*This research was supported in part by the U.S. Air Force 'Under Contract
No. AFOSR-68-1415.
GEi~ERALIZED
CUNULATIVE DISTRIBUTION FUNCTIONS:
THE LINEAR CASE WITH APPLICATIONS
TO
NONPA~~ETRIC
STATISTICS
Gordon Simons*
Department of Statistias
University of North CaroUna at Chapel, BiH
Institute of Statistics Mimeo Series No. 835
August, 1972
I
ABSTRACT
We investigate the behavior of (univariate) cumulative distribution
functions which are defined on an abstract linearly ordered space.
Special
emphasis is given to the study of a class of linearly ordered spaces which
J.H.b. Kemperman introduced into the subject of nonparametric tolerance
regions.
Distribution functions on such spaces can be decomposed.
siderable attention is given to applications.
Con-
In particular, it is shown
how a number of nonparametric statistical procedures can be extended to
include situations of multivariate and time dependent data.
Ga'JERALlZED Wu..ATIVE ~ISTRIBlITION FU\iCTIONS:
THE LWEAR
WE
I.
~'iI1}; .\PPLlCATIONS
TO j,;QI'JPARArl;.1"HIC S-rATISTICS*
by Gordon Simons
1.
Ii'llROlJUCTIDiJ Am SUiil··iJ\RY.
This paper begins a systematic study of
(cumulative) distribution functions defined on an abstract linearly ordered
sp~ce.
J.H.B. Kemperman (1956) has already
sho~m
(in an abstract setting)
the potential such functions have for nonparametric statistical applications.
This potential has motivated our study (even before we were aware of Kemperman's
work on generalized tolerance limits) and it strongly influences and sOEewhat
limits the topics discussed here.
We find that it is possible to extend" the
range of applicability of those procedures of nonparametric statistics which
are based on the so-called "probability inteeral transformation ll to include
situations in which the data ismUkivariate valued. i /
(I~ is sufficient
that the data be expressible in a separable metric space.)
Kemperman requires his linearly ordered spaces to satisfy a mild but
in~ortant countability assumption.
The implications of this assumption'are
more far reaching than his paper suggest.
l1any of the familiar properties
of classical (univariate) distribution functions fail without it.
ass~ption,
The
which is stated with no reference to probabilistic terms, can
be expressed completely in such terms.
He do not always restrict our
*This research was supported in part by the u.S. Air Force Under Contract
No. AFOSR-6S-14l5.
l/ We do not wish to suggest that this possibility is entirely a new idea.
Indeed, Wald showed how to compute nonparametric tolerance limits for
multivariate data in 1943 [il]l
2
attention to spaces for which the assumption holds.
Finally, it should be mentioned that Flavio Rodrigues (1972) has found
a generalized distribution function (defined on a linearized separable
metric space) to be a useful theoretical tool in his \-lark. He is concerned
with demonstrating new relationships between the concepts of weak convergence and convergence in probability.
We shall not discuss his work
further except to indicate points of overlap.
X
points
Cor~CEPTS!
BASIC
2.
and
x
A nonempty space of
is linearly ordered if there has been an exclUsive assignment of
one the relationships
Xl' x2
TERi4HJOLOGY Aim j·JOTATIOiJ.
in
X
2
and x
2
< Xl
to each distinct pair
and the following transitive relationship holds:
3
imply
Xl
symbols "s", ">", and
il
x2 is larger than Xl"
Let
sr~llest
A (B)
we write
Of course,
th~
x
and statements such as llx
and "A < Ell
<
B.
for subsets
=~
order
(B
precedes x
A, B of
We refer to any set such as
1i
2
,
X.
A (including
B as a terminal.
B containing all of the initials (and terminals) of
a-fie~
and its members are called order sets.
has a largest (smallest) point
A
l
0) as an initial and any set such as
a-field
X is called
If
"~Ii
X = AuB with A
X and the empty set
The
and
Xl < x 2
are distinct and Xl < x • Since this
3
3
is a well-known concept, we shall use without explanation the related order
2
< x
Xl < x
= ~).£/
A (B)
X it is said to be closed and
is open if
B (A) is closed.
A and B may both be open and closed at the same time.
Both
£/Since we shall have little need to refer to topology, we let convenience
take precedence over common topological terminology with this term and the
term H open" which follows.
3
may be neither closed nor open, as may be seen when X is the set of
= {x
rational numbers and A
numbers, then
X and
negative integers, then
~
S
K
Likewise, if
X is the set of real
are neither closed nor open.
X is a closed
A linearly ordered space
a
/:f}.
te~inaL
X is the non-
but not a closed initiaL.
X is called a kappa spaae or, for brevity,
spaae if each nonempty initial of X is expressible as a countable
union of closed initials and each nonempty terminal of
as a countable union of closed terminals.~/ vfuen
order a-field
of
If
X is expressible
X is a
K
space
the
B is generated by the closed (alternatively the open) initials
X.
Let
(n,
f)
be a measurable space.
linearly ordered space
A mapping
X from
X is called a random variabLe (r.v.)
the inverse image of each order set
BeB
is a set in
F.
n into the
(in X)
if
This concept
includes the usual reaL random variabLe (r.r.v.).
~
Let P be a probability measure on (n,F). The function
F(x) = P(XSx), xeX, is called the (cumulative) distribution funation
(d. f.) of
F(X)
X.
It is well-defined since ~eB
is a r.r.v. since
{XEX:
F(x)St}
for each XEX.
Further,
is an initial for each real
A d.f. is dense if its range is dense in the real interval [0,1].
F is right continuous at the point
an open initial A with
xeA
xeX
such that
if for each
F(u) < F(X)+E
e > 0
t.
A d.f.
there exists
for each
ueA.
A d.f. is right aontinuous if it is right continuous at every point XEX
for which
+
x
is a proper subset of
X·
A d.f. is disarete if there exists
~/This concept was introduced by J. H. B. Kemperman (1956). Actually, he was
working with a slightly more general ordering by allowing distinct pairs of
points"to ·.be equivale:at and- .notr.order compaTabl~. ..Alll~pf: our results are
expressible in his greater generality.
4
a set of positive constants
{p }
(distinct) points
X
{x}
m
in
X,
F(X)
X.
X €
THE PROBABILITY HJTEGRAL TPJ\iJSFOPJiATIOiJ.
random variable
F is the d"f. of a
If
(for historical reasons) is
integral transformation.
~al1ed the
probability
It is an important fact in the study of nonpara-
metric statistics that
when
such that
=
F(x)
3.
(necessarily countable) and corresponding
m
is a uniformly distributed r.r.v. on [0,1]
F(X)
X is a r.r.v. and
F
is continuous.
We now state and (for completeness) prove the following theorem due
to Kemperman:
Theorem 1.
Let
be a
X
1'. v.
in the
arising from the probability measure
r.r.v. on [0,1] if., and only
Proof.
Let
0 S t S 1,
P(x
A
= {x:
= 0
F(xj S t}
F
is a uniformly distributed
for every
and
B
x€X.
= X-A.
Then
is distributed as claimed if we can show
F(X)
P(F(X)St)
A:J
suppose
as
= x)
X with d. f.
spaae
F(X)
P.
if~
K
+
00
Uk=l ~
~.
Since
where
P(X€~k) tP(X€A)
Similarly, i f
al S a2 S
as
B:J (/J,
k
+
t
k +
Then, even if
then
B
,00
= Uk: l
t.
bk
k
P(X€A)
~
A.
A may be expressed
Then
t ~ F(~)
P(X€A) Steven if
+
= P(X<b ) :
0
=
space, the initial
all belong to
B
= 0,
The converse is inwediate.
K
and hence,
00,
= P(XSb k )
Band
00.
< F(b )
k
X is a
= P(X€A)
l1here
b1~b2~...
A
=
= ~.
all belong to
l-P(X€b ) \; l-P(X€B) = P(X€A)
k
t
and the conclusion follows.
as
5
There is a simple necessary and sufficient condition for
be uniformly distributed on [0,1] which does not require
K
to
X to be a
space:
ThcoreiIl 2.
Proof.
€>O,
F(x)
Let
X
If
F
is dense, then for any
there exists a
= P(X~x)
Since
€>O
4.
~ P(F(X)~F(x»
~
This implies
0
= Ct.
=t
F
X €X
o
for
X.
at which
(Note the curious implication that
Ct
~
F(x)+€.
for every
t
in
is uniformly distributed
X and a family of probability measures
variable with mean
not happen when
= F(y)
To illustrate this, in Section 7, we
X.
except for a particular point
Ct
Thus
X may not contain all the information
which all give rise to the same d.f.
p (X-x )
and
F(x) € [0,1)
In contrast to the distribution functions of real
it reasonably should about
exhibit a r.v.
F is dense.
0
The converse is immediate.
N\PPA SPACES.
F(X)
is a
F(X)
F(x)+€.
P(F(X)~t)
is arbitrary, it follows that
F.
~
~ P(x~y)
P(F(X)<F(y»
random variables, the d.f. of a r.v.
X€X
F.
X€X with
F(x) < F(y)
for which
y€X
the (dense) range of
on [0,1].
be a 1'.V. in X with d.f.
distributed 1'.1'.V. on [0,1] if" and only if,
unifo~ly
~
F(X)
under
P.)
Ct
F(x)
[O,l]}
{p , Ct €
Ct
=0
for all
F(x). 1, and yet
F(X)
is a Bernoull'
Theorem 1 makes clear that this could
X is a r.v. in a K space.
(Consider the case
a
= 0.)
This leads us the the following elementary result:
Theorem 3. Let X be a 1'.V. in the
from the probability measU1'e P.
P(X€B)
on the orde1'
a-fie~d
B.
I(
space
X with d.f.
F
ar'ising
Then F uniquely detel'l7lines the vaZue of
6
Proof. Clearly, F determines the value of P(XeB) on the semi-ring
of sets
B of the form
0+-
x
0+- 0+-
or
x-y
generated by the closed initials.
is
and consequently on the
X
tfuen
is a
K
a-li~8
.
space this a-ring
0
B.
X is a
With the assumption that
K space, we can establish useful
characterizations of dense and discrete distribution functions:
Theorem 4. Let X be a r.v. in a K space X with d.f.
F is dense if:I and only if:I
I.
countable set of distinct points
{x}
m
in X.
= L{ m:x ,}
",x
F(x)
m
Proof.
I
xeX'
L{~l} P(x=xm)
F is disCl'ete if" and only if"
II.
for
P(X=x) = 0
Then
F.
=1
for S011e
and then
P(X=xm), xeX·
is immediate from Theorems 1 and 2.
In showing II, we
need the following lemma:
lemma.
under the Asswnption of Theorem 4:
sup F(x)
x<u
inf P(x<x)
x>u
= P(x<u),
=
F(u)
and
=
sup F(x)
xeX
1.
(The f/SUpil is zero and the ilinr' is one when their index sets are void.)
NOW
suppose
F
is discrete so that
F(x)
From the leu~a, we obtain
F is expressible as
=
L{m~l} Pm~l and
P(X=x)
m
= F(xm)
-
sUP F(x)~
x<x
-
m
Thus
1 -<
t
}
l.{m~l
Pm ~
t{
}
L m~l
and II follows (the converse being trivial).
P{X=xm} ~ 1,
0
Pm.
7
Le:7ima Proof. Let A be the open initial {x: X<U} 'ilhich is expressible
(unless
A
= ~)
Uk: l ~
as
F(x) $ P(X<u)
including the case
for
A=
=
~ 1iu~~ F(~)
sup F(x)
x<u
Since
alSa2~ •••
with
x<u
~.
all in
P(X€A)
A.
=
Then
P(X<u).
the first equality of the lemma follows,
The other parts of the lemma are
Sh01jffi
Corollary 4.1. Let F be a d.f. associated with a r.v.
K
x.
space
Then
for some unique
X in the
aan be decomposed into dense and discrete parts as
F
where
a€[O,l],
are~ respectiveZy~
similarly. 0
F
l
(when a>O)
and F2 (when a<l)
dense and discrete distribution functions of X under
aZtemative probabiZity measures.
.e
Proof. Let P (defined'on (O,FY) be the probability measure which
results in the d.f.
for which
B'
1.
P(X-x) > O.
and
Since
a = P(B').
Pl(X=x) = 0
If
a = 0,
set
l{m~l} P (X=x ) = 1,
If
2
m
O<a<l,
set
for all
F for
x
so
X and let
Further, let
If
= 1,
a
for all
P
z=
x,
P
F
l
B
=
PI
[W€O: X(w)€{x}]
m
=P
so that
x
with complement
= Pl(XSx) = F(x).
Fl(X)
= PZO{$x)
F (x)
2
= F(x).
Since
FZ is discrete according to Theorem 4, part II.
is dense, and
The decomposition follows.
and
P (A)
2
= p(AIB),
A€F.
L{IT~l} P (X=xm) = 1
2
so
Pl(X=x) = 0
F2
is discrete.
0
Corollary 4.2. The d.f. of a r.v. in a
Proof.
denote the set of points
m
is dense according to Theorem 4, part
so that
Pl(A) = p(AIB')
F1
set
{x}
K
space is right continuous.
Clearly, dense d.f. 's are right continuous.
So, in view of the
decomposition given in the previous corollary, we may assume that the d.f.
8
in question is discrete.
ShOlol
the d. f.
There are two kinds of points
is right continuous:
F
be an open initial (expressible as
x
at which one must
The closed initial
(i)
{x: x<v}).
Then, clearly,
+
X
may also
~
F(u)
F(x)+€
+
u€x. (ii) The closed initial +x may not be an open initial nor
equal to the entire space X. Then for any € > 0, F(u) - F(x), which
for all
L{m:x<x
is expressible as
P(X=xm) ,
~u}
m
infinity of
u>x.
for an
will be less than
IJ
We digress briefly before proceeding to the next corollary.
and
be linearly ordered spaces and let
y
(x,y), X€X, y€y,
that the order o-field for
denote the space of pairs
or
XXV
It is easy to check
is a sub-o-field of the product o-field
generated by the order o-fields of
.e
X
linearly ordered (lexicographically) according to:
if
X and
XXV
Let
X and
y
respectively.
Thus, if
Yare random variables (arising from the same measurable space)
in
X and
XXV
is a
y respectively, then (X,Y)
K
space if, and only if,
X and
is a r.v. in
yare
K
XXV.
Furthermore,
spaces.
Corollary 4.3. Let X and Y be random va:l'iabZes in the
respectiveZy, and Zet F«x,y)J deno1;e the d.f. of
X and y,
If the d.f. of Y is dense, then F«X,Y»
K
spaces
(X,Y).
i8 a uniformZy distributed
r.r.v. on (0,1].
Proof. According to Theorem 4, P«X,Y) = (x,y»
(x,y)
€
XXV
(since the d.f. of
from Theorem 1.
Y is dense).
~
P(Y=y) = 0
for
The conclusion follows
U
Remarks
1.
Thi~
corollary is a generalization of a similar result due to
Rodrigues (1972) when working with real random variables.
Flavio
(C.f., D.A.S.
9
Fraser (1953) page 50, Kemperman (1956).)
2.
y is not a
This corollary is still true if
involved.
K
space; the proof is more
We will not need this greater generality.
Corollary 4.4 If X is a r.v. in a
K
space with d.f.
then
F,
is stoahastiaaUy at least as large as a uniform variable on
F(X)
[0,1].
Proof. Without loss of generality, we assume that the underlying
probability space admits a r.v. such as
instance, a uniform variable). Then
on
[0,1]
Y in the previous corollary (for
F(X)
~
F«X,Y»,
a uniform variable
0
by the previous corollary.
F(x-):: sup F(u) = P(X<x» is stochasu<x
ticatically no larger than a uniform variable on [0,1]. Specifically,
Remark. Like"lise F(X-)
F(X-)
.e
S
F«X,Y»
S
(uhere
F(X).
\V'e now proceed to give a probabilistic characterization of
X be a linearly ordered space with order a-field
Let
(X,B).
It will be recalled that
called an atom (relative to
Q)
Q(A) > 0
BeA,
either
Q(rl)
=0
or Q(B)
= Q(A).
point atom if there exists a point
that
{x}€B.
XEA
spaces •
B and let Q
be a probability measure on
if
K
and for each
A€3
is
E€3 with
We shall call an atom A a
for which
Q({x})
= Q(A).
Note
(An atom, in general, does not have to be a point atom
[as we illustrate in Section 7] "lhen one is working with a measurable
space in which all of the single point sets are measurable.)
Theorem
measure
Q on
Proof.
A*
= {x~
5.
X is a
(X,B)
K
spaoe if:I and only if:I every probability
has at most point atoms.
Suppose the probability measure
+
Q(xA)
= O}
and
H*
= {x:
+
Q(xA)
= OJ.
Q has an atom A.
A*
Let
is an intial and
B*
10
a terminal.
If
X
is a
u
may be expressed as
o
=
+
Q(~A)
space, then
k
+
00
l' Q(A*A)
as
k -+
there nrnst exist a point
(since
with
k=l ~
A*UB*.
X
A'll
• in
Q(B*A) = O.
A*~~,
it
~...,hich
case
Since
Q(A)
Q(~) = Q(~) = Q(A) > 0
Tnen
A is an atom), and it follows that
Conversely, suppose that
For when
all in
a 1::;a 2::;···
Similarly,
00.
i
x
Q(A*A) = O.
x€A
and
is not a K space.
Q({x})
= Q(A).
Because of the
symmetry in the definition of a K space, we assume without loss of generality
A which can not be expressed as a
that there exists a nonempty initial
countable union of closed initials.
measure
(X.B)
Q on
relative to which
S be the semi-ring ,of "interva1s
~JO
theoretic difference between
i f for some
.e
x€A, S
Q extends to
B
S.
additive on
~
We proceed to define a probability
-+
xA,
d
A
Let
uhich are expressible as the set-
,
initials.
and let
is a non-point atom.
For
Q(S) = 0
S€S,
let
otherwise.
(as a probability measure) providing
Therefore, suppose
51' 8 ""
2
00 .
Q(S) = 1
Thus
Q(A) = 1.
Q is countably
are (nonempty) disjoint
= L Sk
€ S. The case Q(S) = 0 is trivial, so ~Je
k=l
only need to show that Q(Sk) = 1 for exact one index k when Q(8)
sets in Sand
S
Q on S,
In view of the definition of
such indices.
Now
index for which
-+
for some
S~xA
Q(Sk)
=1
a point
A
= ij
k'
Hence,
~,
~ki'
Q extends to
A is an atom.
x€A.
in each set
from those sets
Thus
there can not be two or more
Therefore, there 'Hill be one such
(i. e., for which
there is an infinity of
Sk'SA,
8
and
= 1.
-+
5k~xA
-+
xA, x€A.
for some
unless
x€A)
But then, by choosing
there results the contradiction
Q(B)
=0
If it were a point atom,
or 1
for every
L€B.
A would have to be a
closed initial and therefore acountable union of closed initials.
0
11
The reader may wish to skip the remainder of this section l;rhich
discusses (without proofs) the properties of
K
spaces.
No subsequent
references are made.
X
Suppose
it too is a
K
Suppose
is a
K
If
y
is a nonempty subspace of
X,
space relative to the ordering given for
X.
X is any linearly ordered space and let
X denote the
X. X
class of initials of
(That is, "initial
n".)
subset of
space.
is linearly ordered by proper set inclusion.
A is less than initial B" means
is a proper
itA
A probably more useful subclass is the class of initials
which are open and / or not closed (or, equally well, the reverse) which
we denote by
X*
and
.e
X*.
X
completes
X**
are all
in the sense that
is isomorphic to
usual ordering,
K
X
All of the initials of
X*
Xx.
X
(If
X
and
are open or closed.
is isomorphic to a subset of
X*
is the rational numbers in their
is ismorphic to the reals.)
spaces if anyone of them is.
X*
Finally,
X, X
We do not know whether a
and
K
X*
space
can have a cardinality in excess of that for the reals.
5.
does in
SOliE
iJOl~PAHJ.\;iETRIC
nonparan~tric
STATISTICS.
Roughly speaking, w"hatever one
statistics with real random variables one can do
with random variables defined on a K space.
Kemperman has given one
example with his paper on generalized tolerance limits.
We give other
examples as we discuss several (generalized) statistics associated with
the names of Kolmogorov, Smirnov, Cramer and von Mises.
~ie
find that
all of these have the same distributions as their classical predecessors.
This is an obvious advantage, for it permits one to use well established
tables in the perforniance of statistical tests.
12
We assume the readers of this section are reAsonably familiar with
the c1assicai concepts and results.
We proceed with an informal exposition,
devoting special attention to "trouble spots H •
For the remainder of this section,
Xl, ••• ,X
denote iid random
n
variables in some linearly ordered space ~/ with COWEon aonjeatured
d.f.
F.
X denotes a
r.v.
in
X
whose d.f. is
F.
Let
...
F (x)
n
(where I
A
... 1 when A occurs and
D
...
n
While it is clear that
D
n
0 when it does not)
sup
IF
X€X
(x) .• F(x)
and let
I.
n
[0,1]
is a real number in
for each point
w in the underlying probability space, it is not immediately clear that
.~
it is a (real) random variable.
k
= lSIc;n
max (n
where
E+
where
X(l) S X(2) S ••• S X(n)
It is easy to check that
- F(X(k)>>' E-
Dn = max (E+,E_)
=
arranged
in ascending order (Horder statisticsl!) and vJhere
(zero if the index set is void).
functions,
each
F(X(k»
F(X )
k
and
Finally, let
uniform variables
and
F(~~-)
n
F(x)
are real random variables (and consequently
Thus
D
is a r.r.v.
denote a generic analog of
D
corresponding to
F(X(k)-)
-D
Since
F(x-):: sup F(u)
u<x
and F(x-) are non-decreasing
Xl' ... , ~\.n
"
is a r.r.v.).
on
[0,1]
"Jith
F
n
n
the true d. f. of such
~/The concepts of independent random variables and identically distributed
random variables in a linearly ordered zpace are the same as for real
random variables.
13
random variables, and let
E+
and
E+
and
E_
be the corresponding values of
E_.
T,'1cororll 6.
If F is the true d.f. of eaoh
1.
D
is dist1'ibuted as
If
F
n
II.
then
n
is the true d.f. of each i~ and X is a
D
n
uniform variables.
D,
n
F is dense, then
D •
is stoohastiaaUy no ZaI'ger than
Proof of I. According to Theorem 2,
define
and if
~:.;r
~'K
D .
n
F(X1), ••• ,F(X )
n
are iid
Identifying these random variables with those used to
Dn
we obtain
Proof of II.
= D.
n
0
Briefly, we use the randomization device illustrated
in the proof of Corollary 4.4 and the subsequent remark.
One obtains
Y 's
where (for definiteness) the
~
iid uniform variables independent of the
each
(.~J
Y ).
k
namely
F«x,y»,
~IS,
is dense.
there are random variables corresponding to
Then
D
n
is distributed as
D.
n
X
is a
n -+ 00)
K
space.
when
we
F
o~it
are
(Xk , Yk)'s,
Relative to the
D
n'
Call then
E
+
,..,
D
n
~
D ,
n
and
o
Quite clearly a Glivenko-Cantelli theoreffi holds (i.e.,
a.s. as
k
and where the d.f. of
so that
,..,
(by part I)
spcroe"
K
is the true d.f. if
F
D -+ 0
n
is dense and / or
the proof.
Cramer (1928), von 11ises (1931) and Smirnov (1936) initiated considerable interest in another class of statistics -- ones based on a
Stieltes integral.
Cramer and von Mises suggested using
(1)
f
X
(Fn(X) - F(x»2 dK(x)
.14
X
with
the real line and
K a suitable non-decreasing function.
Smirnov suggested the modification
(2)
X
where again
f
(Fn(x) - F(x»2
X
is the real line and
~,
~(F(x) dF(x)
[0,1],
defined on
is a suit-
able non-negative (borel measurable) weight function.
we shall not try to interpret (1) in our context~/, but (2) can be
X
interpreted as a Lebesque-Stieltjes integral vnlen
According to Theorem 3,
Alternatively, if
is a K
space~
F induces a probability measure on (X,B).§./
F is dense, it determines a probability measure on the
a-field generated by the closed initials and this measure can be extended
to
(X,S)
by completing it, if necessary.
Our primary concern with (2) is when
F is dense, in which case it
can be interpreted as the expectation of a real random variable for each
point
w in the underlying probability space:
[0,1].
variable on
Y~(w)
P(I
Then Fn(X) = n
is fixed at the value.
~
[~SX] T
I
[F(~)SF(X)]
~
Y = F(X),
-1 \ n
Lk=l I[F(~)SY]
(k. l, ••• ,u).
) S P(Y =
Let
F(~ .
K
» = O.
a uniform
a.s. where each
This is because
Consequently, (2) can be
expressed as the expectation:
~/one might try to extend the definition of the Riemann-Stieltjes integral
using the ilrefinement of partitions;; method. See Apostol (1957), page
192. Alternatively, one might define (1) as a Legesque-Stieltjes integral
by trying to associate a neasure (on the order a-field) with K. This
is not so easy to do as it is when X is the real line. Apparently, one
would need to study the topological structure of X.
£/The integrand of (2) is
lying probability space.
8-measurable
for each point
w in the under-
15
(3)
E(n
-1
~ n
2
lk=l I[F(~)SY] - Y) ~(Y) (w fixed).
(3) can be used for computational purposes:
The statistic in question, when
the
Xk's),
F
is the
conjectur~d
dense d.f. (for
is equal to
1
fo
(4)
-1 ~ n
2
71
(n
lk=l I[F(~~)St]-t) ~(t) dt.-
we state without further proof the following theorem:
Theorem 7.
If
is dEnse and
F
whiah may be aomputed as
eaah
then (2)
(4).
f~ ~(t)dt<oo.
Further, if
F
-1 ~ n
°
(n
I lk=l I[UkSt]
U1 ,U 2 " •• 'Un
- t)
2
are iid uniform variabtes on
(Nowhere have we assumed that
6.
is the true d.f. for
has the same distribution as
l
where
thm (2) is a ,..,..v.
X is a
I(
valued data and multivariate data.
[0,1].
space.)
LWl:.AELY OR(;ERI;~G A nETRI C SPi~CE.
measurements in a separable metric space.
~(t) dt
Frequently, data represents
Such spaces suffice for real
A continuous monotoring of a barometer
for a fixed period of time produces an observation in the separable roetric
space of continuous functions on some bounded closed interval.
In this section, we show that a separable metric space
X
can be
linearly ordered with the following desirable features:
llThiS must be a r.r.v. (or possibly an extended real random variable) since
it is the limit of a sequence of approximating Riemann sums, each ore a r.r.v.
16
(i)
(ii)
The resulting linearly ordered space is a
The r.v.
P (X
(iv)
space.
An observation in X may be interpreted as resulting from a
randdm variiib. Ze ..(iii)
K
= ~,)
X
in' X. ,
X has a dense d.f. unless there are points
x€X with
> O.
Usually, the linear ordering is constructable and not merely
existential.
Test statistics can be evaluated (or adequately
approximated) to permit the performance of nonparametric tests.
(v)
There is flexibility in the construction of the linear ordering.
Presumably, one might seek an ordering which produces a test with
good power against certain specific alternatives.
On the negative side, it may seem unnatural to linearly order multivariate
observations.
The flexibility referred to in (v) may be interpreted by
some as undesirable arbitrariness.
However, most statistical tests are
based on the size of a real valued test statistic.
This has the implicit
effect of "linearly ordering ll the sample space in the slightly 'Vleaker sense
that Kemperman refers to in his 1956 paper.
That is, two points which are
not comparable are considered as equivalent.
By linearly ordering spaces which do not have a natural ordering, we
produce a new class of nonpa:rametria statistical tests.
These should be
evaluated on the basis of their operating characteristics.
Let
X be a separable metric space and
F be the Borel field
generated by the open set (equivalently, by the open balls).
a sequence of progressively refinining partitions
of
(~l)
(a)
m
=
{Ai
i
i
}
l' 2···' m
X with the following properties:
Z
Each
to
Z
We consider
m
F.
contains a countable number of non-void elements belonging
17
(b)
{IAI:
sup
Ae Z }
0
-+
m
A
(d)
Each index
= U
i p .. ' ,im-l
0),
vlhere
denoted the diameter
A
(~2).
ip .. .,im
{i}
m
i
m -+
Ae F.
of an arbitrary set
(c)
as
is a positive integer.
m
The definition of a separable metric space easily guarantees that such
sequences of partitions exist.
{Z } determine a unique linear ordering:
m
The partitions
XEX,
XEA
there exists
unique sequence of indices
a
il, ••• ,im for each
sequence is unique to
m.
i ,i , •••
l 2
For each
such that
Conversely, condition (b) insures that this
x.
(Of course, an arbitrary sequence i ,i , .••
l 2
may not correspond to any point x since .n 00 A
might be void).
m=l i l ,··· ,i m
The sequences which correspond to points in . X can be linearly ordered
lexicographically (as one does with the decimal expansions of the real
~
numbers in
[0,1».
associate with the partitions
m
that we
{Z }.
m
This linear ordering procedure mffices each element
Z
X
This induces the linear ordering of
A
11' ... , i m of
into an I1interval" (the set-theoretic difference of two initials)
and linearly orders
Z
cedes each point of
Al ,3.
Let
m
(m~l).
For instance, each point of
Al ,2
pre-
B denote the order o-field associated with such a linear ordering.
We have the following theorem:
Theorem 8.
X,
Proof that X is a
initial (terminal)
{Z ,~l},
m
as orodered by
K
space.
is a Kspaae and B .. F.
Briefly, any nonempty non-closed
A can be expressed as the countable union
A ..
0+-
U {~: BEZ
m
for some
m~l,
BSA}
18
(A
where
~
-+
U{x : B€Zm
B
for some
m~l,
xB is an arbitrary point in the set
Proof that BSF.
Since
arbitrary closed initial
X is a
+-
x € F.
=
x
where
K
0
B.
space, it suffices to show that an
(The closed initials generate
+-
in
B£A})
n
m=l
the point
Proof that FSB.
x
But
B
m
Bm is the (countable) union of all sets in
F) which precede
S.)
or contain
Zm
0
x.
Briefly, each open ball
(and consequently
B€F
can be expressed as
the countable union
B
and each C'interval")
=
U{C: C€Z
m
for some
~l,
C is in the order a-field
C£B},
B.
0
With this theorem, the claims (i) - (v), mentioned earlier in this
section, are largely apparent:
observation
Certainly (i) is.
X in a metric space
X is generally interpreted to refer
to a random mapping into the measurable space
holds.
(iii) is a consequence of
Concerning (ii), an
~heorem
4.
(X,F).
B=F, (ii)
Since
Concerning (iv),
is quite clear the linear ordering is constructable for finite dimensional
Euclidean spaces.
The requirements for adequateZy evaluating a (nonpara-
metric) test statistic are less than one might initially suspect since the
values of
PCB), B€Z,
m
determine the values of the d.f. of
with the linear ordering) within certain bounds.
upper and lower bounds for
F(x)
lfuen
X (associated
m is large, the
will be quite close to each other.
From a practical standpoint, one does not need to completely specify the
linear ordering.
~
be sufficient.
The specification of a few of the partitions
Finally, property
(v)
Zm may
does not require further comment.
19
Theorems5 and 8 combine to produce an indirect proof of the following
kno~~
If Q is a probability measure on
result:
a separable metria spaae with Borel a-field
if3 and only if3
Q( {x})
= 0 for eaah
XEX.
(X,F)
X is
where
F, then Q is non-atomic
Q
has at most point atoms.
Toe approach we have used for defining a linear ordering appears in
the work of F1avio Rodrigues (1972).
His purpose, like ours, is to define
a liunivariate li distribution function but for a different reason.
COUNTER-EXAHPLES AND CQt'R'iEdTS.
7.
In Section 4, we discuss the
importance and properties of
K
spaces; in this section, we exhibit linearly
ordered spaces which are not
K
spaces and demonstrate some of the possible
consequences.
Let
X be the set of ordinals which are less than or equal to the first
uncountable ordinal
B
(where
AV
=
{A~
n.~/ The order a-field is
A
or
+
c X
A'
+
denotes the complement of
for some
A).
the family of probability measures on
0
P (A)
a
=
l .. a
if A
=.
if nEA'
+
for some
x
~
+
+
a
i f nEA ~ x
I
if x £ A
....
x + in}
+ in}
for some
<X, B)
Let
XEX- in}}
{p, aE[O,l]}
a
denote
which are defined as follows:
XEX - in}
for some XEX - in}
for some
XEX -
un
XEX - {n}.
~/This space has provided many counter-exa~ples in general topology.
Halmos (1950), page 231, has used the space to exhibit an interesting
non-regular measure. M. Bhaskara Rao and K.P.S. Bhaskara Rao (1971)
pursue his example further.
20
(I.e., Mass
a
is placed on the maximal point
placed on each "interval'i of the form
be the
identity map from
n
and mass
[x,n), XE:X-n.)
I-a
is
Finally, let
X
X (i.e. p X(w) • w).
X into
Under each probability measure
P ,
X is identicaUy
the d.£. for
a
given by
=
F(x)
Clearly, the d.f. of
certain sets
AE:B.
The initial
Ol
{
i f XE:X - n
if x =
n.
X fails to identify the values of
This is due to the fact that
X - {n}
P
a
(XE:A)
X is not a
K
for
space.
can not be expressed as a countable union of closed
initials.
Under
P , F(X)
a
= x) = 0
P Q (X
is a Bernoulli variable with mean
for all
XE:X.
This explains why the
K
a.
When
= 0,
a
space assumption is
included in Theorem 1.
This example can be used to illustrate several other counter-examples
associated with our results in sections 4 and 5.
For instance, the
Glivenko-Cantelli conclusion fails for
D + I-a a.s. as
By deleting the point
from
aE:[O,l):
n
n +
X and using the probability measure
on the resultant order a-field, one finds that the corresponding r.v.
has the d.f.
F(x)
00.
P
o
X
= o.
Some concluding remarks:
1.
These examples seem to suggest that linearly ordered spaces which are
not
K
spaces are so unpleasant that they should be excluded from any
theoretical consideration.
Such a position does not appear fully justified.
For instance, theorems 2 and 7 do not require a
It is not difficult to produce dense d.f. 's on
~
product of the real line and the space
K
space--only a dense d.f.
non~
spaces.
X given above.)
(Look at the
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Advanaed CaZauZus.
MathematiaaZ
AnaZysis~
A Mode!'n Approaah to
Addison-Wesley, Reading.
[2]
Chung, K.L. (1968). A Course in ProbabiZity ~eory.
and World, New York.
[3]
Cramer, H. (1928).
Aktuarietids.
Harcourt, Brace
"On the composition of elementary errors".
Vol. 11 13-74 and 141-180.
"Nonparametric tolerance regions".
Skand.
Ann. Math.
[4]
Fraser, D.A.S. (1953).
Statist. 24 44-55.
[5]
Ha1mos, P. (1950).
[6]
Kemperman, J.H.B. (1956).
Statist. 27 180-186.
[7]
Rao, M. Bhaskara and Rao, K.P.S. Bhaskara. (1971).
on [O,n]". Manusaripta Math. 5 195-198.
[8]
Rodrigues, F. W. (1972). "Some. structural relationships between weak
convergence of probability measures and convergence in probability".
Ph. D. Dissertation, University of North Carolina, Chapel Hill.
[9]
Smirnov, N.V. (1936). "Sur la distribution de
Paris. Vol. 202 449-452.
Measure Theory.
Van Nostrand, Princeton.
"Generalized tolerance limits".
2
w ".
WahrsaheinZiahkeitsreahnung.
Ann. Math.
"Borel a-algebra
C.R. Aaad.
SaiD
[10]
von Mises. R. (1931).
Leipzig-Wien.
[11]
Wald, A. (1943). "An extension of Wilks' method for setting tolerance
limits". Ann. Math. Statist. 14 45-55.