+y.
of
E
2
R , are functions of the
For convenience, we shall write them as k(x+y) and
~(x+y).
--
o
This completes the proof.
In terms
36
the inequalities (1.15) become
2(u+v) + 2(u-v)
~
k(u+w) + k(u-w) ,
O:::;w:::;v<oo.
(3.4)
= 2, this condition is equivalent to convexity. Thus k(x+y) is
When k
quasi-monotone (as a function of (x,y)) if and only if it is a convex function
(of the single variable t
= x + y).
In general, k can be no larger than k* defined by
k*(u)
= ~ inf{2(u+v) + 2(u-v)} .
(3.5)
v~O
(Set w = 0 in (3.4).)
Assuming that k* is finite and measurable, k can equal
k* if and only if
inf{2(u+w+a) + 2(u+w-a)} + inf{2(u-w+a) + 2(u-w-a)}
a~O
for all u E lR, 0 :::; w
and v
~
2{2(u+v) + 2(u-v)}
a~O
~
~
v < 00.
w, there exists an a
~
This condition is met if for each u E lR, w
~
0
0 such that
2fu (v) ,
(3.6)
where f u is the even function defined by f u (v) = 2(u+v) + 2(u-v). A suitable
value for a can be obtained if f u is nondecreasing (set a = 0), nonincreasing
(set a
=
v + w), or concave (set a
=
v) on [0,00).
Summarizing, if k* is finite
and measurable, and if for each u, 2(u+v)+2(u-v) is a nondecreasing,
nonincreasing or concave function of v on [0,00), then k* is a suitable and
maximal choice for k.
function
k·~
Under these conditions on k* and 2, any measurable
k* satisfies (3.4).
37
The following are examples of functions 5/, which meet these conditions:
(i) If
51,
= x+
is convex (as a function of the single variable t
y), then
5/,(u+v)+5/,(u-v) is a nondecreasing function of v on [0,00) and k*
= 5/,.
(ii) If 5/, is a symmetric function with respect to some point (u O,5/,O) in R
2
= 25/,0' v ~ 0), and 5/, is concave on [uO,oo),
O
O
then 5/,(u+v)+5/,(u-v) is a nonincreasing function of v ~ 0 for u ~ Uo and is
(i. e., if 5/,(u +v) + 5/,(u -v)
a nondecreasing function of v
on (_oo,u
~
0 for u
~
u
o.
It follows that k* =
51,
o].
The values of k* on (uO,oo) must be evaluated from (3.5) in each specific
case, and it must be checked whether k* is finite and measurable.
instance, if
mean
~
51,
For
is the distribution function of a normal random variable with
and variance
2
0 ,
then k*
is a maximal choice for k.
= 51,
on
(_oo,~]
and k*
= 21
[~,oo);
on
hence k*
4IJ
(This is another example of functions k and 5/,
satisfying (l.15) which can not be separated by a quasi-monotone function m,
i.e., there is no function m which satisfies (1.14) and k
m
~
~ 51,.
The
details are left to the reader.)
When the supports of X+Y and X'+Y' are not the entire real line, the
ranges of u, v and w in (3.6) can be reduced.
Thus k and
51,
will have to
satisfy fewer restrictions, and a wider variety of suitable k,5I, pairs will be
permitted.
(The same can be said when k and
51,
are general functions on
]R2
and
2
the supports of (X,Y) and (X',Y') are proper subsets of R .)
When k and
51,
are functions of x-y (instead of x+y) the analogue of (3.4)
is
5I,(u+w) + 5I,(u-w)
~
k(u+v) + k(u-v) ,
U E
R ,
O~w~v<oo,
(3.7)
(the definitions of u, v and w must be modified appropriately) and the analysis
of (3.7) is similar to that given for (3.4).
38
A generalization to higher dimensions.
A higher dimensional version of Theorem 3.2 can be proven by reducing
the dimension to 2 through conditioning.
-
,.
1 .
J.
( ' ) , a . . -a..
' , a . . ;:0; a. . ,
L:--( a.. ) , "
L. ,
- -a..
1J
1J
functions k and
11
~
of
n
11
1J
1J
;t
Then
(1. 3)
ho lds for any pair of
variables for which the expectations appearing in (1.3)
make sense and which satisfy (1.15) as functions of any two of the n variables
for all fixed values of the remaining n-2 variables.
Proof.
According to the argument in the first paragraph of the proof of
Theorem 5.1 in [3], it suffices to prove the result in the case where a 12 > ai2
and a ij
=
a~j for all (i,j)
Zl'~l
;t
(1,2),(2,1).
Write
are two-dimensional and L: ll is 2x2.
characteristic function that
where
It is easily seen via the
where L:;2 is the self-adjoint generalized inverse of L: 22 and r,* = Ell - L:12 E;2 L: 2l .
Now let RU(n) = R(U l ,U 2) ~ ECn(O,I,~), where U is two-dimensional. Then
l
and
39
Since U(n) is uniformly distributed on the surface of the n-dimensional unit
(See for instance
Lemma 3 in [1].)
Since Rand V are independent, it follows that for all r
~
0
and u ,
2
(3.8)
where
Since
2:*'
r*'
=
= r*
and
suffices to
show that for all r
E.Q,(ll*+r*U (
~
2)
0 and u ,
2
~
~
2:* 2 , l'1'-"2 +ru 2 2: 22
2 )
~
Ek(11*+r*U
I'-"
(2)
2:*'
~
2
~
,
111'-"2 +ru 2 2: 22
2 )
o
and this follows from Theorem 3.2.
Remarks.
(_oo,z],
1.
Z
By letting k and .Q, in Theorem 3.3 be indicator functions of
ERn, one obtains the inequality H ~ H' where Hand H' are the
distribution functions of Z and Z' respectively.
For normal distributions this
inequality is a well-known result due to Slepian [9], and for elliptically
contoured distributions it has been obtained by Das Gupta et al. [3] under the
assumptions that the matrices 2: are invertible and that densities exist.
e
40
2.
Under the assumptions of Theorem 3.3 it is also true that
n
n
P(ZEA) ;::; P(Z' € A) for every set of the form {U€lR , u;::;?}, z € lR. Thus all
results in this section are concerned with random vectors that are
stochastically ordered in the sense described in the beginning of this section.
(This observation may point the way to extensions that are not confined to
Euclidean-space-valued random variables.)
3.
The approach used in proving Theorem 3.3 can be used to extend
the theorem to random vectors Z and Z' with more general distributions than
elliptically contoured.
with distributions Z
2~
For instance the theorem holds for random variables
+
U(k)AR and Z'
2~
+
u(k)A' R, where R is any random
matrix with nonnegative components which is independent of U(k) .
4.
Necessary and sufficient conditions for (1.3)
It is possible to characterize the bivariate distribution functions Hand
H' which satisfy condition (CO).
This is accomplished by a straightforward
generalization of the proof given by Sudakov [11] of a theorem by Strassen [10].
Cast in our context this slight extension of Strassen's theorem says that (CO)
is equivalent to the following condition (Cl):
(Cl)
P(Z€A);::; P(Z'€A') ,
(A,A') € A2
= {(A,A'):
lA,lA' satisfy (l.lS)} •
Thus if (C3) denotes the following,
(C3)
EJI,(Z);::; Ek(Z ') ,
(JI"k) € F , the class of all functions JI"k which
2
satisfy (1.15) and for which the expectations are
defined ,
we have that oonditions (CO), (Cl) and (C3) are equivaZent.
The implications
(CO) => (C3) and (C3) => (Cl) are immediate, and the nontrivial ones are
(Cl)
~>
(C3) => (CO).
41
Two indicator functions k = lA and
~ =
1 satisfy (1.15) if and only if
B
.
. lR 2
A E B and f or every rectangu 1ar set 0 f f our pOlnts
ln
:J-r:
the number of points of {a,c} in B is larger than or equal to the number of
points of {b,d} in A.
By choosing various such pairs of sets A and B one finds
that the inequality in (C3) requires Hand H' to have common marginal
distributions and to satisfy H
~
H'.
In the case of normal or elliptically
contoured distributions Hand H', these two properties imply (C3) as shown in
Theorems 3.1 and 3.2.
The case of more general distributions requires further
investigation.
__
Because of the equivalence of conditions (CO), (C1) and (C3), our goal,
i.e. condition (C3), can be achieved by establishing either condition (CO) or
(Cl).
In Theorems 3.1 and 3.2 condition (CO) is established.
Even though
condition (Cl) is very natural and satisfactory, especially in its relationship
with (C3), it is unfortunately very difficult to verify, and we have failed to
achieve this for any distributions Hand H'.
The equivalence of (CO), (Cl) and (C3) is a special case of a more
general situation which provides new ways of obtaining inequalities of the type
(1.3), and of showing the existence of joint distributions with fixed support
and with certain marginals fixed.
To illustrate the power and novelty of this
approach let us consider a few examples.
Let F be a closed subset of lR4 , and consider an inequality between
functions k and
~
of the following type (simpler than (1.15))
42
(CF)
and the following conditions which depend on F.
(COF)
There exists a random vector (X ,X 2,Y I ,Y 2) whose values are in the set
I
F and which is such that the bivariate marginal distributions of (XI,Y I )
and (X 2,Y 2) are Hand H' respectively.
(CIF)
P(ZE:A)
(C3F)
EQ, (Z)
<:::
<:::
P(Z'E:A') ,
(A,A') E: A2
= {(A,A'): IA,IA' satisfy (CF)} •
E: F , the class of all functions Q"k which
2
satisfy (CF) and for which the expectations are
defined.
(Q"k)
Ek (Z ') ,
By Strassen's theorem, (COF) and (CIF) are equivalent.
Also, if (COF) holds we
have
for all pairs k and
Q,
satisfying (CF), and by taking expectations (C3F) follows.
Thus (COF) => (C3F) and clearly (C3F) -> (CIF). ·Hence conditions (COF), (CIF)
and (C3F) are equivaZent.
H(I)
<:::
By choosing
H'(I) for all increasing sets I ;
and the result includes Theorem lei), .(iv), and (vi) of Kamae, Krengel and
O'Brien [4].
Of course, anyone or both of the inequalities in the
definition of F could be reversed with corresponding results.
If we choose
43
then (C1F) becomes equivalent to
R(A) ~ H' (A) for all squares A
= {(x,y):
and thus (4.1), (COF) and (C3F) are equivalent.
Ix I:;; a, Iy I:;; a} ,
(4.1)
When Hand H' are normal with
zero means, common variances and correlation coefficients p and p' satisfying
Ipl
~
Ip'l, then (4.1) is satisfied and thus (COF) and (C3F) hold, both new
results.
The same is true for absolutely continuous elliptically contoured
distributions EC (O,L,¢) and EC (O,L',¢), where L,L' are as in Theorem 3.2 with
2
2
Ipi
~
Ip'l; in this case inequality (4.1) follows from Theorem 2.1 of [3].
For not necessarily absolutely continuous elliptically contoured
distributions EC 2 (O,L,¢) and EC 2 (O,L',¢), where L,L' are as in Theorem 3.1 with
Ipl
~
Ip' I,
we now give a simple proof of (C3F), and thus also of (4.1), in the
case where the common variances of L,L' are equal.
The approach is through a
construction similar to that of Theorem 3.2 and thus the result is obtained
without using Strassen's theorem.
Also the result is slightly stronger than
that in the previous paragraph in that the elliptically contoured distributions
are not required to be absolutely continuous.
Even though we are assuming for
simplicity of the construction that the common variances of L,L' are equal, no
doubt a similar, but somewhat more involved, construction would be feasible
when the variances are not equal.
Theorem 4.1.
Suppose that
L= [
z
rv
EC (O,L,¢) and
2
z'
rv
EC (O,L' ,¢)where
2
a~
pa
and
Ipl
~
Ip'l.
Then (C3F) holds~ as well as (4.1).
tIJ
44
As in the proof of Theorem 3.2 it suffices to show E~(U(2)A) ~ Ek(U·(2)A')
Proof.
This will be done by defining a random vector (X I ,X 2 ,Y ,Y 2)
I
which satisfies condition (COF); i.e. (XI,Y ) ~ U(2)A, (X ,Y ) ~ u,(2)A', and
I
2 2
(here
~
= 0).
max(lxll, IYII) ~ max(I X2 1, IY2 1)·
The random vectors U(2) and u,(2) can be taken to be (cos 8, sin 0) and
(cos 8', sin 8'), where 8 and 8' are uniformly distributed on intervals of
length 2TI.
Then we can take
where sin 2a=p, sin 2a'=p',
-i ~ a,
a' ~ ~.
We will now determine the joint
distribution of (8,8') so that the marginals will be uniform on intervals of
length 2TI and
max{lcos(8-a) I, Isin(8+a)
I}
~
max{lcos(8'-a') I, Isin(8'+a') /}.
(4.2)
We have
Icos(8'-a') I ~ Isin(8'+a')
I
<~>
cos 2(8'-a') ~ -cos 2(8'+a')
<~>
cos 28' cos 2a'
<~>
sin(8+8'-y)sin(8-8'-B) ~ 0
~
0 <=> cos 28'
~
0
and similarly
Icos(8-a)
I
~
Icos(8'-a')
I
Isin(8+a) I ~ Icos(8'-a') I < > cos(8+8'+B)cos(8-8'+y) ~ 0
I
Isin(8+a) I
/cos(8-a)
~
Isin(8'+a') I <~> cos(8+8'-B)cos(8-8'-y) ~ 0
~
Isin(8'+a')
where fJ = '" _ "", y = '" + ""
Q
to
\.h
\.h
\.h
\.h
(0 ~
I
<-> sin(8+8'+y)sin(8-8'+B) ~ 0
< TI
fJ-'4"'
Q
0 <
< TI)
- Y -2"'
Thus (4.2) is equivalent
45
cos 28' :::; 0
cos 28' ;0: 0
sin(8+8'-y)sin(8-8'-I3) ;0: 0
or
cos(8+8'-I3)cos(8-8'-y):::; 0
cos(8+8'+I3)cos(8-8'+y) ;0: 0
(4.3)
sin(8+8'+y)sin(8-8'+S) :::; 0
The two sets of inequalities in (4.3) determine the set where the support of
the joint distribution of (8,8') must lie.
7f
Let us first consider the case where 0 :::; p' :::; p, i.e. 0 :::; a' :::; a :::; '4.
When a = a', i.e. S = 0, we can take 8 = 8'.
In the general case, S > 0, we
can take 8 to be the following function of 8':
8 = 8' + S
for
7f
= S' + S - "2 for
7f
7f
37f < 8' < 57f _ 13
4 4
-4 : :; 8' < '4 - S
7f
4 -S
,
~
8' < 7f
4
:::;
S' < 37f
4 '
57f _ S
4
~
8' < 57f
4
(4.4)
= S'
-
S
= 8'
-
13 + 7f for
2
for
7f
4 +S
7f<8,<7f+ S
4 4
57f + S
4
,
Since the relationship between 8 and 8' in [-
:::;
8' < 77f
4
57f < 8' < 5n + S
4 4
i-, 7t)
is one-to-one and piecewise
linear with slope 1, if 8' is uniformly distributed on [-
~ , 747f), then so is 8.
Moreover the pairs (8,8') defined by (4.4) satisfy conditions (4.3) and we now
check this for - ~ :::; 8' < ~, in which case cos 28' ;0: 0 (the remaining cases
being similar).
When - ~ :::; 8' < ~ - S, we have 8 = 8' + 13 and thus
sin(8+8'-y)sin(8-8'-I3)=O;o:O, cos(8+8'+S)cos(8-8'+y)=cos 28 cos 20'.;0:0 .
7f
When 4 - S
~
7f
7f
8' :c 4 we have 8 = 8' + S - "2 and thus
sin(8+8'-y)sin(8-8'-S) = sin(28'+S-; -y)sin(-¥) = cos 2(8'-0'.') ;0: 0
e
46
.e
since 0 ::; 8' - a"::;
i,
and
'IT
'IT
cos(8+8'+(3)cos(O-O'+y) = cos(20+i)coS((3+Y-i) = -sin 28 sin 2cx
'IT
since - 4" :S 8
<; -
~ 0
'IT
'8 .
The remaining cases are treated similarly, and Figure II shows the graph
of 8
=
f(8') which achieves the desired properties.
The graph is plotted for
'IT
3 'IT
3 'IT
- 4"
::; 8' < T'
and for T
::; 8 < T7'IT the graph is obtained by shifting the plotted
graph by (7T,7T).
D
If we choose
then (ClF) becomes equivalent to
c
c
H(A ) ~ H'(A ), Le. H(A) :s; H'(A), for all rectangles A = {(x,y): Ixl:s;a, IYI::;b}
(4.5)
and the corresponding conditions (COF), (C3F) and (4.5) are equivalent.
When
Hand H' are absolutely continuous elliptically contoured distributions
EC2(O,L,~)
and EC2(O,L',~), where L,L' are as in Theorem 3.2 with
Ipl : ; Ip'l,
then (4.5) is Theorem 2.1 of [3], and thus (COF) and (C3F) both hold.
If we choose
where for two-dimensional vectors (xl'Yl) ~m (x 2'Y2) means max(xl,y ) ~ max(x ,y )
2 2
l
and xl + Yl = x 2 + Y2 , then (ClF) becomes equivalent to
47
H(A)
~
H'(A) for all measurable Schur-convex sets A
(A is Schur-convex if Z
€
A and Z' ~m Z imply Z'
E
conditions (COF), (C3F) and (4.6) are equivalent.
(4.6)
A) and the corresponding
This is Theorem 2.2 in [6].
All the examples described above have obvious n-dimensional analogues.
~
48
e
e
37T
4""
7T
r--
7T
-
I
4
8=8'+6
---+--.+---f-+--+------t---- 8'
I
I
-.7'
_.Jt'.J
2!.
4
Z"4
I
37T
if
7T
3'ir
14
if
8=8'+6
e'
_.-1
7T
- if
e
8
ex' ::; 0 ::;
ex,
37T
r~,
8=-e;+7T- Y
4:
::
7T
r
'4
~ Z-:(:.J.
__
~
r
I
~~+_~-+---f-_I_::::~--_.,:'!~----'.,..
7T
4 8L
0
,+y
e
L _
7T
7T
7T
1'4
if
e'
----,.+-+---1~~+-------- 8 '
37T
4
-1
-4
FIGURE II
49
References
1.
Cambanis, S., Huang, S., Simons, G.: On the theory of elliptically
contoured distributions. Institute of Statistios Mimeo Series
No. 1246. University of North Carolina at Chapel Hill (1979).
2.
Cambanis, S., Simons, G., Stout, W.: Inequalities for Ek(X,Y) when the
margina1s are fixed. Z. Wahrsoheinliohkeitstheorie ve~. Gebiete
36, 285-294 (1976).
3.
Das Gupta, S., Eaton, M.L., Olkin, I., Perlman, M., Savage, L.J.,
Sobel, M.: Inequalities on the probability content of convex
regions for elliptically contoured distributions. Proo. Sixth
Berkeley Symp. Math. Statist. Prob. 2, 241-264. Berkeley:
University of California Press, 1972.
4.
Kamae, T., Krengel, U., O'Brien, G.L.: Stochastic inequalities on
partially ordered spaces. Ann. Probability 5, 899-912 (1977).
5.
Kemperman, J.H.B.: On the FKG-inequa1ity for measures on a partially
ordered space. Proo. Akad. Wetensohappen~ Ser. A. 80, 313-331
(1977).
6.
Nevius, S.E., Proschan, F., Sethuraman, J.: Schur functions in
statistics II. Stochastic majorization. Ann. Statist. 5, 263-273
(1977).
7.
Roberts, A.W., Varberg, D.E.:
1973.
8.
Schoenberg, I.J.: Metric spaces and completely monotone functions.
39, 811-841 (1938).
9.
Slepian, D.:
Strassen, V.:
New York:
Academic Press,
The one-sided barrier problem for Gaussian noise.
Teoh. J. 41, 463-501
10.
Convex Funotions.
Ann. Math.
Bell System
(1962)~
The existence of probability measures with given margina1s.
Ann. Math. Statist. 36, 423-439 (1965).
11.
Sudakov, V.N.:
The existence of probability measures with specified projections.
Math. Notes Aoad. Soi. USSR 14, 886-888 (1973).
12.
Tchen, A.H-T.:
Inequalities for distributions with given margina1s.
Teoh.
Report No. 19, Dept. of Statistics, Stanford University (1976).
13.
Widder, D.V.:
1946.
The Laplace Transform.
Princeton:
Princeton University Press,
UNCLASSIFIED
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4. TITLE (end Subtitle)
TYPE OF REPOfilT .. PERIOD COVERED
TECHNICAL
Probability and Expectation Inequalities
6. PERFORMING O.,.G. REPORT NUMBER
..
Mimeo Series No. 1263
7.
8. CONTRACT OR GRANT NUMBER(.)
AUTHOR(.)
Grant AFOSR;75-2796
Grant MCS78-01240
Stamatis Cambanis and Gordon Simons
••
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PERFORMING ORGANIZATION NAME AND ADDRESS
AREA A WORK UNIT NUMBERS
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I'. SUPPLEMENTARY NOTES
I •. KEY WOfilDS (Continue on rever.e .Id. 11 nece .. ary end Idenllly by block numb.r)
probability inequalities, expectation inequalities, normal distribution,
elliptically contoured. distributions, stochastic orderings, quasi-monotone
functions, n-monotone functions
20. ABSTRACT (Continue on re ..er•• • Ide If neceuary end Identify by block numb.r)
This paper introduces a mathematical framework within which a wide
varjety of known and new inequalities can be viewed from a common perspective.
Probability and expectation inequalities of the following types are
considered: (a) P(ZEA) ;::: P(Z'EA) for some class of sets A, (b) Ek(Z) ;::: Ek (Z')
for some class of functions k, and (c) Et(Z) ;::: Ek(Z') for some class of pairs
of functions t and k. It is shown, sometimes using explicit constructions of
Z and Z' , that, in several cases, (a) < > (b) < > (c) ; included here are cases
,
.
·e
DO
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1473
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20.
UNCLASSIFIED
of normal and elliptical1Y1contoured ;distributions. A case where
(a) => (b) <~> (c) is studied and is :expressed in terms of "n-monotone"
functions for (some of) which integral representations are obtained.
Also, necessary and suffic~ent conditions for (c) are given.
•
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