*This research was partially supported by the Air Force Office of Scientific
Research under Grant #AFOSR 72-2386.
t
A dissertation submitted in partial fulfillment for the requirements for
the degree of Doctor of Philosophy under the direction of Professor
Gordon D. Simons.
STATISTICAL TESTS BASED ON THE
LEVY AND PROHOROV METRICS t
by
Charles H. Alexander*
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 965
November, 1974
CHARLES HAYNES ALEXANDER, JR.
Statistical Tests Based on the Levy
and Prohorov Metrics. (Under the direction of GORDON D. SIMONS.)
The author develops non-parametric one-sample and two-sample
goodness-of-fit tests using as test statistics the Levy and Prohorov
distance between empirical distribution functions.
Computational
procedures are described for computing the test statistics.
Recur-
rence equations are described for computing the distribution of the
two-sample test statistics, using results about the maximal matchings
in certain graphs.
The asymptotic distribution of the one-sample
test statistic is expressed in terms of the distribution of fluctuations in the sample path of the Brownian Bridge stochastic process.
Tables of these distributions are given in the appendix.
The power
of the tests against certain alternatives is discussed, and the results of simulations comparing the power with that of the KolmogorovSmirnov test.
ACKNOWLEDGEMENTS
I wish to thank my advisor, Professor G.D. Simons.
His suggest-
ions and encouragement combined with a remarkable aversion to mathematical nonsense have assisted the author immensely throughout.
More
than an advisor, he has proved to be an educator in the broadest sense
of the word and it has been a rewarding experience to work with him.
My thanks also go to the members of the faculty of the Department
of Statistics who have contributed so much to my graduate education,
particularly the members of my examination committee, Professors Hoeffding,
Cambanis, Chakravarti, and Wegman.
I am grateful for the financial support provided by the Department
of Statistics and also for the assistance of the Department of Mathematics,
especially Professors Cannon and Kelly who have been members of my committee.
Let me here add to the praises of June Maxwell who has ably, speedily,
and most of cheerfully typed this manuscript under combat conditions.
Finally, I would like to thank my fellow students for their contribution to my education, particularly N.l. Fisher, whose help in the preparation of this manuscript has been most appreciated.
ii
TABLE OF CONTENTS
Chapter I.
Introduction
1.1
Outline and Summary----------------------------------------- 1
1.2
Weak Convergence
1.3
1.4
The metrics dL and dp ----------------------------------- 5
Asymptotic Distributions Based on the Wiener Process--------12
1.5
Terminology from the Theory of Graphs-----------------------14
Chapter 2.
2.1
of Measures------------------------------- 4
Computational Procedures
Computation of
dp(P,Q) for General
S when
P and
Q
have Finite Support--------------------------------------16
2.2
Computation of
dp(P,Q) when
2.3
Computation of
d -----------------------------------------24
L
Chapter 3.
3.1
3.2
P has Finite Support---------20
Null Distribution for the Two-Sample Test Based on dp
3.3
Introduction------------------------------------------------27
Matchings in the Graphs (X,Y,f )----------------------------29
z
Definition of a Mapping a: X + X --------------------------41
3.4
Classification Scheme for Elements of
3.5
Recurrence Equations for K(u), u
3.6
Reducing the Recurrence Equations---------------------------53
3.7
Computating of
Chapter 4.
4.1
4.2
4.3
E
X--------------------43
C -----------------------44
#D.1 (Z,s)------------------------------------55
Asymptotic Distribution of the Test Statistics
Asymptotic Distribution for
dp ----------------------------62
Asymptotic Distribution for d ----------------------------74
L
Asymptotic Continuity of f(WO)-----------------------------75
Chapter 5.
Statistical Properties
5.1
Related Test Statistics-------------------------------------86
5.2
Power of the Test Based on
iii
dp ------------------------------88
iv
Appendices
A.1
Tables of the One Sample Distribution Function--------------97
A.2
Tables of the Two Sample Distribution Function-------------101
A.3
Bounds on
Pr{f(WO) s c} ----------------------------------110
Bibliography-------------------------------------------------------113
CHAPTER 1.
Introduction
1.1
Outline and Summary
Let
(S,p)
be a metric space.
and let
peS)
S be the class of Borel sub-
Let
sets of
S
be the space of all probability measures on
(S,S).
A number of ways of defining a metric on the space
peS)
have
been suggested and it seems natural to use such a metric to construct
tests of statistical hypotheses.
For a one-sample goodness-of-fit test,
the metric may be used to measure the distance between the empirical
measure of a random sample and the probability measure from which the
sample is hypothesized to have been drawn.
(By "empirical measure" is
meant the measure which puts equal mass at all the sample points and
zero mass elsewhere.)
The corresponding two-sample test for identical
distribution would use as a test statistic the distance between the
empirical measures of the two samples.
Kolmogorov-Smirnov test, where
metric on
Q(_oo,t]I .
S
The standard example is the
is the real line and the uniform
peS) is used, i.e., the metric
sup Ip(_oo,t]
-oo<t<oo
The Cramer-von Mises test is based on a directed distance
between the empirical distribution function
00
bution function
~
(.)
F,
dC(F ,F) =
n
is a weight function.
ric in its arguments.)
nJ
_00
(de
dK(P,Q) =
F
n
and the null distri-
(F (x)-F(x))2~(F(x))dF(x),
n
is not metric since it is not symmet-
Other tests based on metrics on
been suggested in [1] and [26].
where
peS)
have
2
The so-called "Levy" metric,
dL , which is defined for
k
S = E
(k-dimensional Euclidean space) and has as its metric topology the
topology of weak convergence of probabili ty measures, and the "Prohorov"
metric,
dp ' which generates the same topology on
general class of metric spaces
peS)
for a more
S, have been given little discussion in
the context of hypothesis testing.
Dudley [12] remarks that there
I
exists no practical means for computing the value of the test statistic
for the test based on
dp '
Zolotarev [32] suggests that a barrier to
the use of tests based on
d
L is the lack of knowledge about its
characteristic function, which presumably prevents the development of
the asymptotic distribution theory along the lines of that for the
Kolmogorov-Smirnov test.
An additional difficulty is that while the statistical tests based
on
d
K
are distribution-free for continuous null distributions, the
analogous tests based on
dp
and
d
L
are not.
In the case
we can construct a distribution-free one-sample test by using
d
L
dp
or
to measure the distance between the uniform measure on [0,1] and
the empirical measure of
{X.}
1
function and
test would use
1
or
{s./N}
1
F is the null distribution
where
denotes the sample.
dp
cal measures of
{F (X. )} ,
The corresponding two-sample
to measure the distance between the empiriand
{r./N} , where
and
1
are the respective ranks of the two samples among the pooled observations,
and
N
= m+n.
Alternatively, the two-sample test statistic could be the
distance between the empirical measure of
measure of
{l/N, ... ,N/N} .
{s./N}
1
and the empirical
3
In what follows, we will discuss the computation of these test
statistics based on
dp
and
d
L
, their distributions under the null
hypothesis, and some of their statistical properties.
For
dp ' exact
distributions of the two-sample test statistic are computed for the
case of equal sample sizes, corresponding to the results of Massey [21]
for the Ko1mogorov-Smirnov test.
We are unable to give similar results
for the one-sample test, as are found by Birnbaum and Tingey [6] for
the Kolmogorov-Smirnov test.
The asymptotic distributions of the test
statistics are related to the distribution of particular fluctuations
in the sample path of the
"Brownian Bridge" process.
As with
d ,
K
there is a single asymptotic distribution for the one-sample and twosample test statistics.
We have not succeeded in finding a closed-form
expression for this distribution, although some crude upper and lower
bounds have been computed.
statistics based on
d
L
The asymptotic distribution of the test
is closely related to that of the statistics
d , and can be given explicitly.
K
In Chapter 1, we give definitions and a brief description of weak
based on
convergence and of the metrics
d
L
and
dp '
Results and relationships
are collected here which will be of use in later chapters.
contains computational procedures for
putes
For
dpCP,Q) for measures
P
and
dp
and
d .
L
Q is arbitrary.
One algorithm com-
Q which each have finite support.
l
S = E , a more efficient algorithm computes
finite support and
Chapter 2
dpCP,Q) where
P has
In Chapter 3, recurrence equations
are derived which are used to compute the exact null distribution of the
two-sample test statistic for equal sample sizes.
Chapter 4 contains
the results on the asymptotic distributions of the test statistics.
4
Chapter 5 consists of a brief discussion of the statistical properties
of the tests, including the results of simulations comparing the power
of these tests with the power of the Kolmogorov-Smirnov test against
normal location and scale al ternati ves, and against "Lehmann" nonparametric alternatives.
An appendix contains tables of the distribution
of the two-sample test statistic obtained using the methods of Chapter
3, and the results of Monte Carlo estimation of the exact distribution
of the one-sample test statistic for sample sizes 5, 10, 20, 40, 60,
and 80, as well as a description of the computer programming methods
used.
1.2 Weak convergence of measures.
As before, let
(S,p)
be a metric space and let
of all Borel subsets of
S.
measures on (S,S).
{P}
Let
Definition 1.2.1:
n
{P }
n
(a)
If
(b)
k
S = E
then we say that
and
F
n
and
peS)
f
P be elements of
F
peS).
is said to converge weakly to
F(x) = P{y:y~x}
~
be the a-field
be the space of all probability
Jf dP + Jf dP
S
n
S
defined on S.
P => P) if and only if
n
real- val ued function
Let
S
P (written
for every bounded continuous
and
if and only if
F (x) = P
n
n
P => P.
{y:y~x}
(Note:
n
, n=1,2, ... ,
if
Yi ::; xi ' i=l, ... ,k.)
(c)
Let
{X}
n
T be a metric space and
and
T the Borel subsets of T.
X are random elements (i.e., measurable mappings) from a
probability space (n,A,p) to (T,T), we say that
PX-
l
n
that
=> PX{X }
n
If
l
as probability measures in
converges in distribution to
peT).
X.
X =>X
n
if and only if
In this case we say
o
5
Definition 1.2.2:
For each
A.
Let
AES
if
P(oA) = O.
P E peS).
A
~
S , let
oA
denote the boundary of
P-continuity set
is called a
(Note that oA E S, since
oA
if and only
0
is closed.)
We will usually use the following equivalent characterization of
weak convergence instead of the original definition.
The next two
Propositions are found in [5], where additional equivalent conditions
for weak convergence are given.
P => P
Proposition 1.2.3:
P-continuity sets
before, then
k
XE E
F
n
F
(S,S) to (T,T).
some sequence
+
{h} and
+
are defined as
for every point
F(x)
o
x.
h
n
{x } of points of
n
is separable,
T
n
S
E
T the
be measurable functions from
E be the set of all
Let
(Since
h (x).
F (x)
F
for all
T be a separable metric space and
Let
T.
Borel a-field of
and
n
peA)
+
n
{F}
is continuous at
Let
Proposition 1.2.4:
and
if and only if
=> F
such that
k
S = E
If
A.
P (A)
if and only if
n
XES
for which there exists
such that
x
n
x
but
h (x )
n n
E S.)
P => P
and
P (E) = 0,
If
+
n
o
then
Proposition 1.2.4 frequently appears through its corollary
if
is a continuous function from (S,S) to (T,T), then
h
implies
n
d
L
Definition 1.3.1:
by
P => P
n
P h-1 => Ph- 1
1.3 The metrics
peS)
that
dx(P,Q)
and
dp •
(a)
= inf{E>O:
Let
S =
E
k
.
P(_oo,x] - E
Define the metric
~
Q(_oo,x]
~
d
X
P(_oo,x] + E
on
6
for all
(b)
Let
dL(P,Q)
x
(c)
Let
E
= Ek .
Define the metric
inf{E>O: P(-OO,X-E] - E
=
all
on
S
~
d
L
P (S)
on
~
Q(-oo,x]
by
P(-OO,X+E] + E; for
k
E }.
(S,p)
peS)
be an arbitrary metric space.
Define the metric
dp
E
dp(P,Q) = infh>O: Q(F) ~; P(F ) + E; for all closed sets
by
F ~ S}, where
FE
{y
=
S: there exists
E
x
E
F such that
p(x,y)
E}.
<
o
dp
(Note: the definition of
Q.
Let
However, suppose that
is not apparently symmetric in
Q(F)
G ~ S be closed, then
GE:
~
P(F E: ) + E:
is open.
P and
for all closed sets
Since
F
S.
~
G ~ S _ (S_GE:)E: ~ GE:
and Q(S_GE:) ~ P((S_GE)E:) + E:, we have the inequalities
peG) ~ P(S_(S_GE:)E:) ~ Q(G E) + E:, hence
appears in [30].
between
P
dp(P,Q) = dp(Q,P).
Previously the definition of the Prohorov distance
and
Q was usually given as what, in our notation would be,
min(dp(P,Q), dp(Q,P)).
This latter definition is more general since
peS) 1 Q(S),
when dealing with a larger space of measures so that
dp(P,Q)
may not equal
Proposition 1. 3.2:
if and only if
(b)
If
S
=
dp(Q,P).)
(a)
dp(Pn,P)
k
E , P =>P
n
Let
-+
S be separable and complete.
is given in [14].
P => P
n
O.
if and only if
dL(Pn,P)
The proof of (a) first appears in [24].
-+
O.
A proof of (b) for k
=
1
Bhattacharya in [3] observes that essentially the
same proof works for
(P(S),d p )
This proof
k > 1.
Prohorov [24] shows that the metric space
is separable and complete when
S is separable and complete.
7
These assumptions about
1.3.2 shows that
S
and
dp
o
can be weakened (see [5]).
d
L
are equivalent metrics, i.e.,
peS), when
they generate the same topology on
S
=
k
E .
However, they
do not generate the same uniformities, as shown by the following exDefine
ample from [11].
Then
j = 1, ... ,no
P
n
and
dL(Pn'~)
=
Qn
by
Pn (2j) =
lin , while
~(2j+l)
=
dp(Pn'~) = 1/2.
lin,
The
topologies generated by these metrics are relevant to their statistical properties.
Two examples to this effect
remark in [15] that
are suggested by Hampel's
gives a "literal quantitative description" of
dp
"rounding of the observations" and "the occurrence of gross errors."
Let
Example 1. 3. 3:
be real numbers and define
and Yi = Xi ' for i = 2,3, ... ,n.
measures of
{x.}
and
1
{y.}
Let
P
n
and
respectively.
1
~
be the empirical
Assume that
minlx.-x. I
Then dp(Pn'~) = dL(Pn'~) = min(E,l/n), while
i!j 1 J
dK(Pn,Qn) = lin. In this case, when E < lin, d
and dp give a
o
<
E
<
L
0
better idea of the size of the "error" in the first observation.
Example 1.3.4:
Now define
i = k+l, ... ,n.
Let
E
>
y. = x. + E, i = 1, ... ,k, y. = x.
1
kin.
1
Then
1
1
for
dp(P,Q) = kin, while
n
n
o
dL(P ,Q ) = lin.
n
n
Neither
d
L
nor
dp
generates a distribution-free test.
In
particular, neither metric is invariant under transformations corresponding to changes of scale in the data.
This lack of invariance must
hold for any metric which generates the topology of weak convergence
on
8
The following Glivenko-Cantelli-type result is due to Varadarajan
[31].
Proposition 1.3.5: Let
and let
Then
P
n
==:>
peS), where
E
S
separable and complete,
IS
be the empirical measure of a sample of size
n
P
P
a.s.
P
Hence
n
from
P.
a.s., and for
•
o
The definitions which we have given for
most often found in the literature.
dp
and
d
are those
L
They are actually not the most
convenient for our purposes and the following results show that cerFS ]
Let
tain modifications can be made.
{y
=
S:
E
x
E
F
dp(P,Q)
=
such that
p(x,y)~d.
(a)
Proposition 1.3.6:
inf{s
(b)
>
>
P, Q
k
peE ).
E
(a)
d (P ,Q)
L
Q(_oo,x]
~
For each closed
~
For each
x
P(_oo,x-s'] - s'
E
~
peS).
E
Then
for all closed
Then
P(_oo,x-s) - s
0:
Proof:
(b)
P, Q
0: Q(F) ~ P(Fs]) + s;
Let
infh
Let
F
P (F
s'
~
F
.
S}
s:.
=
P(_oo,x+s] +
for all
S
E } •
s',
) +
for each
s
>
0
and
s'
>
s.
k
E
P(_oo,x-s) - s
~
P(-oo,x-s] - s,
for each
s > 0
o
Definition 1.3.7:
of
s
For given
by
su~(P(-oo,x-s)
XEE
P, Q
E
peS), define the function
s
Vp(P,Q;s) = sup(P(F) - Q(F ])) , the supremum
being taken over all closed
max{
k
E
S,
~
s' > s .
Vp(P,Q;s)
x
F
s:. S.
Define
- Q(_oo,x]),
VL(P,Q;s)
=
sup (Q(-oo,x] - P(_oo,x+s])}.
xEEk
o
and
9
Proposition 1.3.8:
(a)
For
c
[0,1], dp(P,Q) :::: c
E
if and only if
VpCP,Q;c) :::: c.
(b)
For
Proof:
c
[0,1], dLCP,Q) :::: c
E
(a)
By 1.3.6 (a), if
Conversely,
if and only if
Vp(P,Q;c) :::: c, then
~ c implies for c'
dp(P,Q)
+ c', for all closed
F
S.
~
>
c
Similarly,
dpCP,Q):::: c.
that
c
P(F):::: QCF '])
But for each closed set
c' ]
c]
lim Q(F
) = QCF ), so
c'+c
i.e., VpCP,Q;c) :::: c.
(b)
VL(P,Q;c) :::: c.
F,
for all closed
lim PC_oo,x-c')
and
P(_oo,x-c)
=
c' -tC
F
S,
c
lim P ( _00, x+c ']
c'+c
=
o
P(-oo,x+c] .
The next two results show that when
the Euclidean metric, then in defining
consider all closed subsets of
S
dp
= El
or
S
= [0,1]
under
it is not necessary to
S, but that a smaller class of sets
will suffice.
Definition 1.3.9:
Let
F be the class of all closed subsets of [0,1]
which can be expressed as the union of finitely many disjoint closed
intervals.
Let
if we write
b.1+ 1 - c.1
and if
c
>
m
F
26
F
(26
be the class of all members of
as the union of the disjoint intervals
for i
f 1, then
Proposition 1.3.10:
only if
)
1, ... ,m-l;
c
m
Let
<
1 P, Q
c
sup (P (G) - QCG ]))
GEF(2c)
and if
b
l
f 0, then
F
such that
I. = [b., c.]
J
J J
b
l
o.
E
>
0;
o
PC[O,l]).
~ c.
Then
dp(P,Q):::: c
if and
10
Let
Proof:
Vp(P,Q,c) =
the set
c
m
~
E
>
It suffices to show that
c
sup (P(G) - Q(G ])).
GEF(2c)
G(F)
1 - <5.,
be the union of
{b. }
where
C
(G(F)/] = F ]
But
° be arbitrary.
1
For each closed set
F
with every interval [co ,bo 1]
1
c
P (F) _ Q(F ])
~
0
The proof of the same result for
dp(P,Q)
dp(P,Q)
and that
P,Q
E
1
peE )
is similar except
F are allowed to consist of the unions of finite
or semi-infinite intervals.
follows that
~
c
As immediate corollaries to 1.3.10 it
if and only if
c
sup(P(F) - Q(F ]))
FEF
dp(P,Q)
tion of
~
~ c,
=
inf{E > 0: P(F) ~ Q(F E])
+ E
for all F E F} =
inf{E > 0: P(F) ~ Q(F E])
+ E
for all F E F(2E)}
It follows from Definition 1.3.1 (and the fact that
dp(Q,P))
Then
c
P (G) _ Q(G ])
G(F) E F(2c).
that the sets of
such
1+
{co} are as in Definition 1. 3. 9.
1
and
F -c G(F) , so
and
Fe [0,1], let
~ dL(P,Q)
dp(P,Q) =
that for
P,Q E peEk),
dL(P,Q),
since the inequalities which must, in the defini-
dK(P,Q)
dp ' be satisfied for all closed sets
and also that
F need only be satis-
fied for sets of the form (_oo,x] or [x,oo) in the definition of
No such general inequality exists between
dealing with measures in
P([O,l])
d
and
K
and when
P
=
dp .
d .
L
However, when
U, the uniform meas-
ure, the following relationships hold.
Proposition 1.3.11:
Let
S = [0,1]
Lebesgue measure restricted to
S.
and
Let
p(x,y) = Ix-yl.
V E peS)
Let
be arbitrary.
U be
1hen
11
Proof:
x E [0,1], IU[O,x] - V[O,x] I
If for some
and hence
~
dp(U,V)
To show
E.
dp(U,V)
choose a closed set
such that
Therefore,
~
dK(U,V), suppose
F
such that
~
V(F)
c 2<···<bm
~
c
m
~
I.
J
2E
0, then either
>
dK(U,V)/2.
0 < E < dp(U,V).
>
E
U(F )
E.
+
FE F(2E), as in Proposition 1.3.10.
of the disjoint intervals
b2
~
dp(U,V)
~
Then we may
We may choose
Write
F as the union
= [b.,c.], j=l, ... ,m, where 0
J
F
J
b
~
1
~
c
1
<
1.
We have one of three cases
(i)
E < b1
and
c
(ii)
E < b
and
c
(iii)
0
1
= b1
m
m
A = 2m-2
= 1,
and 1 = c
U(F£) = U(F) + 1.£,
Then
< 1 - E;
or b
1
=0
and c
m
< 1 - £
m
where A = 2m in case (i) ; A = 2m-1 in case (ii) ;
in case (ii); and hence
m
I
V(l.) ~ U(F£) + £ = (1.+1)£ +
j =1
I
U(I.)
j =1
J
Then for some j,
Suppose case (i) holds.
and hence
m
2£ < (V-U)(lj)
~
1
~
j
~
J
m,
V([b ,1])
m
~
m-1
I
j =1
U([b ,1]) + E, so that
m
V(l.) ~ 2(m-l)£ +
J
and hence for some j, 1
~
j
Suppose case (iii) holds.
V([b ,1])
m
~
~
j=2
I
J
U(I.) + 2£
J
m
1.
Then
£ , or
J
~
U(lj) + 2£ , so that
~
QK(U,V)
2(m-2)£ +
c
U(l.)
j=l
m-l
~
~
dK(U,V)
Then either V([O,c ])
l
m-1
V(l.)
>
m-1
m-1, V(I )
j
U([b ,1]) + £ , so that
m
I
J
2dK(U,V)
Suppose case (ii) holds and for definiteness that
either
V(I.)
L U(l.)
j;:2
J
~
U([0,c 1 ]) + £
£, or
2dK(U,V)~2£.
or
12
and hence for some
j, 2
~
j
~
V(I.)
m-l,
~
J
In all three cases
s
~
UCI.) + 2s
J
dK(U,V).
so that
Letting
s
t
dpCU,V),
o
the proof is complete.
This proof is a modification of that of Dudley in [12], where it
is shown that
~
dKCU,V)/2
~
dpCU,V)
2dK(U,V).
at least very nearly the best possible..
in
Indeed, if V is the measure
P([O,l]) which places unit mass at 0, then
dp(U,V)
=
1/2.
points
° and
dpCV Ck ) ,U)
k ~ 2.
Let
1
dp(U,v(k))/dK(U,V(k))
dp(U,V) = dK(U,V)
>
+
1.
shows that if
n
U, then
from
dKCU,V) = 1 , while
places mass 1/2k
dK(V(k) ,U)
=
1/2k.
Hence as
k +
00
We have no example in which
Un
is the empirical measure of a sample of
n~E(dp(Un'U)) is bounded and bounded away from
Dudley [12] also gives asymptotic upper bounds on
where
V
Vn
then
at 1/k,2/k, ... ,(k-l)/k,
zero.
E
at the
0.
1.3.11
size
l/k
and mass
1/(2k+l) , while
=
VCk )
If
The bounds given are
E(dp(Vn,V)),
peS) , for a more general class of metric spaces
is the empirical measure of a sample of size
n
from
S, where
V.
1.4 Asymptotic distributions based on the Wiener process.
Definition 1.4.1:
such that for
=0
t
(a)
E
and var(W(t))
tl~···~tk ~
The probability measure
W defined on
[0,1], Wet) is normally distributed with
= t,
=0
(W(O)
a.s.) and such that for
1, the random variables
W(t ) - W(t _ )
k
k l
are independent,
It follows that
E(W(t)W(s))
=
°
C[O,l]
E(W(t))
~
to
~
W(t l ) - W(t )' W(t 2 ) - W(t l ), ... ,
O
will be called Wiener measure.
min(s,t) , for s,t
E
[0,1].
13
(b)
and
The measure
WO
E(WO(t)Wo(s))~
Bridge.
on qO,l]
s(l-t)
We may define
(Alternatively
is defined as
Wet)
for s
WO
by
with
WO(t)
WO(t)
0
s
$
t E [0,00)
t
$
1.
and then
WOes)
The "sample path"
1.)
$
$
and hence
{W(l)
Wet)
WO(t)
may also be thought of as being distributed as
conditioned on the event
WO
Wet) - tW(l), 0
=
may be defined for
(l-s)W (s/ (l-s)),
E(Wo(t)) = 0,
normal,
will be called the Brownian
t
$
is a continuous function on [0,1],
is also.
WO(t)
Wet)
= O}.
has an important role in deriving the asymptotic distribution
for a number of test statistics, including the Kolmogorov-Smirnov,
making use of Proposition 1.2.4.
This was first recognized by Ooob [10],
and the proof was made rigorous by Oonsker [9].
application of t his method are [4] and [27].
Other examples of the
The following results are
found in [5] and [10].
Proposition 1.4.2:
let
U.
o
U
n
(a)
t
$
U be the uniform measure on [0,1] and
be the empirical measure of a random sample of size
Then the process
$
Let
1,
X
n
defined by
converges weakly to
WO
X (t )
n
= n .ok (Un [ 0 , t ]
2
n
from
- U [ 0 , t ]) ,
in the Skorohod topology on
0[0,1], the space of all functions on [0,1] which are continuous from
the right and have left-hand limits.
position 1.2.4 that
tion to
(b)
Let
ntdK(Un,U)
=
In particular this means by Pro-
sup Ix (t)1
O::;;t::;;l n
converges in distribu-
sup IWO(t)1
O::;;t::;;l
U and
m
{r./N} ,where
V
n
{s.}
be the empirical measures on
and
{r.}
1 1 1
{s./N}
1
and
are the ranks among the pooled obser-
vat ions of two random samples from the same continuous distribution
14
and
N
= m+n, m and n being the respective sample sizes.
m and
Y
m,n
n
(t)
=
provided 0
Y
m,n
approach infinity, the process
.k-
(mn/N)2(U ([O,t])-V ([O,t]))
m
n
<
a
~
min
~
b
<
Let
Proposition 1.4.3:
b}
converges weakly to
o
MO
sup WO(t)
O::;t::;l
=
Pr{ sup IWO(t)1 ::; b}
O::;t::;l
00
(c)
<
mO
= 1 + 2
I
(_1)k exp (_2k 2b 2 ), b > 0 •
exp(-2k 2c 2 )
I
=
k=-oo
c
inf WO(t).
O::;t::;l
k=l
00
mO ~ MO ::; b}
where
=
2
(b)
Pr{a
and
1 - exp(-2b ) , b > 0.
Pr{MO
=
defined by
00.
(a)
<
Then as
= b-a,
b
>
0
00
I
exp(-2(b+kc)2) ,
k=-oo
and
a
<
o
O.
These results agree with earlier results proved by Kolmogorov and
Smirnov.
Billingsley proves (c) by arguments about a sequence of
random walks which approximate the process
Woo
Doob uses a more
general result about the probabilities of various level crossings by
the WO process, using the Inclusion-Exclusion principle to obtain the
expression given for Pr{a
me ::; MO ::; b} .
<
1.5 Terminology from the Theory of Graphs.
sets.
A finite simple
graph~
r,
pairs Cx,y), with initial vertex
A matching is a subset of
r
Let
X and
Y be finite
is a collection of arcs, or ordered
x
E
X and terminal vertex
in which each element of
y
E
Y.
X Crespo Y)
appears at most once as an initial (resp. terminal) vertex.
A maximal
matching is a matching of maximum possible cardinality; there is not in
15
general a unique 11l;lxim:1l
11latchin.I~.
matching in the graph (X,Y,r)
The 1lI111lhcl'
will be denoted
deficiency of the graph (X,Y,f) is defined as
max(#(F) - #(f(F))), where
F~X
that (x,y)
E
f},
f(F)
and for any set
=
{y
E
llj"
;In:s
in a lila xi lila I
N(X,Y,r).
D(X,Y,f)
Y: there exist
The
=
x
E
F such
S, #(S) denotes the cardinality of
S.
Proposition 1.5.1:
in [23].
N(X,Y,f) = #(X) - D(X,Y,f).
This result is found
o
CHAPTER 2
Computational Procedures
2.1
Computation of dp(P,Q)
for general
S when P and Q
have finita support.
By Proposition 1.3.8(a), we know that
if
Vp_(P,Q;c)
~
c.
dp(P,Q)
~
c
For purposes of testing statistical hypotheses, we
need only to check whether a single inequality
where
c
a
we first test whether
dp(P,Q)
determined that
k
o and
Z -1, and
~
1/4
dp(P,Q)
or
~
dp(P,Q)
a.
c
a
holds,
If instead we
to any desired accuracy,
liZ, then, depending on the result,
dp(P,Q)
~
k
(i/Z , (i+l)/Zk]
3/4, etc., until finally it is
dp
E
k
chosen large enough to provide the desired accuracy.
It is therefore our purpose to compute
of
~
is the critical value of a test of size
wish to compute the actual value of
whether
if and only
for some value of
Vp(P,Q;c)
i
between
for any given value
c.
If
P has finite support, i.e.,
then it is clear that
P(X)
= 1,
where
c
Q(F ])).
Vp(P,Q;c) = sup(P(F)
X
= {xl, ... ,xn },
This supremum
F~X
is a maximum, so there exists a subset
peG) - Q(G c ])
~
X such that
~ P(F) - Q(F c ]) for all F ~ S.
be able to determine the set
on the measure
G
Q.
When
l
S = E , we will
G without placing further restrictions
For general
S, we are able to solve the problem
using Proposition 1.5.1, but only when
Q also has finite support.
17
Let
put mass lin
P
n
at each of the points
and
Define
Qn put mass lin at each point of Y = {Yl""'Yn}
let
the graph (X,Y,r c )
p(x. ,y.)
1
]
c,
~
by letting the arc (x.,y.)
1
where
p
is the metric on
]
S.
r
E
if and only if
C
Then
is just lin times the deficiency of (X,Y,r ), and hence by Proposition
c
1.5.1,
Vp(Pn ,Qn ; c)
=
n
-1
(n-N(X,Y,r c )).
There does not seem to be a
method for finding the number of arcs in a maximal matching in a graph
except by constructing a maximal matching.
There are published algor-
ithms for finding a maximal matching in any graph.
The standard method
has been the "Hungarian algorithm" (see [2] or [18]).
A new and more
efficient algorithm has been proposed by Hopcroft and Karp ([16]).
X and
11m
Y need not have the same cardinality.
at each of the points
each of the points
,
n-l#(r (F)))
c
m = n.
m =
Suppose
and n
Yl'··· 'Yn·
Then
Vp(P
On
,~;c)
=
P
m
put mass
put mass lin at
-1
sup(m #(F)
m
F~X
where the graph (X,Y,r ) is defined as in the case
c
m and
=
xl" .. ,xm and let
Let
n
have greatest common factor
and
P2 q , where
Define the graph (X' ,Y'
,r~)
q, so that
are relatively prime.
as follows.
X'
and
Y'
Then
each consist
of PlP2 Q vertices,
and
Y'
Define
f'c
= {y11 ' Y22' ... ,y1PI' Y21' Y22' ... ,y2p 1' ... ,yn l' ... ,ynp 1}
as follows.
For each
i
and
j, 1
~
i
~
m and
1
~
j
~
n,
18
let (x. ,y. ) E f'
lr JS
c
for all values of
X and
Y'
are connected by an arc in
f
c
Suppose
Xl"
f'({Xol, ... ,x. })
C
1
IP2
#(G
Let
1P2
1
x., then
f~,
whenever the "originals"
{x·l'· .. ,x. } n GO ~ ~, then
1
IP2
To see this observe that for each r, 1
= f'({x.
})
c
lr
}nG~~
and hence, if
U
~
x. EGO'
lr
{X.l' ... ,X. }))
1
IP2
G
if and only if
X'
~
such that for each
{X.l' ... ,X. }
1
IP2
c
G.
i,
Then we
have just shown that
#(f' (G))) = max (# (G) - # (f' (G))) .
c
GEG
c
max (# (G)
G~X'
For each set
F*
~ X,
let
G* =
U
{x. ' ... ,X. }, in this
{i:x.EF*} 1 l
IPZ
1
way defining a 1-1 correspondence between
sets of
X.
That #(G*) -
the definition of f'.
PI'
Y,
may be assumed to contain
1
G be the class of all sets
{X.l' ... ,X.
~
c
{X.l' ... ,X. }) - #(fc'(G
O
1
IPZ
oU
s
Le., if
GO'
2
~
X
~
GO
{XiI" .. ,x ip } ~
P2 and 1
Max(#(G) - #(f' (G)))
contains any "copy" of
all "copies" of
~
•
G~X'
Proof:
r
consists of PI copies of each point of
and the "copies" are connected by an arc in
Proposition 2.1.1:
~
Intuitively, X' consists of P2 "copies" of
if arid only if (xi'Yj) E f c '
each point of
rand s, 1
#(f~(G*))
G and the class of all sub-
= PZ#(F*) - pl#(fc(F*)) follows from
Hence
max (# (G) - # (f' (G)))
GEG
c
= max(p2#(F)
F~X
- PI (#fc(F)))
o
19
We can thus find the value of
PlP2qVp(Pm'~;c)
by applying
the Hungarian or Hopcroft-Karp algorithm to (X' ,Y' ,f').
However,
c
unless PI and P2 are relatively small, the sets
X'
and
Y'
will
contain so many points that the procedure will be excessively timeconsuming, even though the special form of (X' ,Y',f')
allows some
c
steps to be saved in applying the algorithms.
The procedure we have described also extends to the case where
P (resp. Q) does not give equal probability to the points in X (resp.
Y), but the probability of each point is a rational number.
and
case there exist positive integers
that
In this
such
Vp(P,Q ;c) is equal to a constant times the expression
m n
I
I
maxi
P·XF(X.) q.xF(yo)/
Fs=.X i=l 1
1
j=l J
J
denotes the indicator function of the set
where
In this case we will let
i = 1, ...
,m,
and let
j = l, ...
,n,
f'c
Y'
X'
consist of
consist of
q.
J
being defined as before.
F.
po "copies" of
1
"copies"
of
x
i
'
Yo,
J
With the obvious modifi-
cations of the proof of 2.1.1, we can show that for this graph (X' ,Y'
,f~),
m
n
max(#(G) - #(f' (G))) = maxl L poXF(x.) - .L qJoX f (F)(YJo))
Gs=.X'
c
Fs=.X i=l 1
1
J=l
c
In this case, depending on the values of
Pl,···,Pm
and
it may be impractical to find a maximal matching in (X' ,Y'
for small values of
special case
l
S = E
m and
n.
ql, ... ,qn'
,f~)
even
In the next section we will treat the
1
and take advantage of the total ordering on E to
develop a more efficient and more generally applicable algorithm.
20
2.2 Computation of dp(P,Q) when P has finite support.
Let the support of
xl S x s ... sx .
n
2
X = {xl' ... ,x } where
n
Q be an arbitrary measure on El . As described
Let
P be the set
before, we wish to find a set
~ P(F) - Q(F c ])
F ~ X.
for all
erization of such a set
Definition 2.2.1:
If
G ~ X such that
P(G-F) - P(F-G) ~ Q(G c ] - FC])
is as bad as
worse than
We shall use an equivalent charact-
G:
P,Q, and
Let
peG) - Q(G C ])
c
be given.
-
Let
G
~
X and
F
Q(F c ] - GC ]), we shall say that
F, and if the strict inequality holds we shall say
F.
If
G is as bad as every subset of
G
G is
0
(Note, if the support of
1
let
X.
X, we shall call
G a worst set.
closed:
~
P
S = E , c = 1/4, let
is infinite, the worst set may not be
P put mass 1/4 at the points
-1/4 and 1/4 and be equal to Lebesgue measure on the set (-1/4,1/4).
Let
Q put mass 1/2 at the points -1/2 and 1/2.)
peG)
maximize
A worst set must
Q(G c ]).
Note that if
F
~
than F) if and only if
G
~
X, then G is as bad as F (resp. worse
c]
c]
-1
-1
Q(G
- F ) S n #(G-F) (resp.< n #(G-F)).
This will be the criterion of relative badness in most of what follows.
Our algorithm for finding a worst set will examine the points
xl, ...
in order to see whether they belong in a worst set.
,X
n
The
basic idea is the following.
Proposition 2.2.2:
value of
i
Let
i
range from 1 to n.
such that either
Let
i'
be the least
21
(If
(a)
Q({Xl, ... ,Xi}C]) < i/n,
(b)
Q({xl,···,x.}
1
i = n, replace
c]
- {x.1+ I}
{x.1+ I}
must hold for i = n. )
c]
or
c]
~
)
(Note that either (a) or (b)
~.)
by
i/n
=
If (a) holds for i
. , , then
xl must be
1
included in every worst set; if (b) holds for
a worst set containing
contain
Proof:
=
xl' there is a subset of
.,
1
,
then if
G
is
G which does not
and which is also a worst set.
Assume (a) holds.
I n H
I n H
1
Let
~
H
I = {xl' ..• ,x '}. If
i
I n H is worse than H. Suppose
then (a) implies that
~ ~
, but xl i H.
Let
Let
be the smallest element 6f
c]
c]
Q({xl, ... ,x _ l } - {X } )
j
j
Since by definition of i',
H U {xl'" .,X j _l }
xj
X.
is worse than
H.
<
I n H.
.
(J-l)/n ,
xl i H implies that
Thus
H
is not a worst set.
Assume (b) holds.
clearly
H- I
est value of
Let
H
is as bad as
j, 1
:::;
j
.,
:::; 1
,
~
H.
X.
Suppose
If
H, then
H.
If
I
I 1:. H, then let
j'
be the great-
such that
x.
J
E
xl
E
~
H and
c]
c]
n [H - {xl'···,x }]
= ~
{Xl'···,X j }
j
If no such element exists, let
If
j' = i', then
j' = i'.
Let
J = H - {Xl' ... ,X '} .
j
Q({Xl, ... ,xj,}C] - JC]) ~ Q({Xl, ... ,xj,}C]
l}c]) ~ j'/n , by definition of
i', since (b) holds. (This could
fail unless xl
X , because we might not have Q({x , ... ,x ,} c] ~
n
l
j
c]
c]
Q({x l , ... 'X '}
- {x., I} ).) It follows that J is as bad as H, since
j
J +
#J ~#H - j', while Q(H c ]) - j,'/n ~ Q(H c ]) - Q({Xl, ... ,xj,}C] _ JC])
{X.,
J +
~ Q(Jc]) , so Q(Jc]) _ Q(H c ]) :::; n-l(#J - #H).
22
If
jl < iI, then
as bad as
G
n
. I•
=
will be a worst set.
Let
GO
= 1. I ,
G1
= {xl}
and let
In general we will have
1
Assume inductively that
I.e.,
Gk
{Xl}
Gk
=
is
c ... -cGn
GO -c G1Let
= ~.
Gl
=
such
GO if (b) in
if (a) in 2.2.2 holds for
G s= {x1' . . . , x }
k
k
is contained in any worst set and that
G - ({xl,···,x } - G )
k
k
is also a worst set.
satisfies the conditions which 2.2.2 shows are satisfied by
U
{xk+l' ... ,xn }
i
i
range from
1
to
n-k+l.
Let
i'
be the least such value
such that either
c]
(a ' )
Q({xk+l,···,xk +i }
(b I)
Q({xk+l'···'~+i}
Then if (a l
i
J
Hence in seeking a worst set, we need only consider points of
Let
of
G
k
G is a worst set, then
if
Again
•
o
2.2.2 holds for i
i
1
H.
We next wish to define a sequence of sets
that
. I
jl < iI, by definition of
since (a) cannot hold for
- G~])
c] _ GC]
k
holds for i = iI, we let
)
x + i. G + .
k i
k l
iI, then
(a ' ) and (b l
)
since
GO
<
U
{
i/n,
xi + l
x +
k l
}c]
E
or
)
>_
i/n .
G+ ;
k l
if (b l
)
holds for
((a) and (b) are really special cases of
= ~.)
By the same argument as in 2.2.2 (restricting consideration to sets
H such that
{Xl' ... ,x k } n H = Gk ) we can show that
G +l
k
is contained
in any worst set and that there always exists a worst set which contains
By induction,
be a worst set.
G
n
will necessarily
23
Having defined
each point
x +
k l
G , our algorithm will then be to check for
k
whether (a') or (b') holds for i
struct the set
G+
k l
noting that if
(a') holds so that
Q({xk +2 ,··· ,xk+i'-l}
i'
>
1
implies that
implies that
accordingly.
c]
c]
- Gk +l )
Q({x k + l }
<
tested individually.
i', and to con-
The algorithm can be improved by
~+l E
(i'-l)/n
c] - GC ])
xk +2 "" ,xk+i'-l
=
2:
k
Gk + l , then we will have
(provided
Hence,
lin.
are all in
i' > 1). Indeed,
G
X
k+ l
E
Gn
and they need not be
n
This improvement could be significant if the
evaluation of the measure
Q is time-consuming.
The algorithm described in this section is sufficiently efficient
to permit the use of Monte Carlo methods to estimate the distribution
function of test statistics based on
pose, the case of a measure
effect replacing
P
by
n
when
S
=
l
E .
For this pur-
P which puts mass lin at each of
points is sufficiently general.
probabili ties at each of
dp
If instead,
n
P places rational-valued
points, we can compute
d p (P ,Q)
by ln
a measure which places equal mass on an approp-
riate number of "copies" of each point of the support -- here "copies"
are regarded as distinct observations taking the same value.
the case of general
S
, for S
=
l
E
Unlike
this does not require a signifi-
cant increase in the amount of computation since all the points in each
set of "copies" can be tested at the same time; if any is included in
the worst set then all must be.
(By contrast, in the general case our
procedure was to find a maximal matching -- in which only some of a set
of copies might appear as initial vertices -- and not to find a worst
set,)
24
2.3 Computation of dL .
Let
S
=
l
E , let
P,Q
E
peS), and let the support of
P be the set
In this case the computation of
dL(P,Q) is simple.
~
VLCP,Q;c)
c
E
[0,1]
c.
By Proposition 1.3.10,
Therefore, as with
whether
VL(P,Q;c)
~
dL(P,Q)
~
c
if and only if
dp ' to be able to find out for each
c, permits computation of
dL(P,Q) to
any desired accuracy.
v (P,Q;c)
L
is defined as
maxi sU~(Q(_oo,x-c) - P(-oo,x]), su~(P(_oo,x] - Q(_oo,x+c])}.
XEE
XEE
As a function of
x,
Q(_oo,x-c) - P(-oo,x]
is non-decreasing on the
interval (_oo,x ) and on each of the intervals
l
P(-oo,x] - Q(-oo,x+c]
[x.1 ,x.1+ 1), 1
~
i ~ n-l.
is non-increasing on each of these intervals.
Hence
sup (Q(_oo,x-c) - P(_oo,x]) = max (Q(_oo,x.-c) - P(_oo,x.)) ,
-oo<x<oo
l~i~n
1
1
and
sup (P(_oo,x] - Q(-oo,x+c]) = max (P(_oo,x.] - Q(_oo,x.+c]) .
-oo<x<oo
l~i~n
1
1
Thus
VL(P,Q;c)
can be computed by evaluating the P-measure of 2n sets
and the Q-measure of 2n sets.
If
cases of
P
P
is not a discrete measure, it may be possible in some special
and
Q to determine points
sup (P(_oo,x] - Q(_oo,x+c])
-oo<x<oo
and
x
at which
sup (Q(_oo,x-c) - P(_oo,x])
-oo<x<oo
must
occur, provided that a simple functional form for the distribution functions associated with
P
and
Q can be found.
Specific examples can
readily be suggested, for example, the case where the two density
functions are unimodal.
However, a useful general characterization of
25
cases in which the maximizing values can be readily determined is not
apparent.
If
S
= Ek ,
we can generalize the method described in this section,
although the number of points to be examined increases as
the method impractical even for moderate values of
coordinates of
x.
(1)
(k)
Xi , ... ,x i ' for each
as
1
y~j) ~ y~j) ~ ... ~y~j)
{xi j ) , ... , x~j ) }
Let
Z
~
i = 1, ... ,n.
Let
k
E be the set of all points of the form
(2)
1
i ,i , ... ,i
k
l 2
Write the
denote the ordered values of the set
y.
where
k.
k
n , making
(k)
, ... ,y.
1
2
)
k
are integers between 1 and
P(_oo,x] - Q(_oo,x+c]
n.
is non-increasing in each coordinate of
x
in any rectangle of the form
[y~l) 'Y1~1) +1 ) x [y~2) 'Y1~2) +1 )x ... x[y~k) ,y~k) 1) ,
11
1
i , ... ,i
k
1
where
12
2
1k
are integers between 0
and
n,
meaning of replacing the left-hand endpoint by
_00
YO
1k +
being given the
Hence
sup(P(-oo,x] - Q(-oo,x+c]) = max(P(-oo,y] - Q(_oo,y+c]) .
k
yEZ
XEE
Likewise,
coordinate of
Q(_oo,x-c) - P(_oo,x]
x
in each of the rectangles described and hence
sup(Q(_oo,x-c) - P(_oo,x])
xEE
is a non-decreasing function of each
= max(Q(-oo,y-c)
k
- P(_oo,y)) .
yEZ
As before, the problem is reduced to evaluating the P-measure and
Q-measure of a finite number of sets.
If
#(Z)
can be as large as n
Q has finite support, the problem can be treated by the
k
26
methods of Section 2.1, dealing separately with the two quantities
(a)
in£{ E: > 0: P ( -00, x]
Q(-oo,x+c]
~
c;
for all
x
E
k
E }
(b)
infh > 0: Q(-oo,x-c) - P(-oo,x]
~
c·,
for all
x
E
k
E }
-
and
Let
X = {xl'···,x } be the support of
n
be the support of
Q.
if and only if
~
max(P(-oo,x]
y.
J
P
and let
Y = {Yl'''·'Yn}
We then define a graph (X, Y, r )
c
x. + c, so that
1
Q(_oo,x+c]) .
nD(X,Y,r )
C
Thi s deals with (a) .
=
by
(x.,y.)
J
1
E
r
c
max(P(F) - Q(r (F)))
c
F.<::X
For (b), let the arc
XEZ
(y.,x.)
J
1
E
r
C
if and only if
y. + c < x. , and again
J
1
nD(Y,X,r) =
C
max(Q(F) - per (F))) • max(Q(-oo,x-c) - P(_oo,x)). In each case, the
c
F.<::Y
XEZ
deficiency may be found by the Hungarian or the Hopcroft-Karp algorithm.
The extension to the cases of unequal numbers of atoms for
P
and
Q, or for atoms not all the same size but rational-valued, is done in
the same way as in Section 2.1.
CHAPTER 3
Null Distribution for the Two-Sample
Test Based on dp
3.1
Introduction.
As described in Section 1.1, the null distribution of our two-
sample test statistic based on
tribution of
dp(~
n
,v ) where
n
r /2n, ... ,r /2n and s/2n,
n
l
out replacement from {l,2,
~
and
(r./2n,s./2n)
and only if
f
E
J
1
oL"
x=
by letting
c
J
and v
are the empirical measures of
n
<rn being a random sample with-
,2n} and sl<
<sn being the elements in order
F or each cE[O,l], we define the graph
{r l /2n, ... ,r /2n},
n
Y
{sl/2n, ... ,sn/2n}
=
!r./2n-s./2nl ~ c, i.e., if
if and only if
1
i. p. (2nc), where
~
jr.-s·1
1
c
n
is,,for.equal sample sizes, the dis-
,sn/2n, r l <
of {l,2, ... ,2n} - {r , ... ,rn}.
l
(X,Y,f)
dp
J
"i. p." denotes "integer part
The following can be stated immediately based on the results of
sections 1.3 and 1.5.
Proposition 3.1.1:
For positive integers
p,n, and z,
be the number of distinct samples
{rl,.··,r }
n
that
z/2n
D(X,Y,f / ) = p.
z 2n
Pr{dp(~
n
,v )
n
~ c} =
Then for
Pr{D(X,Y,f / )
z 2n
~
~
and
let
A(p,n,z)
{sl' ... ,sn}
such
c < (z+1)/2n,
z/2}
D
Proposition 3.1.1 can be generalized to the case of samples which
are not real-valued.
However,. for
S
=
1
E , the structure of
sufficiently simple to permit the evaluation of
A(p,n,z);
f
c
is
the general
28
case seems unmanageable,
does the case of unequal sample sizes.
RS
rc
In the remainder of this chapter, we will discuss the features of
which permit this evaluation to be performed using the simple relationship expressed in Proposition 3.1.2 below.
Let
X be a countable set such that each element
unique classification
determine for each
K(x)
in a "classification set"
u E C the number
are classified as
u, i.e.,
K (u)
K(u)
perty that for each
-1
x,y E K (u)
u,v E C,
#{a-l(x) n K-l(v)} = #{a-l(y) n K-l(v)}
the number of elements
depends only on
T(u,v).
(If
u
and
C.
We wish to
-1
X which
To this
(u) ) .
a: X ~ X having the proimplies that
a (z) = x
v, not on the choice of
K(u) = 0, let
and
x.
X
E
K
K(z) =
-1
(u)
V
Call this number
T(u,v) = 0).
L K(u)T(u,v),
K(v) =
Proposition 3.1.2:
# (K
Thus for each
z E X for which
of X has a
of elements of
is defined as
end we predicate the existence of a mapping
x
for each
v E C.
UEC
Observe that
Proof:
that
K(w)
K(a(w))
=v
and
K(u)T(u,v)
K(a(w))
= u.
is the number of elements
But
-1
K (v)
= u}
n
L {w:
K(w)
=V
such
and
UEC
0
For a fixed
values of
=
wEX
our problem of computing
~,
and
p
A(p,n,z)
for various
can now be expressed in terms of Proposition 3.1.2.
00
U {(X,Y): (X,Y) is a partition of 0,2, ... ,2n}
n=O
X and Y with #(X) = #(Y) = n}.
The case n = 0 we
X will be the set
into subsets
take to represent a partition of the empty set, which we will identify
with the empty set.
Note that the sets
X and
Yare now sets of
29
integers, not multiples of 1/2n.
Each element
u
tuple of integers
of the classification set will consist of a 5-
u = (k,n,d,s,t) = (k(u),n(u),d(u),s(u),t(u)) , des-
cribing various properties which hold for each partition (X,Y) classified as
In particular
n
has the meaning of the previous paragraph.
r z is similar to
is the deficiency of the graph (X,Y,r ).
z
d(u)
r
u.
and will be defined below, as will the parameters
z/2n
The mapping
whenever
a: X + X will be defined in such a way that
n(u)
~
i.e., we define
n(u) = 0;
for all
n(v),
except when
and
v
K(K(~)) =
in
u
= K(~),
1.
C, then
Hence, if
K(u)
t.
T(u,v) = 0
in which case
Only for the classification u
a(~) =~.
we define
u
sand
T(u,u)
= K(~)
= 1,
is
T(u,v) can be computed
can be found for all
u
E
C.
In the next section we define a classification set and a mapping
a
We then have
Ll I
A(p,n,z) =
K(k,n,p,s,t) .
k s t
in such a way that
T(u,v)
can be computed for all u and v.
3.2 Matchings in the graphs (X,y,rzl.
Our ultimate interest is in partitions (X,Y) of {1,2, ... ,2n} into
sets
X and Y of equal size.
For use in later proofs we wish to
describe properties which apply more generally to partitions of
{1,2, ... ,n+n'}
into sets
The positive integer
X and
Y with #(X)
=
nand #(Y) = n'.
z will be considered to be fixed and we will
hereafter use the convention that for any partition (X,Y) under consideration,
r z will denote the graph such that
Ix-yl ~ z
for
x
E
X and
y
E
Y.
We write
,
(x,y)
r z (X,Y)
E
r z whenever
if the partition
30
is not clear from the context.
We shall deal with
D(X,Y,r)
z
by looking at maximal matchings in
(X,Y,r ), using Proposition 1.5.1.
z
in
D(X,y,r z )'
Since we are really interested only
rather than the matchings, it will be convenient to
consider a single "canonical" maximal matching, uniquely defined for
each partition (X,Y), and having certain special properties, which we
will describe in Proposition 3.2.14 following some discussion.
Let
Definition 3.2.1:
#(X)
n, #(Y)
=
=
nl
and let
,
matching and write
(X,Y) partition
X+(m)
and
=
are not necessarily in order.)
=
=
= Y-Y+(m).
first element of
X-X+(m),
0
X+
y+
and
and not
In fact, we will always
X+
is connected to the
y+ , second to second, etc.
Proposition 3.2.2:
Let
m be a maximal matching in (X,Y,r ).
z
m* ~ r
there exists a maximal matching
Y+(m*) , and
such that
z
X+(m)
=
Then
X+(m*),
yl(m*) < Y2(m*)< ... <y (m*)
M
Follows from Lemma 3.2.3.
Let
Lemma 3.2.3:
(x.,Y.,) E r
11
(The elements
{Yl(m)'''.'YM(m)} , X-em)
be able to assume that the first element of
Proof:
be a maximal
z
Define
in the particular matching which produced them.
=
so that
{(xl (m)'Yl (m)), ... ,
We will be chiefly interested in the sets
Y+(m)
mEr
xl (m) < x2(m)< ... <~(m).
{xl(m), ... ,~(m)} , Y+(m)
V-em)
Let
M = N(X,Y,r z ).
m as the set of arcs
(xM(m),yM(m))} , such that
Yl (m), ... ,yM(m)
{1,2, ... ,n+n'}
z
and
and
y., E Y and
(x.,Y.,) E r.
J
J
x.,x. E X be such that
J
Z
If
x.
1
J
1
~
x.
J
and
y.,
J
~
y., , then
1
31
(x. ,y. I)
r
E
J
1
and
Z
(x.,y.,)
J
r.
E
1
Z
(The roles of
X and
Y can be
interchanged. )
Proof:
It follows from the inequalities
Ix.-Y., I
J
J
Ix. -y. I I
and
J
$
1
and
mer
-
$
Z
Z
o
Let
M(x,Y,r)
Z
such that
A matching in
Y+(m)
Z
z.
Definition 3.2.4:
matchings
$
and the assumption
Z
$
Ix.-y"
I
1
1
be the collection of all maximal
o
Yl (m)< ... <yM(m)
M is uniquely determined by its sets
and vice versa.
X+(m)
and
However, for a given partition (X,Y), there
may be a number of matchings in
M(x,Y,r ).
Z
We will introduce a schematic notation to be used in describing
the partitions and matchings under consideration and use it to give
some simple examples showing considerations involved in choosing a
"canonical" matching.
the sets
Z ~
X = {2} and
yxy
Y
will denote a partition of
= {1,3}.
Here
1, we have two matchings, denoted by
mean respectively {(2,1)}
and {(2,3)}.
always assume that the elements of
i.e., the matching is in
M.
X+
=1
n
++yxy
and
and
n'
{l,2,3}
= 2.
into
For
-++
yxy , by which we
In using our notation we
and
y+
are connected in order,
++yxy
Of the two matchings given,
will be
chosen as the "canonical" one, because it is "earliest" (the +' s occur
+-+earlier), in a sense to be made precise. From among yyxy,
--++
+-+yyxy, the matching yyxy is the "earliest." However, in
--++
yyxy
in
the matched
x
and
-++yyxy, and
-++yyxy and
yare as close together as possible"
yY~y, there is an intervening element of
Y-.
+-+yyxy
while
will be said
not to be "compressed," while the other two matchings will be said to
be "compressed," a notion to be made precise below.
In this case, the
32
"canonical" matching will be chosen to be the' "earlier" of the two
-++yyxy .
"compressed" matchings, namely
"compressed" matchings assign
+' s
It will later be seen that
and
-' s
in a useful pattern.
The choice of the "earliest" of the "compressed" matchings is made for
definiteness.
Corresponding to the notion of earliest matching, we now intro-
graphic ordering.
In this ordering we say that
,~(m)'YI
(Xl (m),
xl (m) = xl(m')
but
but
x 2 (m)
<
Le., if
than
xl(m) < xl(m'), or
xl (m) = xl(m')
x 2 (m'), or
Let
m be the least element of
respect to the ordering just defined and let
xl(m)
if the vector
x 2 (m) =
and
x (m) < x 3 (m'), etc.
3
Proposition 3.2.5:
Then
m < m'
(m), ... ,yM(m)) is ZexicographicaZZy smaZZer
,~(m'),YI(m'),''''YM(m')),
(xI(m'),
x2 (m')
which we shall call the Zexico-
M(X,Y,f z )
duce a total ordering on
~
xl(m'), ... ,xM(m)
~
m'
be any element of
~
xM(m'), yl(m)
M(X,Y,f) with
z
~
YI (m')""'YM(m)
M.
yM(m').
(Note that the definition of lexicographic minimality only implies that
Xl (m) ~ xl (m').
+++-+
is yxxxy , m'
Proof:
if
>
{(3,5), (4,1)}
xl(m)
~
xl(m').
Further,
(since
YI (m')
E
or y. (m)
J
be the least index,
j
>
~
3.)
YI(m)
~
yl(m').
y. (m').
J
existence of such a
2
Y+(m)
~
j
Indeed,
replace
V-em)) and get a matching
which is lexicographically smaller than
Let
z
provided
yl(m') , then by Lemma 3.2.3, we can in
Y (m')
I
by
m ~ M the result may not hold, for example if m
could be
Necessarily
Y (m)
I
YI (m)
If
in
M
m.
~
M, such that
x.(m)
J
>
x.(m'),
J
We wish to arrive at a contradiction by assuming the
j.
Suppose first that
x. (m) > x. (m' ) •
J
J
Since
j
33
is the first index of its kind,
Hence
J-
x. (m')
E X-em).
Similarly,
J
We have two cases:
a)
y. (m)
>
y. (m' ) ;
J
b)
y. (m)
~
y.(m') .
J
J
~
x. l(m)
x. l(m')
y. l(m')
J-
~
x.(m')
<
J-
y. l(m)
<
J-
x.(m).
<
J
J
y.(m).
J
J
In case (b), we can apply Lemma 3.2.3 to show that (x.(m'), y.(m))
J
because
x. (m) >
replace
x. (m)
J
J
by
and
y. (m)
M, contradicting the definition of m.
(a), we contradict the maximality of
E
J
Y-(m), as was shown above for
E
Y+ (m)
by
x. (m').
Since either (a) or (b)
J
x. (m)
Y-(m).
x. (m)
J
x.(m'), but
J
~
x. (m' ) •
J
> y.(m').
Then as before
J
J
By Lemma 3.2.3 we may replace the element y.(m) in
So assume that
J
In case
m, since in this case we must have
leads to a contradiction, we must have
y. (m')
Z
J
smaller matching in
y.(m')
r ,
Hence we can in X+(m)
~ y. (m') .
J
J
x. (m'), which is in X-em), to get a lexicographically
x. (m')
J
E
J
~
J
y.(m)
J
y.(m'), obtaining an earlier matching than
m, contradicting
J
o
the definition of m.
We now define the class of compressed matchings.
M' (X,Y,r z) be the class of all m E (X,Y,r z)
such that X+(m) + Y+(m). U [min(x. (m),y. (m)), max(x. (m),y. (m))] ,
Definition 3.2.6:
Let
00
where
i=l
1
[i,j] denotes the integers between
1
1
i
and
j
1
inclusive.
none of the intervals [min(x. (m),y. (m)), max(x. (m),y. (m))]
1
1
1
1
(I.e.,
contains any
o
The lexicographic ordering on
subset of
M.
M induces an ordering on M'
We wish to show that the least element
m of
M'
as a
in this
34
ordering satisfies the stronger conditions of Proposition 3.2.5 when
compared to any other element of
M'.
To do this we first give an
algorithm which associates with any element
c
m
m E M a unique element
M'.
of
Let
Definition 3.2.7:
m E MCX,Y,r).
z
We define the element
c
m EM'CX,y,r z ) by the following algorithm.
Let mCO) = m. Suppose mCO) , ... ,m Ci) have been defined.
(i)
m
(i)
m
EM', let
c
m
m(i); we have completed the algorithm.
M', we define m(i+l)
~
as follows.
Let
If
If
be the least index,
j
I s j s M , such that [minCx.(m Ci )), Y.Cm Ci ))), max(x.(m Ci )),
J
y.(m Ci )))]
J
!
J
X+Cm(i)) + Y+(m Ci )).
J
We have one of four mutually ex-
clusive cases:
a)
there exists an x E X-Cm Ci )) such that x. Cm Ci))
<
b)
there exists an x E X-(m Ci )) such that y. (m Ci))
<
c)
there exists a
y E Y-(m(i)) such that x. Cm Ci ))
there exists a
y E Y- Cm Ci )) such that Y.Cm Ci ))
d)
x
<
x
<
<
y
<
<
y
<
J
J
J
J
y.CmCi))
J
x.Cm Ci ))
J
y.Cm Ci ))
J
x. Cm Ci ))
J
The cases are mutually exclusive because if, for example, Ca) and Cc)
both hold, then Cx,y) E r
z
, contradicting the maximality of m.
x Cor y) need not be unique.
such
Let
XO Crespo yO) be the largest
x Crespo y) in case Cb) Crespo Cc))
Crespo y) in case Ca) Crespo Cd)).
Define
and the smallest such
x
l
mCi + ) in cases Ca) and
Cb) Crespo Cc) and Cd)) by replacing, in X-Cm Ci )) Crespo Y-Cm Ci ))) ,
x.Cm Ci )) by XO Crespo Y.Cm Ci )) by yO).
J
J
0
We must show that the procedure described in 3.2.7 eventually
produces an element of
M'.
To do this, define the function
g on M
35
by
M
gem)
=
L Ix. (m)-y. (m) I
. 1
1
1=
We claim that (unless
m
(i)
E
1
M')
g (m (i)) > g ((i+l))
m
it will follow that since the function
the algorithm will continue until
g
g
is bounded below by
0,
can no longer be decreased.
Our claim is an immediate consequence of the following result.
Lemma 3.2.8:
(i)
In case (a) above, let
l
~
0
be the integer such
that
Xl (m
Then
(i)
) <... <x. (m
J
(" 1)
m 1+
(i)
) <... <x. l (m
J+
consists of the pairs
(i)
)
x
<
(Xl (m
<
(i)
x j + l + l (m
), Yl (m
(i)
(i)
)< ... <~(m
(i)
)
)), ... ,
(i)
(i)
(i)
(i)
(x·l(m
),y.l(m
)) and if l> 0, (x. l(m
),y.(m
)), ... ,
JJJ+
J
(i)
(i)
(")
(x. l (m
), y. l 1 (m
)), and (for l ~ 0), (XO, y. l (m 1 )) and
J+
J+
J+ (i)
(i)
(i)
(i)
(xj+l+l(m
)'Yj+l+l(m
)), ... ,(xM(m
),yM(m
))
(ii)
In case (b) above,
m(i+l)
is identical to
m(i)
except that
the arc (x.(m(i)),y.(m(i))) is replaced by (xO,y.(m(i))).
J
J
J
The cases (c) and (d) interchange the role of X and
Y but are
otherwise identical.
Proof:
(ii)
x
E
(i) is obvious when one follows through Definition 3.2.7.
Because
X-(m(i))
j
is the least integer such that there is an element
such that
y. (m(i)) < x < x.(m(i)) , we cannot have
J
x < x. l(m(i)) < x.(m(i)) , because then
JJ
The result is then obvious.
J
y. (m(i))
J-l
< x < x.
J -1
(m (i) )
o
36
Let
Proposition 3.2.9:
mal element of
Proof:
y. (m)
~
1
c
y. (m )
1
Proof:
since
Let
M' (X,Y,r)
be the lexicographically miniz
E
m'
M'
E
be any other element.
Then
The result will be an immediate corollary of the following.
Lemma 3.2.10:
and
M'.
mO
~
Let
m'
E
M'
and
y. (m'), i==l, ... ,M.
M such that x.1 (m)
mE
c
x. (m ) ~ x. (m')
Then
1
1
~ x. (m')
1
and
1
y. (m' ), i= 1, ... , M.
1
It suffices to prove the conclusion for
m(2) == (m (1)) (1), etc.
m
(1)
instead of
We prove this result for
c
m
m(1) in the
cases (a) and (b) above, the cases (c) and (d) following by interchanging the roles of
X and
Suppose (a) obtains.
x. (m') < y. (m')
J
J
B)
x. (m') > y. (m')
J
J
Let
the smallest element of
Z+l elements of
J+
m' EM' (X, Y, r ),
if
z
7
l(m')
~
and hence of
then
x
between
x.(m) and
J
X in the half-open interval
J+ L-
in the interval
Z be as in 3.2.8(a).
X-em)
x. l(m), ... ,x. 7(m), and
p+L-+
We distinguish two further cases:
A)
Suppose (A) holds.
X
Y.
xO.
J
7 l(m)
<
J
is
there are
(x.(m),xO], namely
J
~
y. (m)
J
>
must be elements of
y. (m'), and because
J
(x.(ml),xO].
Of course, if
J
x.(m')
J
X
X+(m'). But because
Z elements of X+ (m')
XO , we have at most
X in the interval
x. (m) < x. (m' ) .
J
J
XO
y.(m),
XO
x. (m') < XO , it follows that all elements of
J
(x.(m'),xO]
p+L-+
Since
Since
~
Hence, if
XO , then
x. (m')
J
x. (m) < x. (m').
J
J
37
In either case, then,
and
x. 1 (m)
J+
x ° s; x. Z(m') .
x. (m' ) , ... , x. Z(m)
J
J+
s;
x. Z 1 (m') ,
J+ -
s;
Therefore, in making the replacements of arcs
J+
described in 3.2.8(i), we have
x. (m I
J
) , ••• ,
x. Z(m (l))
J+
s; x.
J+
Z(m') ,
and
Suppose (B) holds.
Since
XO
<
y.(m)
J
s;
y.(m')
J
<
x.(m') , we
J
again have
x.(rn(l))
J
s;
x.(m'), ... ,x Zemel))
J
j+
s;
x. Z(rn').
J+
Suppose that case (b) obtains.
Only one arc, (x.,y.) is changed
J J
(1 )
in going from m to m . But x. (m(l)) s; x. (m) and y.(m(l)) =
J
J
J
y. (m) , so x.(m(l)) s; x. (m') and y . (m (l)) = y. (m' ) .
0
J
J
J
J
J
c
mO = m , where
Proposition 3.2.11:
m is the minimal matching in
M(X,Y,f z )'
Proof:
The matching
(X,Y).
o
Follows immediately from 3.2.10.
mO will be our canonical matching for the partition
For each (X,Y), the sets
uniquely determined.
X+(mO), Y+(mO), X-(mO), Y-(mO)
The choice of
are
mO as canonical matching establishes
a certain pattern in the arrangement of the elements of these sets.
example, let
X+
+ y+
{S,6,7,8}
z = 2
and consider
+
Y
and OO,1l,12,13L
would lie
max(x. (mo),y. (mo))].
1
of
X or
The elements of
are arranged in two segments of consecutive elements, namely
tain as many elements of
x
----++++-++++---
xxxxxxyyyyxyxyyy
For
1
In general, such segments must each conas of
+
Y , since otherwise an element of
in one of the intervals [min(x. (mo),y. (mo)),
1
1
The segments are separated by one or more elements
Y , but clearly an element of
X
cannot be adjacent to an
38
element of
Y-.
Other remarks can be made, but we wish to proceed more
formally and collect the results in Proposition 3.2.14.
Definition 3.2.12:
the function
For each partition (X,Y) of {1,2, ... ,n+n'} define
f = f(X,Y)
on {O,l, ... ,n+n'} by
f(O) = 0
f(j) = #(Xn{1,2, ... ,j}) - #(YnU,2, ... ,j}),
I
Let
be the interval of integers
Z+Z'
Z'
Y,
{1,2, ... ,n+n'}, Z of
0 s jo s n+n' - 1, Z
where
for j=1,2, ... ,n+n'.
[jO+l, jO+Z+Z'] consisting of
consecutive integers from the set
of
and
~
0
and
Z'
~
O.
X and
I will be
said to be non-switching if either
f(j)
fU O)
0,
for all
j
E:
[jO+l, jO+Z +Z' ]
f(j)
fU O) s 0,
for all
j
E:
[jO+l, jO+Z + Z']
The interval
I
~
will be said to be strictZy non-switching if the strict
o
[jO+l, jO+Z +Z').
inequality holds for all
j
Examples:
switching
E:
xxyyyx
non-switching (not strict)
non-switching (strict)
Note that if
elements of
jO+2Z
E:
or if
[jO+l,jO+2Z]
X and
xxyyxy
xxxyyy.
is a non-switching interval containing
Z elements of Y, then jO+l
E:
Z
X if and only if
Y.
Definition 3.2.13:
The elements
Let
(X,Y)
1,2, ... ,n+n'
partition
{1,2, ... ,n+n'}
as before.
will be divided into the sets
Al < Bl < A2 <... <Ar < Br < Ar+l , for some value of r, where each set
+
+
A. -c X- + Y and each set B. 5:. X + Y . Let ql be the least element
1
1
of
X+ (mO) + Y+ em ° ).
Let
B
l
be the largest non-switching interval of
39
which has
as its left-hand endpoint.
be the first element (if one exists) of
X+(mO) + Y+(mO)
greater than the right-hand endpoint of
B
l
non-switching interval of
Let
hand endpoint.
Let
Al
q.
X+(mO) + Y+(mO)
B.1
and
1
A
r+l
B be the largest
2
which has
q2
as its left-
and let
be defined as the interval [l,ql)
the right-hand endpoint of
which is
be defined in this manner until the
i=2,3, ... ,r, be the interval of elements of
and let
and let
Let
X- + Y
A., for
1
which lie between
and the left-hand endpoint of
B.1- 1
be the interval of elements to the right of
Al ,A 2 , ... ,A r +l
or all of the sets
B •
r
B. ,
1
Any
o
may be empty.
Proposition 3.2.14:
(a)
For i=l, ... ,r,
X as of
Y.
For i=2, ... ,r,
(b)
qi-l
(c)
Either
(d)
If
B.1 consists of the same number of elements of
if
,
A. = ¢
1
then
q.
1
E
X if and only if
Y.
E
A.1
~
X- (mO)
A.1 i ¢ , then
A.1
or
q.
1
E
~
Y (mO)
for i =1, 2, . . . ,r+ 1.
X if and only if
A.1
c
-
X,
for i=1,2,oo.,r.
Proof:
B_
i l
(a) and (c) are as noted above.
(b) holds because otherwise
would not be the largest non-switching interval starting at
qi-l'
For (d) we need a preliminary Lemma.
Let the interval
Defini tion 3.2.15:
l elements of
matchable
X and
under
r
z
l
of
Y.
i f the graph
I
I = [jO+l,jO+2l]
consist of
will be said to be completely
(InX, InY,r )
z
has zero deficiency.
0
40
Lemma 3.2.16:
Let
be a strictly non-switching interval which is
I
r .
completely matchable under
jO+l
InY.
~
for i=2, ... ,l,
la.-b.
1
Hence, since
and
I
I
~ z,
<
b.1- 1
1
a.1
Let
f
be as in 3.2.9.
be
Then for i=2, ... , l, a.1
f(b i _ ) - f(jO)
l
0
X.
E
InX
be the elements of
the elements of
z
[jO+2, jO+21-l] is completely
Assume for definiteness that
Let
r .
Then
r z.
matchable under
Proof:
z
<
b.1- 1; otherwise
would not be strictly non-switching.
since
<
But
is completely matchable under
I
b.,(a.,b.
1)
1
1
1-
E
r.
Z
The matching
o
Continuation of 3.2.14:
Assume that (d) does not hold for some
and for definiteness that
q.1
E X
and A. c Y.
1 -
strictly non-switching interval starting at
We can form a matching
q.
1
smaller than
m
E
J
Then
be the largest
J S B..
1
mO
except that
A., the last element of J
is
1
unmatched, and the remaining elements of
Then
1
m which is the same as
is matched to the last element of
as in Lemma 3.2.16.
q .•
Let
i
M' (X,Y,r),
z
J
are matched to each other
but
m is lexicographically
mO , contradicting the definition of m.
°
o
(As an illustration of this last point, take r=2, i=l and z-3, and
---++++++
--++++-++
consider yyyxxyyyx.
The matching shown is not mO because yyyxxyyyx
is a lexicographically smaller compressed matching.)
41
3.3 Definition of a mapping a: X ~ X.
Let
X
n
=
{(X, Y) : (X, Y)
is a partition of
{1,2, .•. ,2n} such that
00
# (X) = # (Y) = n L
= (X,Y)
x
E
B (X,Y,r)
r
z
X
n
Then
and let
is the set
will be that element of
B
r
X = U Xn
n=O
j
B
r
Xo =
where
defined in Definition 3.2.13.
X . which is the same as (X,Y)
n-J
----++++--++++++--
to have been obtained by removing
B
a(X,Y)
a(X,Y)
from (X, Y).
r
will be said to have been obtained by
a(X,Y)
a(x)
xxxxxxyyyyyxyy.)
convenient to adopt some suggestive terminology.
r
except that
underlined, then
r
In describing the relationship of (X,Y) to
of B ) into
a(x)
(In our schematic notation
B
with
xxxxxxyyyyyx~yy
Then
Ar+l have been "pushed for-
ward" and "relabeled" in the obvious manner.
will be the element
Let
= #(XnBr(X,Y,r z)) = #(YnBr(X,Y,r z)) , where
has been "removed" and the elements of
if z = 2 and x is
{~}.
i~serting
B
r
r
z
will be said
Conversely, (X,Y)
(or really a copy
q (X,Y,r ) - 1.
after the integer
it will be
For purposes
of proof we shall wish to consider in the same sense "removals" and
"insertions" of
arbitrary numbers of
X-type and
Y-type elements, not
just completely matchable intervals.
We now return to Proposition 3.1.2.
cisely of every element (X' ,Y') of
a
-1
(X,Y) will consist pre-
X which we can get by inserting an
interval into (X,Y) such that the inserted interval coincides exactly
with
B (X' ,Y' ,r).
r
the form
element of
Our interest will be in sets of elements of X of
z
-1
-1
a (X,Y)nK (v), in particular in how the deficiency of each
a-I(X,Y), which is given by one of the parameters of its
classification
v, relates to the deficiency of elements of
K-l(K(X,Y)), all of which have the same deficiency.
42
There is one further complication.
of going from a(XI,yl) to (XI,yl)
Consider the inverse procedure
by insertion of an interval.
Depend-
ing on the location of the insert, it may be the case that not just any
B (X I , YI) .
interval can be inserted so' as to coincide with
we may insert
(XI,yl) =
we insert
xxyy
-++++-++++--
xy
, and have
after 6 to get
a(XI,yl) = (X,y).
-++++++++-
xxxyyx~y
For example,
after 6 to get (for z = 2)
into (X,y) = xxxyyxyy
xxxyyx~yy
r
(for z
But if
, we then have
= 2)
a(XI ,V') =
xxxyyy, which is not the original element (X,Y), i.e., B (XI ,yl ,r )
r
z
does not coincide with the inserted interval.
In this example,
xxyy
is intuitively "long enough" to separate
element 6 (in X) from element 7 (in Y) so that after the insertion
these elements cannot be part of the same matched segment, while
is not long enough.
xy
However, length of the inserted interval is not a
sufficient condition for the interval to be inserted in such a location.
Definition 3.3.1:
A partition (X,y) of
{1,2, ... ,2Z}
with #(X) = #(Y)
will be said to be separating if
is completely matchable under
rz
a)
(X,y)
b)
TIle partition (XI,yl) obtained by inserting (X,Y) after 1
into xy (assuming for definiteness that lEX;
if lEY,
then insert into yx) is not completely matchable under
o
r z.
Thus for z = 2, the interval
is not completely matchable.
completely matchable.
xy
xxyy
is separating because
is not separating because
(Note that for z = 3,
-++++-
xxxyyy
xxyy
is
xxxyyy (and also xxyxyy)
43
are separating but
xyxyxy
is not, although it is of the same length.)
3.4 Classification scheme for elements of X.
Suppose that
a)
(X, Y)
b)
(X,Y,r)
z
c)
B (X, Y,r )
r
z
E
X
n
has deficiency
ends with
X or Y, s
consecutive elements of either
1,
#Ar+ l(X,Y,r)
= t.
Z
d)
Then
~
s
d,
X is divided into three sub-
K(X,Y) = u = (k,n,d,s,t), where
sets corresponding to
k = 0,1, or 2 according to which of the following
conditions is satisfied.
~
For some i*, 1
A)
A = ¢
r
B)
A
and
element of
A
r+ l i ¢ and
C)
1
A.
1
= ¢
r(X,Y,r Z ), A.*
i ¢
1
A.*
1
1 +
=
¢, ... ,
if and only if the final
A
r+ l i ¢ and
is classified as
B
r
X.
c X if and only if the final element of
A
r+1 -
Y.
for all i, 1
~
.
1
<
-
r.
k
=
0 when (D) holds or (A) and (B) hold.
k
=
1 when (A) and (C) hold.
k
= 2
when (C) holds, but (A) does not hold.
Examples:
(B
0:
k
k = 1:
k
and
Z
is classified as
D)
~
B.*(X,Y,r) is a separating interval.
= ¢ , or
r+l
i*
=
2;
r
is underlined)
Z =
3.
----++++++-++++++++---
(A) and (B) hold.
i*
yyyyxxxxxxxxxxyyy~yy
(A) and (C) hold.
i*
----++++++--++++--yyyyyyyxxxxxxyyxxxx
(C) holds,
yyyyyyyxxxxxxxyyy~xx
-++++++----++++++++---
2.
=
(A) does not.
2.
B
r
44
The remaining case, namely (A) and (D) fail to hold but (B) holds,
b~cause
is not possible
A.~ *
1-
and
c/>
A.*
1
~ +
of the nature of mO
= c/>, . . .
,Ar
i*
If
= c/>.
=
To see this, let
r, then if
is not
separating, the interval consisting of the last element of
elements of
B.*
1
able interval.
than mO.
of
If
and the first element of
Ar+ l' is a completely match-
Thus we can construct a matching of larger cardinality
i*
<
r, then the interval consisting of the last element
A.*,
all the elements of
1
B.*
1
and the first element of
is a completely matchable interval.
add the last element of A.*
1
of
A.~ * , all the
Then, using Lemma 3.2.16, we may
X+
to
+
+
and take out the last element
Y
B.*
l' matching instead the elements of
1 +
and last, to each other only.
lexicographically smaller than
B.*
l'
1 +
B.*
1 except for the first
1 +
M'
The result is a matching in
m.
°
which is
This is a contradiction and so the
remaining case cannot occur.
The value of
mO(X,Y,r)
z
if
k
is crucial in determining what can happen to
B (X,Y,r)
r
z
is removed, in particular, whether the de-
ficiency will change, as will be seen in the next section.
3.5 Recurrence equations for
Definition 3.5.1:
Let
K(u), u
DO(Z,s)
consisting of
Z elements of
an element of
X and end with
E
C.
be the set of all non-switching intervals
X and
s
Z elements of
Y, which start with
consecutive elements of
Y.
Let 0 1 CZ ,x)
be the set of members of DOH,s) which are separating and let D (Z,s) =
2
DO(Z,s)-D l (Z,s).
Let DbCZ,s) ,Di CZ;s), and D2CZ,s) be the corresponding
sets of intervals which start with an element of
As an example, let
K(X,Y)
=
(0,6,4,2,4).
z
0
Y.
----++++----
= 2 and (X,Y) = xxxxxxyyyyyy.
In this case any element of
Then
Di(3,1) may be
45
inserted immediately after the 10th position to give an element (X' ,Y')
of
-1
K (1,9,4,1,2)
such that a(X' ,Y')
(X,Y).
=
Di(3,1)
yield distinct elements (X' ,Y').
(X' , Y' )
in
= 1.
Br(X' ,Y' ,r z )
because
u
=
must be separating when
It follows that there will be a 1-1 correspondence between
-1
-1 -1
K (1,9,4,1,2)na (K (0,6,4,2,4))
where
Further, for any element
-1
-1 -1
K (1,9,4,1,2)na (K (0,6,4,2,4)), it is clear that
Br(X' ,Y' ,r z ) E Di (3,1),
K(X' ,Y')
Distinct elements of
(0,6,4,2,4)
and
v
=
and
Di(3,1), i.e., T(u,v)
(1,9,4,1,2).
=
#Di(3,1),
The general situation is
the following.
Proposition 3.5.2:
Let
(X,Y) E K-l(u), with the last element in
(for definiteness).
Let
VEe
a)
Suppose
(resp. key)
d(u)
= 2),
dey)
=
>
with
0,
n(v)
and
t(u)
= #Di(Z,s(v))
T(u,v)
>
Let Z = n(v) - n(u).
n(u).
t(v).
>
Y
Then if
key)
=
1
(resp. #D (Z,s(v))). This corres-
2
ponds to the previous example.
b)
Let
(resp. k(u)
d(u)
= 1)
= dey)
and
key)
(resp. #Db(Z,s(v))). If
T(u,v)
= #DO(Z,s, (v)).
element of
c)
Let
d(u)
k (u)
= t(v)
t(u)
(resp. key)
= t(v) = 0
For example,
= dey) = O.
= 0),
and
(X,Y)
>
=
O.
Then if
T(u,v)
= dey)
d(u)
k(u)
Then if
k(u)
(or
DO(Z,s(v))
The resulting (X' ,Y')
>
0, then
----++++----
xxxxxxyyyyyy , insert an
= key) = 0,
T(u,v).
as appropriate) after the
will have
key)
=
0, since
O.
Let
dey)
= k (v) = O.
=
d(u)
+
s(u)
In this case
and
T(u, v)
t(v)
=
=0
= #Db(Z,s(v))
(X,Y) is completely matched and we insert
Db(Z,s(v))
final position.
d)
=1
t(u)
In this case
an element of
=
0 , and
Db(Z,s(v)) after the 8th position.
#DO(Z,s(V)).
dey)
>
t(u)
+
= #D l (Z, s (v)).
s(u)
and
For example if
46
----++++----
(X,Y) = XXXXXXYYYYYY , insert a separating interval after the 6th position, i.e., immediately after the last element of
e)
of
u
Br (X,Y)nX.
If none of the above relationships holds between the parameters
and
(X' ,Y') of
Br (X' ' Y' , r z )
v, then
a
-1
(X,Y)nK
-1
(v) = ¢ , i.e., there is no element
X which is classified as
v from which we can remove
to get an element (X,Y) ln
-1
K
(u),
We postpone the
proof of 3.5.2 until after the following corollary of 3.5.2 and 3.1.2.
Proposition 3.5.3:
z
n-l
L
K(O,n,d,s,t) =
L
#OO(l,s)
=1
K(l,n-l,d,i,t)
i=l
n-l
minlz,t)
L
#01 (Z,s)
(if
d = 0)
+
L
l=l
+
n
L
K(O,n-l,d-i,i,t-i)
i=l
z
L K(O,n-l,O,i,O)
#OO(Z,s)
l=l
i=O
(The first line includes classifications such that (b) of 3.5.2 holds;
the second includes classifications such that (d) holds; and the third
and fourth such that (c) holds.)
We will write below
+
K(·,n,d,s,t) = K(O,n,d,s,t)
+
K(l,n,d,s,t)
K(2,n,d,s,t).
z
n-l
K(l,n,d,s,t) =
L
#DO(l,s)
l=l
l
l=l
K(O,n-l,d,i,t)
i=O
n-l
+
l
#01 (l,S)1
L
z
L K(·,n-l,d,i,t')
t'>t i=l
(The first line comes from classifications such that (b) holds.
second comes from those for which (a) holds.)
The
47
n-l
I
= L #02(Z,s)
K(2,n,d,s,t)
z
L K(·,n-l,d,i,t')
t'>t i=l
Z=l
o
(Corresponds to the case when (a) holds.)
Proof of 3.5.2.
(a)
For
u
and
v
related by
key)
we wish to show that for each (X,Y)
E
= 1,
d(u)
=
dey), and t(u) > t(v),
-1
K (u), there is only one way to
insert an interval into (X,Y) to get an element (X' ,Y') of
that the inserted interval coincides with
a(X' ,Y')
(X,Y)
=
(X,Y).
r
0i(Z,s(v))
such
(v)
z
t(v)
0i (l,s(v)) into
Ar+ l(X,Y,r).
z
elements of
It will be apparent from the argument that any element of
can be so inserted into (X,Y).
-1
B (X' ,Y' ,r ), i.e., such that
This is to insert an element of
immediately in front of the last
K
0i (Z,s(v))
Necessarily for distinct elements of
inserted at the same location into (X,Y), the resulting
elements (X' ,Y') are distinct.
These statements hold independently of
the choice of (X,Y) for any (X,Y)
K-l(u), given the choice of
E
will demonstrate a 1-1 correspondence beween a
-1
(X,Y)nK
-1
(v)
v.
This
and
°iCZ,s(v)).
Given the values
t(u)
and
t(v), provided that the inserted element
does actually correspond to B (X,Y,r ), it must have been inserted after
r
z
the first t(u) - t(v) elements of
0i (Z,s(v));
is in
if it is in
02(Z,s(v))
Ar+ l(X,Y,r)
z
and it must be in
00(l,s(v)), we would violate 3.2.l4(b).
If it
we would violate (A) above, since by the next Lemma
the inserted elements are in this case matched only to each other by
(and hence
Lemma 3.5.4:
I
=
(i)
[jO+l, jO+2Z]
Let
Ar (X' ' Y' ' rz )
(X, Y)
is nonempty.)
partition
{l, 2, ... ,n+n' }.
be a non-switching interval containing
Let
l
elements of
48
Z of Y, and starting with an element of Y (we could equally
X and
well choose
mO
X).
to elements of
to elements of
to
If no elements of
I
=
Xn[jO+2Z+l,n+n ' ]
InY, then no elements of
Yn[l,jO]'
InX
are matched by
mO
are matched by
(For our purposes, we will apply the Lemma
Br (X I ' yl , r z ) - - the inserted interval. )
(ii)
If no elements of
Yn[l,jO]
InX,
then no elements of
InY
mO
are matched by
m°
are matched by
to elements of
to elements of
Yn[jO+2Z+l,n+n ' ] .
Proof:
X
j
jO+1
Let
be the first element of
x.
J
is matched to
Y+(m O ) ,
E
to an element
ment using
since
X'
J
X' I
J
jo
I
of
or some earlier element
mO
X+
to
to elements of
of
is compressed.
such that
InX
in place of
switching, each element of
matches
y+
InX
Xn[jO+2Z+I,n+n ' ]
InX.
x'
J
Then
of
Y, then we must have
But then
I
>
x ..
If
jO+l
must be matched
Repeating this argu-
J
x., it will follow that since
I
J
InY 'must be in
Y+(m O ) .
in order, the elements of
greater than
are mapped to
matched to a proper subset of
InX,
X.
J
,
InY
is nonmO
But because
can only be matched
since by hypothesis no elements
InY.
This implies that
which is impossible.
InY
is
This contra-
diction establishes (i).
(ii)
If the interval
I
is non-switching as defined above, as we run
through the elements starting at the left-hand endpoint, then it is also
"non-switching" as we run through the elements in reverse order, starting
with the right-hand endpoint.
in the argument take
direction
(ii)
is thus the same as (i), except that
the elements in reverse order, i.e, going in the
n+n l , ... ,2,1.
o
49
It follows that in case (a) the inserted elements can be matched
B (X', YI , r ).
only to each other and therefore will comprise
there are no elements of
(b)
DZ(l,s(v))
z
Indeed,
X after the inserted inverval, since it is
This is true for intervals
inserted in the middle of Ar+ l(X,Y,r).
z
in
r
as well as
Di(l,s(v)), hence the result when
= 2.
k(v)
We need a preliminary Lemma.
Lemma 3.5.5:
X and
Let
n' of
(X,Y) partition
Y.
Suppose
formed by inserting
Proof:
n+n'
E
{1,2, ... ,n+n'}
Y-(mO(X,Y,r z )).
Let
It suffices to prove the result for j = 1;
N(X' ,Y',r ) = N(X,Y,r ) , then
z
z
by Proposition 3.2.9.
easy to see that
n
elements of
(X',Y')
Y-type elements at the end of (X,Y).
j
elements one at a time, using the result for j
If
into
So as sume
N(X' ,Y' ,r z )
~
for j
>
be
Then
1
insert the
= 1.
mO(X,Y,r) = mO(X' Y' ,r )
z '
z
N(X' , Y' ,r ) = N(X, Y,r ) + 1. (It is
z
z
N(X,Y,r z ) + 1, since only one Y-type
element has been added to (X,Y).)
If
mO(X,Y,r )
z
and hence since
~
mO(X' , Y' ' z
r ) , then
m'
defined as
matching in
X, and hence since
mO - {(x*,n+n'+l)}
(but of course not in
does not include
x*
minimal matching
m in
initial vertex.
Y+ (mO(X' ,Y' ,r ))
z
E
mO EM', (x*, n+n'+l) E mO(X' , Y' 'r)
z'
is the n-th and final element of
The matching
n+n'+l
Then
as an initial vertex.
M(x,Y,r z )
where
X = X', of X'.
is then a maximal
M(X ' , Y' ,r )),
z
+
and
m'
By Proposition 3.2.5, the
also does not include
[x*+l,n+n'] £ Y (m)
x*
since
x*
as an
Ix* - (n+n') I
~
z,
50
and if one of those elements were in
x* EX-(m)
n+n'
E
Y-(m), it could be matched to
to increase the cardinality of
In particular
Y+ (m).
c
By Proposition 3.2.11,
in the interval
c
by another element of
and hence
Y
m(n+n')
m with terminal vertex
(see Definition 3.2.7),
m
m
M and so there are no elements of
m(n+n'),n+n'l (where
initial vertex of the arc in
constructing
and therefore since
= m
is lexicographically minimal in
Y-(m)
m.
n+n'
n+n'
E
denotes the
n+n'), then in
will never be replaced
Y+ (m 0 ).
This contradicts our
o
hypothesis.
As in (a), if the inserted elements do indeed form
Br (X' ' Y' , r z )
we can conclude that the insertion must be made at the location claimed,
B (X, Y, r ).
i.e., right after
r
It will thus suffice to show that the
z
elements of an interval inserted at this location are mapped only to
m0 (X' , Y' , r
each other by
z
).
Consider now the elements of
Bi*(X,y,r z )·
(X,Y)
up through and including
Suppose for definiteness that
A*
i
~
X, so that B * begins
i
with an element of
X and ends with an element of
the element after
B.*(X,Y,r)
1
z will be an element of
first element of
B.*
l(x,y,r Z)
1 +
interval, depending on whether
of
Y'
of
A.1 *
Y.
Then in (X' ,Y',r ),
z
Y'
or the first element of the inserted
i * = r).
By 3.5.4 (ii), if this element
were matched to an earlier element of
X'
by
would have to be also matched to an element of
the assumption that
type element of
B.*
1
is separating.
or subsequent to
B.*
1
any element to the left of
(either the
B.*
1
Hence by
m0 , some element
violating
B.*,
1
3.5.4 (i) no
can be matched by
But by 3.5.5, since
mO
Y' to
51
XnB.*
no element of
1
can be matched by
mO
to any elements to the left
B.*,B.*
1, ... ,B r (X,Y,r)
11+
z and the
inserted interval are matched only to each other, this being the minimal
of
Hence the elements of
B.*.
1
compressed matching for the whole right-hand portion of the partition
(X,Y) starting with
The existence of a 1-1 correspondence between
and either
DO(l,s(v))
k(v) = 1)
(if
or
a
-1
DO(Z"s(v))
(X,Y)nK -1 (v)
(if
k(v) = 0)
follows.
(c)
In this case, both
(X,Y)
(X' ,Y')
and
are completely matched, so
any completely matchable nonswitching interval starting with the right
type of element can be inserted at the end of (X,Y).
(d)
In the previous cases, the insertion of an interval has left the rest
of the matching unchanged.
In case (d) the situation is more complicated.
We first note that there are four possibilities for (X' ,Y'):
(i)
Ar (X' ' Y' , r)
z
both
~
and
X' or both
(ii)
A.1
= ~
(iii)
Ar
= ~ ,
r+ 1 (X' , Y' , rz )
are of the same type, i.e.,
A
~
Y' , or
Ar -I-
~
and
Ar+ 1 = ~.
for i = l, ... ,r+l.
but some earlier set
A.1 -I-
~.
Ar+l
is arbitrary in
this case.
(iv)
and
A
r+l
are of opposite type, and both are nonempty.
Case (ii) corresponds to (c) above.
and case (iii) corresponds to (b).
how the parameters of
K(X~Y')
Case (i) corresponds to (a)
Only in case (iv) have we not seen
and K(X,Y)
are related.
(Note:
will have been proved when we show that (iv) corresponds to (d)).
(e)
In case (iv), when
of
A + (X' ,Y' ,r z )
r l
will be the last
B (X' Y'
r
'
52
is removed, some of the elements
r )
'z
will be matched to elements of
elements of
s(u)
B (X,Y,r ).
r
Hence the inserted
z
B (X' Y' r )
r
' 'z
interval which is assumed to become
Ar(X' ,Y' ,r z ) , and
must be inserted in
the location described in (d), i.e., just before the last
of
B
r
(X,Y,r ).
z
because
B
are of opposite type.
s(u).
elements
The inserted interval must be separating if it is to
r (X',Y',r),
z
coincide with
s(u)
Further, we must have
((iv) implies that
A (X' Y' r )
r
' 'z
and
A l(X',Y',r)
r+
z
k(u) = 0
and
t(v) = t(u)
k(v) = 0, by 3.2.14 (d).)
It remains to be shown that
d(v) = d(u) + s(u)
and that when a
separating interval starting with the appropriate type element is inserted in the location described, the inserted elements are matched to
each other.
The following example shows what might happen when such an
insert is made.
z
Let
= 3.
++++--++++++--
(X,Y) = xxyyxxxxyxyyyy
++++---++-++++++----
(X' ,Y') = xxyyxxxxyxXXXyyyyyyy
suppose for definiteness that
X , so that
of
(1)
A + (X,Y,r )
z
r l
Let
I
denote the interval of elements
Since the inserted interval, which we will call
J.
YoJ
can be matched by
mO(X' , Y'r
, z)
We argue
is separating,
to an element earlier
0 .
Consequently, as in 3.5.4(i), we can see that no element of
can be matched to an element earlier than
(3)
J,
Br oX.
This follows from 3.5.4 (ii), as shown in (b) above, since
XA
(X' Y' r ) =
n r+l
' 'z
(2)
Y.
starts with an element of
Br (X,Y,r z ) up to and including the last element of
no element of
than
~
r (X,Y,r)
z
B
If
A (X,Y,r )
r
z
loY
I.
1 0, then it follows by 3.5.5 that elements of
+
53
B (X,Y,r)
r
z
of J , the
elements of
and later are matched only to each other.
#B (X,Y,r )/2 - s(u)
r
z
elements of
YnI
are mapped only to each other by
XnI
mO(X' , Y' , r z )
hence the deficiency of
Thus the elements
and the #B (X,Y,r )/2
r
z
mO(X' ,Y',r ), and
z
is increased by
(the part of the matching earlier than
unchanged.)
over that
s (u)
B
remains
r
mO is the minimal compressed matching, it is clear
Since
XnI
that the last element of
E
X-(mO(X' ,Y',r )), and that the elements
z
of the inserted interval are matched only to each other.
(4 )
If
Ar (X,Y,r Z ) =
then the sets
8,*,8,*
1'" .,B r
1
1 +
But in this case as in (b) above, since
to each other.
ting, no elements of
elements of
0 ,
B,*
1
1
is separamO to
or later elements can be matched by
B.*
1
or earlier, and hence the elements of
A. *
are adjacent
B,*,B.*
1,···Br- 1
1
1 +
are matched only to each other, giving the same conclusion as in (3).
o
3.6 Reducing the recurrence equations.
The following result places limits on the range of the indices of
summation in the equations of Proposition 3.5.2.
Proposition 3.6.1:
o (X, Y, r Z)
Then
Suppose
Proof:
Let
(X,Y) partition {1,2, ... ,2n},
i. e. , N(X,Y,r )
: : ; n - z,
Z
N(X,Y,r )
Z
<
z.
~
Then
z, 1 : : ;
Z ::::;
where
from being matched to each other.
z
). < z,
# ( I ) < 2 z.
2.
must contain at
Using
X
and
3.2.16, we can, as before,
show that there must be a non-switching separating interval, I.
N(X , Y, r
~
n.
least one separating interval in order to keep the elements of
Y-
n
Since
54
However, a non-switching interval
l
elements of
Y cannot be separating if
the elements of
l < z.
InX
z.
in front of
I
~
(in
l + 1
r z)
element.
i - 2
l
Hence
a
b ..
So
(a. 1 b.)E
r Z.
1-
1
'1
of
l
elements of
l < z.
and those of
elements of
to the right of
l - i
~
1
be al< ... <a
Suppose for defini teness that
there are at least
least
I
To see this, let
InY be
2
to the left of
a.
and at
Ia.1- 1- b.1 I
Hence
I
1
~
i
~
l,
~ 2l - (l- i) - (i - 2) -
If we insert an
X-type element
, for
l
i,
and a Y-type element at the end of
to the new
b l <·· .<b
For each
l < bl ·
I
X and
and
X-type element
I, then we can match
to the new
Y-type
o
is not separating.
We can reduce the number of variables appearing in 3.5.2 by collecting terms in the following manner.
Define
Vl(n,d,t) =
I
z
I
K(",n,d,i,t')
t'>t i=l
z
L K(O,n,d,i,t)
i=l
z
L K(l,n,d,i,t)
i=l
min(z,t)
V (n,d,t) =
4
L K(O,n,d-i,i,t-i)
i=l
The equations of 3.5.2 then become:
n-l
K(O,n,d,s,t) =
L #D O(l,s)V 3 (n-l,d,t)
l=l
n-l
+
L #D l (l,s)V 4 (n-l,d,t)
l:l
n
+
(if
d=O)
I
l=O
#D (l,s)V (n-l,O,O)
o
2
55
n-l
I
K(l,n,d,s,t) =
#Oo(l,s)Vz(n-l,d,t)
Z=l
n-l
L #01 (l,s)V l (n-l,d,t)
+
l=l
n-l
I
K(Z,n,d,s,t) =
l= 1
Our initial conditions are
case
n =
V (0,0,0)
4
t
#Oz(l,s)V (n-l,d,t)
l
VZ(O,O,O) = 1, corresponding to the
o,
(0,0,0) = 0, V3 (0,0,0) = 0,
°= 0.and If(X,Y)V. =(p, d, t)andare VIknown
for i=1,Z,3,4 and all
1
and all p
~
puted for all
n-l, for some
n , then once
k,d,s, and t, we can compute
and all values of
d
and
t.
and
K(k,n,d,s,t) have been comV.(n,d,t)
1
Thus in evaluating
tions, it will never be necessary to
d
for
i=1,Z,3,4
the recurrence equa-
store the values of
K(n,k,d,s,t).
This reduction from four indices to three makes evaluation by electronic
computer feasible for values of n
#0. (l,s)
values of
1
up to several hundred, once the
are known.
3.7 Computation of #D;(l,s).
Using an argument based on 3.1.Z we will be able to find
i=O,l,Z,
for values of
up to 12.
The limit of
facilities.
l
#0. U,s),
1
up to several hundred and for values of
z = 12
z
is established by computer storage
The method will suffice to construct tables of our distri-
bution function up to about n = 70, after which we cannot compute the
upper tail.
After Sections 3.1 - 3.6, we have remaining only the problem of
enumerating for a given value of
switching partitions (X,Y) of
z, the completely matchable non-
{l,Z, ... ,2l}
such that
#(X) = #(Y)
and
S6
such that the interval is separating, and also those such that the
interval is non-separating.
cult to handle directly.
and
#D (l,s), where
3
The "separating" question will be diffi-
#DO(l,s)
However, we can instead enumerate
D (l,s)
3
DO(l,s)
is defined as the subset of
consisting of strictly non-switching partitions.
#DZ(l,s)
Proposition 3.7.1:
Proof:
For
(X,Y)
{1,2, ... ,2(l+1)}
ning of
let
t(X,Y)
formed by inserting an
t(X,Y)
D3 (l+1,s+1), with
t-
(X' ,Y')
be the partition of
X-type element at the begin-
Y-type element at the end.
mapping onto a subset
elements of
#D (l+l,s+1).
3
DZ(l,s),
E
(X,Y) and a
non-separating,
~
D3 (l+1,s+1).
E
of
l
E
t
D3 (l+l,s+1).
Then, since (X,Y) is
is clearly an injective
In fact,
t
is onto all of
being defined by removing the first and last
D (l+1,s+1).
3
So defined,
t-l(X' ,Y')
is com-
pletely matchable, by 3.2.16, and is clearly non-switching (though not
necessarily strictly non-switching). t-l(X' ,Y') is not separating since
(X' ,Y')
is completely matchable.
I f we can find
we find
#D U,s)
Z
~
#DoCl,s)
Hence
-1
(X' ,Y')
E
#D 3 (l,s) for all
and
#D 3 (l+1,s+1)
t
and
#Dl(l,s)
~
D (l,s).
2
land
0
s , then
#DO(l,s) - #DZ(l,s)
Let us first address the problem of finding for a given value of
#DO(l,s)
for all
land
s.
We use the ideas of 3.l.Z.
Let
Yl
be the class of all non-switching completely matchable partitions of
00
{l,Z, ... ,Zl}
such that
Define the mapping
#(X)
~
a: Y + Y -
l
l l
#(Y)
(for
and
1
l~l)
E
X.
Let
by letting
Y = U Yl
l=O
a(X' ,Y')
be
z,
57
the element of
Xl
element of
a(y?)
=
obtained by removing from
YZ- l
and the last element of
Let
(X,Y)
Y ,
namely
E
YZ
example, if
z
=
4
r2
= IX Z_S +2
and
(X, Y)
= 4, r 2 = 4, r 3 = 3.)
integers.
- YZ
-s+ 2 1 ,
Let
u
(X,Y)
...
= Ix-Yzi.
,rs
Z
=
r
l
=
For
5, s
= 3,
= (Z,s,r l ,··· ,r s )'
(Z,s,r , ... ,r )
l
s
=
by
r l ,··· ,r s
XXYXXYXYYY , then
Then let
Y as follows.
be a vector of positive
The following conditions are necessary and sufficient for
C.
E
(1)
Z :2 s,
(2)
r.::; z, for
(3)
Z :2 r
Proof:
l
and
successive elements of
s
Define
YZ-s+l'YZ-s+2""'YZ
Proposition 3.7.2:
r
{0}
=
YO
for elements of
C
Suppose (X, Y) ends with
IX Z_S +l - Yz--s+ll,
U
We let
Y' .
the last
0 .
Define the classification set
rl
(X I , Y' )
S
:0;
1
l
z.
1
i
s-l, and
:0;
s.
rs
:2 r :2·· .:2r ·
2
s
The necessity of (1) and (2) are obvious as is the necessity of
since between
:2 r :2 ... :2r '
s
2
s-l elements of
rl - rs
# (Y)
:0;
X and
elements of
r -r
1
s
and
elements of
Y not including
Y.
YZ-s+l
there are
So we have located
YZ-s+l""'Y'
Hence
rl:o;
= l.
To show sufficiency, we describe an algorithm by which an element
(X,Y)
E
YZ can be constructed with u as its classification vector.
Start with an
X-type element, followed by
in front of these elements an
Y-type elements.
s
Y-type elements.
X-type element followed by
In front of these elements, insert an
Insert
r _ - r
s l
s
X-type element
58
followed by
insert an
r _ - r _ Y-type elements, ... , in front of these elements
s 1
s 2
X-type element followed by r 1 - r
Y-type elements. We now
2
have assembled
r
1
-
s
Y-type elements and
X-type elements.
s
X-type elements in front of these.
ment of
Y
at the beginning.
Z-r 1
The
{l,2, ... ,2Z} is clearly non-switching
If
Z > r1
re~ulting
clearly the initial element of
(to itself);
s
the first
and has classification vector
Y
x
Z-s+l
and
u.
This follows from
is completely matchable
Z-r 1
X-type elements inserted, one at a time, by
the algorithm can be matched to the right-most
since
insert an ele-
partition of
We need only show that it is completely matchable.
(2) :
'
Insert
s
are matched, then the
YZ-s+l
Y-type elements;
r -s
1
elements of
inserted at the same time (which will be immediately in front of
can be matched respectively to
x
X
Z-s+l
)
o
YZ-r +l' ••• 'YZ-s.
1
For
-1
K
(X' ,Y')
EO
Y, u,v
C,
EO
we have
(v), we may obtain every element of
element after one of the last
Y-type element at the end.
s(u)
#a(X' ,Y') = 1.
Let
1
elements of
ficient conditions for
Y and inserting a
It will be clear that for each
u, v
EO
C.
(X,Y) in
a- (X,Y) by inserting an X-type
T(u,v) = #a-l(X,Y)nK-l(v) = 0 or 1, depending on
Proposition 3.7.3:
For
v
EO
C,
u.
The following are necessary and suf-
T(u,v) = 1.
(1)
Z(v) = Z(u) + 1
(2)
s(v)
~
(3)
For
i=1,2, ... ,s(v)-1, we have
s(u) + 1
r s ( V ) -1+
. 1 (u) + 1- r S ( V) -1. (v) .
59
(1) and (2) are clearly necessary (consider the definition of
Proof:
a).
Note that in performing the insertion described to get an element
(X' ,Y')
a
E
-1
(X,Y)nK
-1
for each of the elements
(v),
of (X,Y), we have inserted exactly one
ment and the corresponding element
-1
X-type element between that ele-
x , x _ ' ... ,
l 1
or
x
of
l-s(v)+l
X.
(u), we can make the appropriate insertion to ob-
Then given (X,Y)
E K
tain an element
(X',Y')
a
E
-1
(X,Y)nK
-1
(v), if and only if
(1),(2) and
o
(3) are satisfied.
Proposition 3.7.4:
~
l
Let
Then
z.
Let
# (U) =
for some
min (l ~z) -sO
L
j=l
z
Proof:
#(U) =
l
j=s
#{u
J
Let
(*)
C: s (u) = s
E
l
r 1 (u) = j}.
we wish to evaluate
j,
r 1 (u)
0'
=
Let
j} .
j1 = j-r 2 ,
sO-l
i=l
and
o
Thus for given
U. = {u
C: s(u) = So
E
=
u
=
#(U.) , where
J
(l,sO,j,r 2 , .. • ,r
r s _1-50.
so
)'
Then
o
and
j. = j-s o
1
It is clear that, conversely, for any integers
jl"" ,js -1
o
which
satisfy (*), the vector
s -1
sO-l
l
i=2
by Proposition 3.7.2.
jl'· ··,js -1
o
o
j.,sO +
1
It follows that
satisfying (*).
example, [13]).
l
i=3
j., ... ,so +
1
#(U.)
is the number of sequences
j +SO-2J
This number is (
.
' (see, for
J
J
60
Z < z, then we need only consider j
If
So
such that
~
j
Z.
~
o
Using the notation of 3.1.2,
Once the values of
K(u) for all u
E
T(u,v)
e
compute the values
Let
U E
e
es
e, let
v, we can find
We will not actually
T(u,v), but will use them implicitly as follows.
es ,
w = (r , ... ,r )
s
l
For two elements
s(u) = s.
we say that
w < w'
j(u)
(ri, ... ,r~).
denote the rank of
by three parameters,
j*(j(v),sOs(v))
(l(u),s(u),j (u)).
esO
be the rank in
u = U(v)-l,sO,rl' ... ,rs ).
o
is at most one such vector; if
u
Let
E
e
3.7.3
in this
can now be described
and let
VEe
of that vector
By
For each
es (u)
in
A classification
and
when the vector
is lexicographically smaller than
lexicographic ordering.
that
and
K(K-l(~)) = 1.
by starting with
of
U E
u
be the set of all possible values of (rl(u), ... ,rs(u)) where
such that
(rl, ... ,r s )
are known for all
(rl, ..• ,r
So )
such
it can be seen that there
T(u,v) = 0, there will be no such vector.
Then
z
L
K(l(v),s(v),j(v)) =
K(l(v)-l,s,j*(j(v),s,s(v))),
s=s(v)-l
with the convention that
The array
1
~
j'
~ #
(U ,)
s
K(O,O,O) = 1
j*(j' ,s,s')
for
(U, as in 3.7.4)
s
1
~
s'
K(Z,s,O) = 0 otherwise.
and
~
z
and
s'-l
~
s
~
z
and
was determined using 3.7.3 essent-
ially by brute force, though systematically.
The pattern established
by the lexicographic ordering permits this.
The practical limitation
61
on the value of
z
which can be handled is due to the size of this
array.
To evaluate
#D 3 (l,s), we use the same procedure except that in
3.7.2 we have the additional condition that
this,consider the sequence
(X,Y)
s > 1
2
a(X,Y), a (X,Y), ... ,a
if
l
>
1.
l-l (X,Y),0
To see
If
is strictly non-switching, then no element of this sequence ex-
cept
al-l(x,Y)
may be a partition which ends with .... xy, i.e., such
that
s(u) = 1.
It can then be seen that
sufficient condition for (X,Y)
E
The problem of enumerating
if
is also a
l > 1
0 .
3
DO(l,s)
of enumerating the set of matrices of
certain conditions.
s > 1
can be rephrased as a problem
D's
and l's
which satisfy
However, this equivalent problem does not produce
a more satisfactory solution, and we will simply give the statement of
this problem.
To enumerate the set of
lxl matrices M such that:
consists only of O's and l's;
(1)
M
(2)
The non-zero elements in each row are consecutive, with the
last 1 being on the main diagonal;
(3)
For i=2, .. . ,l, the first non-zero element in row
i
never
appears in an earlier column than the first element of row
i-I;
(4)
The sum of the elements in each row is less than or equal to
z.
Equivalently we may replace (4) by:
(5)
M is symmetric;
(6)
For i=l, ... ,l, the total number of non-zero elements in the
i-th row and i-th column (no element being counted twice) is
less than or equal to
z.
CHAPTER 4
Asymptotic Distribution of the Test Statistics
4.1
Asymptotic distribution for dp '
We use results from Sections 1.3 and 1.4 to get some preliminary
lemmas.
Let
F be defined as in 1.3.9.
[b. ,c.]
then the intervals
intervals of
F.
1
will be referred to as the "component"
1
For each
F E F and
If
F E
F, we are interested in the total number
of left-hand endpoints, excluding 1, of component intervals, plus the
number of right-hand endpoints, excluding
q (F)
lim
]J
°
°
denotes uniform measure on [0,1].
]J
1.3.9.
Specifically we define
(F -F)
0+0
where
0.
F(20) be defined as in
Let
F(20)= {F E F: q(F) = q(Fo])}.
Note that
Define the function fon
D[O,l] by
1
fey) = ~~¥ q(F)+l
r
J
dy(t) ,
for all yED[O,l].
F
k
(Note: for
F
=
U [b. ,c.], we define
. 1
1=
For each
° > 0,
1
1
f dy(t)
F
define the function
k
=
I or (c.)
. 1
1=
1
- y (b. )) . )
fee) on D[O,l] by
1
63
f(O) (y)
that
1, and let X => X.
=
F
Let {a } be a sequence of numbers such
n
n
Then. f(an ) (X )=> f(X).
n
By Proposition 1.2.4, since
y E C[O,l], Y
that for
.
n
an 4- 0.
Proof:
sup
q(F)+l
FEF(20)
Let X and X be probability measures on qO,l] with
Lemma 4.1.1:
X(C[O,l])
f dy(t)
1
=
n
~
It
X(C[O,l])
f(CXn)(y) ~ fey).
0[0,1] implies that
Y in
n
(a )
fey) ~ limsup f
We first show that
1, it suffices to show
=
n (yn) , i.e., that given s > 0,
n -)o<xl
for
n
sufficiently large,
f
sup
1
dy(t)
q(F)+l
FEF
F
~
q(F~+l f dyn(t)·
sup
(a )
FEF n
This will be done by showing that for
+ s
..
F
n
sufficiently large depending
(a )
only on
y, then for each
and on
F E F, there exists a set
G EF
n
such that
q(F~+l f dy(t)
dy (t) +
n
E:
•
F
G will be constructed from
F
as in Proposition 1.3.10.
I.e.,
k
if
F
=
U [b.,c.]
. 1
1
1=
as above, then
1
with every interval
[O,b ]
l
if
b
l
~
an
[c.1 ,b.1+ 1]
and with
G is defined as the union of
such that
[c ,1]
m
if
b.1+ 1 - c.1
c
m
~
1 - a
~
n
2a
n
F
and with
•
Then clearly
G E F(2an ).
Remark 1:
q(F)
~
q(G).
(Clearly
each component interval of
val of
F
GE F
so
q(G) is defined.)
G contains a point of some component inter-
and hence must contain the whole component interval.
component interval of
F
Indeed,
Each
intersects precisely one component interval of
64
G.
If a component interval of
which has an endpoint at
G contains a component interval of
0 (resp. 1), then that component interval of
G has an endpoint at 0 (resp. 1).
component intervals of
Thus the endpoints in (0,1) of
G can be placed in 1-1 correspondence with the
endpoints in (0,1) of a subset of the component intervals of
course, a component interval of
vals of
F.
(Of
G may contain several component inter-
F.)
Remark 2:
< 2a.
n
F
G- F
consists of at most
q(F) intervals each of length
(These will be referred to as component intervals although in
general they are not closed intervals and
G - F i F.)
Indeed, it is
evident that for each component interval of G - F, at least one of its
endpoints must also be the endpoint of a component interval of
that endpoint is 0 (resp. 1), then we must have
b
l
= cl =
=
l
c
l
=
If
0 (resp.
1), in which case the endpoint is counted once, as a right-
hand (resp. left-hand) endpoint.
intervals of
Choose
G- F
n
Therefore the number of component
is less than or equal to
q(F).
sufficiently large that
sup
t ,t E:[0,1]
l
b
F.
Iy(t l ) - y(t 2 )1 < ~
and
sup
IYn(t) - y(t)1 <
O~t:::;l
2
I.
It l - t 2 1<2an
This is possible because
y
E
C[O,l] implies that
uous and further that convergence to
vergence in the uniform metric.
y
in D[O,l]
y
is uniformly continis equivalent to con-
65
q(F~+l I I riYn (t)
f dy(t) I + q(F~+l I J dy(t)
-
G
G
J dy(t) I
-
G
.
F
By remark 2,
Iy ( t 1 )
s up
2 q (F)
t
l
,t
d
2
- Y( t 2)
O,l]
I :; 2 q (F) 1
.
..
It -t !<2an
l
!
I!
d Y ! :;; q(G)
dYn
2
sup Iy (t) - y(t)1
O:;;t:;;l n
E
q(G)%:;; q(F)-3
<
•
Therefore,
Iq(F~+l J dy(t)
-
1
q (F)+ 1
F
IG dYn(t) I
:;; E
•
Hence,
1
(*)
q(F)+l
f dy(t)
1
<
- q(F)+l
F
J
dYn (t) +
E
•
f
y (t)
G
fG Yn (t)
We have two cases:
(i)
case (ii), replace
G by
~ 0
and
(ii)
G
n
< O.
In
(2a )
G' =
Ill.
Then
G' E F
n
(ii) and (*)
imply that
1
r
q(F)+l J
F
dy(t)
:;; E
=
1
q(G')+l
f
dYn (t) +
E.
G'
In case (i), (*) and Remark 2 imply that
,
In either case we thus have the necessary inequality, showing that
(an)
fey) :;; limsup f
n-?<lO
(y ) .
n
We complete the proof by showing that
f
·
1Imsup
n-?<lO
(a )
n (Yn) :;; fey)
.
66
As above, given
F
E
€
and choosing
n
sufficiently large, then for any
F,
I I dYn(t) - I dy(t) I ~ q(F)€
F
,
F
so
J
dy (t)
n
~
J
dy(t) + q(F)€ .
F
F
Consequently,
1
dY n (t)
q(F)+l
(2a )
F
FEF
n
sup
I
<
;~
I
q(F;+l
~
1
dy(t) + €
q(F)+l
(2a )
F
FEF
n
sup
dYCt)
I
+
fey) + E •
E "
(a )
Since
€
is arbitrary, limsup f
n-l-<iO
n (Yn) ~ fey).
o
Recall from Proposition 1.3.9 that
Pr{
sup
FEF(2c)
+ c;
Letting
].1
denote uniform measure on [0,1], let
the empirical measures of the sets
ri,···,r m and
sl, .. ·,sn
1
N
at each of the points
Proposition 4.1.2:
1
and
and
\In
be
{s. IN} , where
1
are the ranks among the pooled observations
of two i.i.d. random samples from
m
n
In other words,
as -:-;i.!N m
+:-:'\I •
n
N
{r. IN}
].1m
].1, and
N
=
m+n.
Let
be defined
is the measure which places mass
1 IN, 2 IN , • • • , 1.
67
Proof:
Let
c
be a continuity point of the distribution of
f(WO).
cn -~ }
~
for all
1,
~(Fcn-?]
For
= q(F)cn-~,
_ F)
since for such a set
there will be no overlap at the ends of the component interval.
F
Hence
the last expression is equal to
1:
Pr{n2(~
sup
Pr{
=
n
(F) -
~
~
for all FE F (2cn
(F) ) ::; (q(F)+I)c;
I
q(F)+1
Jdxn (t)
s c}
= Pr{f(cn
-~
-~
)}
)(X) s c}
n
F
FEF (2cn- )
1/
where
X (t) = n'2 (~ [0, t] - II [0, t]). By 1. 4. 2,
4.1. I
cn - 2)
f(
(X )
n
and hence by
n
1:
n
~.=:>
f(WO).
Hence
lim Pr{n ~dp(lln,ll) s c}
n-7<X>
lim PrU(cn
n-7<X>
-~
) (X ) s c}
n
o
Pr{f (WO) ::; c}
-~
Proposition 4.1. 3:
Proof:
constants
dp(llm'Y )
N
=">
f(WO)
o
We treat separately the cases (i)
a
and
(m~)
to show that
sequence
(;)
(m. ,n.)
1
1
b; (ii)
-~
~
n
dp(llm'Y )
N
-+ 0;
=")0
(iii)
f(Wo)
~
n
-+
as
m,n -+
<
a s -m s b
n
<
00,
for some
This will suffice
00
as m,n -+
00
00;
indeed, any sub-
must contain a sub-subsequence satisfying (i), (ii),
or (iii), so weak convergence would hold for that sub-subsequence.
68
(i)
Let
bution of
n)
0 (m,n) = (=u
1111'1
f(WO).
h2.
Let
for all
1
n(F) satisfies - N
1
- ~(I)I ~ N.
Thus
+
for all FEF(2co(m,n))} ,
~n(F)
n(F)
Multiplying through by
<!
- N
Pr{
sup
FEF(2co (m,n))
since for any interval
'
= O(~)
o(m,n)
-1
q(F~+l J dym,n (t) ~
F
, uniformly in
I~[O,l],
F.
, we get
q(F) (im)t O(N -h-2);
r
=
FEF(2co(m,n))}
Pr{~N(~ m(F) - v n (F)) ~ q(F) (co (m,n) + n(F))
+ co(m,n);
where
be a continuity point of the distri-
By 1.3.9,
+ co(m,n);
=
c
for all
c +
FEF(2co(m,n))}
(;)to(N- t );
for all FEF(2Co(m,n))}
is defined in 1.4.2. By 1.4.2, as m,n +
in such a way that
Y
m,n
(i) is satisfied, Y
~ WO , so by 4.1.1, f(co(m,n))(y
) ~ f(WO).
m,n
m,n
where
Since we have assumed
00
0
<
m
a ~ n'
(m)hi 20(N -h-2)= O(N -h-2). Hence since c
is a continuity point of the distribution of
f(WO) ,
69
(ii)
In this case we consider
::; cm
-,~
}
- lJ (F))
+
-h:
cm 2;
-h:
Pr{lJm(F) - lJ(F) ::; (yN(F
=
cm
+
cm 2]
-h:
)) + lJ(F
cm 2]
h:
m
+
c; for all FEF
(2
cm
-J:?)
},
I ::; n (F) <
- N
- !N '
-J:?
)
m
c
s
F
is as defined in 4.1.2.
X
Hence if
q(~)+l I dXm(t)
sup
FE F(2cm
Since
m
-
n
-+
is a continuity point of the distribution of
f(WO), then
c} .
= lim
m,n~
(iii)
-F)
-J:?
J:?
Pr{
where
-h:
) -lJ (F
Pr{m (lJ (F) - lJ(F)) E q(F) (c+m2n(F))
=
h
were
cm 2]
Let
o(m,n)
=
h:
n 2 /m.
+
co(m,n)
for all
FEF(2co(m,n))}
=
70
Pr {(F)
].lm
- YN (F)
<_
(YN(F c8 (m,n)]) _ ].l(Fco(m,n)]))
+ (].l(F) - YN(F)) + ].l(Fco(m,n)]_F) + co(m,n)
for all
=
pr{~(YN(F) - vn(F)) ~ q(F)n(F) + (].l(F) - YN(F)) + c8(m,n)(q(F)+I);
for all
h
were
FEF(2c8(m,n))}
FEF(2co(m,n))},
I ~ n (F) -<_1
- N
N'
+ co(m,n)(q(F)+I); for all FEF (2co (m 'n))
}
=
k
Pr{n 2 (].l(F) - vn(F))
~
n
q(F)n(F)(2+ ffi)o
-1
(m,n) + c(q(F)+I)
for all FEF(2co(m,n))}
=
Pr{
sup
(2c8 (m,n))
q(F~+1 J dXn(t) ~
n(F) (2+ ;)o(m,n)-l + c},
F
FEF
n(F)(2 + ~)o(m,n)-l
m
= O(n~/N) = 0(1).
Hence in the case
~
=
lim Pr{dp(].lm'Y N) ~ c ~ }
m,n+co
Proposition 4.1.4:
Proof:
(..lir~dp(].l
,v )
mn
m n
=>
=
n
Pr{f(WO) ~ c} .
f(WO)
We treat separately the cases (i)
m
_+00
as m,n
o< a
+
00.
~ ~ ~ b <
n
o
00
and
71
TIl
(i i)
-
ro 1e s
0
-+
n
f
Let
m
TIl
The case
0.
an J
o(m,n)
-
-+
n
n
m
i. e. ,
00
-- -+
n.
=
(~)-1:
Pr{dp(~ ,v ) ~ co(m,n)}
(i)
0, follows due to the symmetric
=
m n
Pr{~ (F) ~ v (Fco(m,n)])
m
n
+ co(m,n); for all
Pr{~ (F) TIl
=
v
n
FEF(2co(m,n))}
(F) ~ (v (Fco(m,n)]_F) _ ~(Fco(m,n)LF))
n
+
~(F
co (m n)]
'
-F) + co(m,n);
for all
FEF(2co(TIl,n))}
Pr{o(m,n)-l(~ (F) - v (F)) ~ o(m,n)-l(v (Fco(m,n)]_ F)
m
-
n
~(F
n
co (m n)]
'
-F)) + c(q(F)+l);
for all FEF(2co(m,n))}
Pr{
q(F~+l
sup
FE F(2co (m,n))
(Ym,n(F) - Zm,n(F))
~
c}
where
YTIl,n (F)
and
o(m,n)
-1
(~m(F)
-
vn(F))
Z
(F) = o(m,n)-l(vn(Fco(m,n)]_F) _ ~(Fco(m,n)]_F))
m,n
Y (t)
m,n
and
=
Y
m,n
~
=
WO
o(m,n)-l(~m[O,t] - vn [O,t])
as m,n
f(co(m,n)) ~/ f(WO).
showing that
-+
00
where
a
<
a
is a random process on 0[0,1]
~
m
-
n
~
b
<
00
,
by 1.4.2.
Hence
We can thus complete the proof for case (i) by
sup
(F~ + 1 Zm,n (F) ~ h(WO) , where
q
FEF(2co ( m,n ) )
hey)
=0
72
for
y E 0[0,1].
For each
y E 0[0,1], let
I
(_Nr~
hm, n(y)
sup
(q(F)+l)-l
dy
CO
F Er=C2co (m,n))
F (m,n)]_F
mn
It suffices to show that
Since
I (.Ji.)
-~ (v n -ll) I ~ (l+b) In~ (v n -ll) I,
mn
and
h:-
n 2 (v -ll)
~ WO
n
by 1.2.5, it suffices (as in Lemma 4.1.1) to show
such that
that f0r any y E C[O,l] and {Yn} in 0[0,1]
J
1
q(F)+l
dYn
+
y
n
+
y,
0, for each F E F(2co(m,n)).
FCO (m,nt F
Let
m and
n
be sufficiently large that
sup
O~t~l
Iy
n
[0, t] - y [0 , t]
I
£:
< -
3
and
E:
sup
IY(s) - y(t)1 < '3 .
Is-t I<co (m,n)
For any
FEF
(2co (m,n))
f
1:(
Cu
F
f
1:()]
Cu
F
m,n
dyn -
-F
m,n
dy
)]
~
q(F)
-F
.
f
sup
ly(s)-y(t)1 < q(F)t '
Is-t!<co(m,n)
dy < 2q(F) sup
Fco(m,n)]_F
as in the proof of 4.1.1.
Iy [O,t] - y[O,t]1 < 2q(F)
O~t~l n
Hence
1
q(F)+l
<
£:
•
~,
73
-~
1.
Pr{dF,(Jl m,v n )
(i i)
-"2
} = Pr{Jl (F)
m
~
cm
+
cm -~ , for all FE F (2cm
~
Pr{ Jl (F) - Jl (F)
m
+ cm
(v
-~
n
(F
cm
~
v (F
n
cm "]
-~
)
)}
-~
])
; for all
Pr{X (F) - Z
(F) ~ (q(F)+l)c;
m
m,n
for all FEF(2cm
-~
)}
where
and
X (F)
m
Z
m,n
(F)
We wish to show that
converges in probability to
But
0.
q (F) sup (v [0, t] O~t~l n
for any set
-~
FEF (2cm)
~
sup O<:tl~···~tp~l
°,t])
,
So
(~)~ (n ~ sup (v [O,t] - Jl[O,t])) ,
O~t~l
n
which converges in probability to
Proposition 4.1.5:
]J [
For p
=
1,2, ...
n
0, since
define
~ -+ 0.
n
o
T (WO) as
p
!-W°(t )-WO(t )+ ... +(-l)P+lWo(t ) I. For
l
2
p
c
>
0,
74
{f(WO) ~ c}
Proof:
~
T
p-l
that
~
3c, ... ,T
p
~
~
3c, ... ,T
=
{Tl(WO) ~ 2c, T (WO) ~ 3c, ... } ~
2
p-
P
~
(p+l)c} .
Tl (WO) ~ 2c,
Tl ~ 2c, T2 ~ 3c, ... ,
We wish to show that this implies
(p+l)c}.
Assume that
1
>
p, T.
~
1
(i+1)c
for i
=
By the triangle inequality,
J-P
pc, T (Wo)
pc, T ~ pc. Then T ~ (p+l)c.
p
P
T. ~ (i+l)c for all i; assume for purposes of induction that for
some j
T.
~
l(WO)
~
(j-p+l)c,
it follows that
1,2, ... ,j-l.
T.
T + T.
~
J
T.
J
P
~
J-P
(j+l)c.
Since
T
P
~
pc
and
o
Pr{T (WO) ~ 2c, T (WO)
l
2
~ 2c} = Pr{T (WO) ~ 2c} , and
Pr{T (WO) ~ 2c, T (WO) ~ 3c} directly
l
2
2
O
to give rough upper and lower bounds on Pr{f(W ) ~ c}. The joint disUsing Proposition 1.4.3, we can evaluate
tribution of
(T ,T ,T ) seems intractable.
l 2 3
4.2 Asymptotic distribution for dL .
Proposition 4.2.1:
a)
~
n dL(~n'~) ~
sup I~O(T)I
O:;;t~l
-~
b)
(m~) dL(lJm,Y N)
-~
c)
Proof:
(;)
dL (~m' \in)
sup
~
I~Wo (t) I
O~t~l
sup I~O(t)1
~
O~t~l
We shall prove only (a), since the proofs are completely anal-
ogous to 4.1.3, 4.1.4, and 4.1.5, respectively.
a)
Pr{dL(~n'~) ~
cn
-h:
2}
=
7S
+ en
for all
Pr{-2cn -Jz
0n (x)
~
~(x)
.-
[a,l]}
x
E:
~
Zcn
-Jz ;
Zc}
Pr{ sup
-.~;
-+
a<;t~l
for all x
€
[O,l]}
I
Pr{ sup WO (t)
Osts1
I
<;
Zc} ,
o
by 1. 4.2.
4.3 Absolute continuity of f(WO).
Definition 4.3.1:
U.= u(k)
1
i
For
k >- 1
For
and i=l, ... ,k, define
inf
WO (t)
i-1<t< i
k - - k
=
p s k, define
max (max
U.
- V.
+ U. - ... + ( -1)
. <k 1 1
1Z
1
1 <'
3
-1 <. " <1 1
p+1
U.
1
p
max
1~i1<" .<ip~k
1
+.. . + (- 1)
U.
V.
1
1
p+l
) i f pis odd,
V.
1
Z
p
p
and
U.
max (max
<' <" .<l. -<k 1 1
1 -1
1
p
V.
+ U.
1
1
max
V.
. < ••• <1. -<k 1 1
1 ~11
U.
.
Remark 4.3. Z:
apparent that
P
1
2
- ... +(-1) p+1 U.
1p
3
+ ... +(-1)
p+1
2
For a continuous sample path
WO (t),
) if P is even.
U.
1
p
tEe
o
[a, 1],
may be defined in the same way as
it is
76
t l ,t , ... ,t (in the defini2
p
the restriction that no two of the points
tion) are in the interval
Definition 4.3.3:
For
[ik'
j+l]
k
' f or any
c > 0, let
A(k,c)
o
j.
denote the event that
o
max
l:o;i:o;k
We will establish the absolute continuity (with respect to Lebesgue
measure) of
f(WO)
by showing the following.
is absolutely continuous for 1
(1)
(II) For p
~
T(k) (Wo)
P
As
(III)
k +
(Note:
c
if
2
T (Wo) - T 2(WO)
p
p-
= Tp(WO)
, where
TO
>
c
= O.
Pr{T (WO) - T 2(WO)
pP
00, Pr(A(k,c)) + 1.
A(k,c) occurs then
k
Ti )
c}
>
+
p
is
k
T .
l
=
For fixed
1.
T(k) (Wo) =?T (Wo)
particular that
ability arbitrarily close to 1, for
and
k.
:0;
Of course,
+ 0,
It is not hard to show that
P
:0;
p
arbit~arily
c, as
as k
+
00, in
small with prob-
sufficiently large.
However,
even when combined with (I), this weak convergence does not imply the
absolute continuity of the distribution of
Tp(WO).)
(II) and (III) together imply
(IV)
Given
E >
(Indeed, choose
0 , for
c
Proposition 4.3.4:
k
sufficiently large,
first, then choose
We need the following lemmas:
=
T } > 1 P
E.
k.)
(I) and (IV) together imply that
continuous.
Proof:
Pr{T(k)
P
f(WO)
is absolutely
77
Lemma 4.3.5:
Let
Pr{X
X be a random variable on some probabi Ii ty space.
X ,X , ... be absolutely continuous random variables such that
1 2
f X}
k
Proof:
k
~
Let
for all
for
Let
k.
0
as
~
k
Then
00.
X is absolutely continuous.
A have Lebesgue measure zero.
Choose
>
E:
o.
sufficiently large.
Then
Pr{X
PriX E A}
:0;
Since
is arbitrary,
E:
k
A}
E:
=
0,
Pr{XkE A} + Pr{Xk f X} <
E:,
o.
PriX E A}
0
T (WO) is absolutely
It follows that if (I) and (IV) hold, then
continuous for each
Lemma 4.3.6:
Let
p
p.
X ,X , •..
l 2
probability space and
each rcmdom variable
be a sequence of random variables on some
N a positive integer-valued random variable.
Xi
is absolutely continuous, then
X is
N
absolutely continuous.
Proof:
For any Lebesgue-measurable set
A,
00
Pr{X
But if
N
E A}
L PriXn
E A,N=n}
n=l
A has Lebesgue measure
Lemma 4.3.7:
0,
PriX
a. s.
n
E A, N=n}
as p
o for
-+
00
•
absolutely continuous, assuming (I) and (IV) are true.
(n+l)-lT (Wo)
n
for some
n.
o
Before proving 4.3.7, we show that it will imply that
4.3.7, with probability one,
each
f(WO) is
Indeed, by
f(WO) = sup (p+l)-IT (WO) =
l:O;p<oo
p
n, depending on Woo
If
78
Let
N = N(WO)
be the least such value of n (define
N in any manner
on the probabi Ii ty zero set where there may not be such a value of
Then by 4.3.5 and 4.3.6,
uous.
f(WO) = (N+l)-lTN(WO)
n).
is absolutely contin-
We complete the proof of 4.3.4 by proving 4.3.7.
Proof of 4.3.7:
Since
Mk(WO)
=
sup
IWO(s)-WO(t)l, 0 ::; s, t ::; 1.
O::;ls-tl::;l/k
is almost surely a uniformly continuous function on [0,1],
WO(t)
Let
Let
0::; t
l
< t <... <t ::; 1.
2
n
The
n
points
tl, ... ,t
n
can be
partitioned into pairs and singletons such that for each pair,
(t"1
t.1+ 1)' both points
[1k
j+l)
' k
k
singletons.
t,1
and
It easily follows that for any
n::;
Mk(WO) ~ 0
n
~
k
T + n - k M (W 0)
k
2
k
(n+l)-lTn ~ ~
n+l T1 (WO) +
finite and
are in one of the intervals
k-l
[-r-,
1], and such that there are at most
'-0 , ... , k - 2 , or
' J-
T
Hence
t,1+ 1
1
-4
Mk(WO).
Since
Tl
(Wo) is
a s.
'
a.s., it follows that (n+l)-lTn(WO) ~ 0 a.s. as
D
We will prove (II) first and then (I) and (III).
If
Proposdtion 4.3.8 :
T (WO) - T 2(WO) >
p
p-
C
and
A(k,c) occurs, then
a.s.
Proof;
Since
assume that
WO(t)
is a.s. a uniformly continuous function on [0,1],
WO(t) is uniformly continuous.
such that
Then we may choose points
79
Since by hypothesis,
t
P
:0;
p-
T (WO), it follows that
p
sion for
for 1
p
i
2(WO) > c, we must have
From the triangle inequality applied to the
< ... <t .
1
T (WO) - T
p-l.
:0;
Thus if
T (WO)
p
expres-
T
(WO) + \WO(t.)-WO(t. 1)
p-2
1
1+
:0;
T (WO) - T
(WO) > c, then
p
p-2
c.
>
If in addition
and
are
i
A(k,c) occurs, this implies
j -1 < t
-
k
i
<
t i+l ~< ik
Then by
o
remark 4.3.2,
Proposition 4.3.9:
Proof:
is absolutely continuous for each p
P
:0;
k.
itself absolutely continuous, it suffices to fix the values
.
.<l
p
+(_l)p+lv.
1
:0;
k
and (if
p
V. - U. + v.
12
13
+
u. - ...
11
12
13
(It will be seen that the same
p
11
u. - v.
is even) show that
is absolutely continuous.
proof applies for
p
T(k) (WO)
Since the maximum of finitely many absolutely continuous random
variables is
1 -< 1. <.,
1
- ... +( -1)
p+l
U. , and for both cases when
I
p
is odd.)
i <... <i :0; k be fixed for the remainder of the proof.
1
P
We wish to consider the conditional joint distribution of
i -1
i
° i 2 -l
° i2
given WO (-f--), WO ( ~), W (-k-)' W (lZ),···,
Let
1
:0;
i.
The values
Let
il:=
.-1- and
k
i. 1-1
J+
k
l
be the points in order of the set
II
r
i.
i.-l
--L= l or ----1- = l, j=l, ... ,p} , where p
k
k
might not be distinct.
:0;
r
:0;
2p.
Y(WO) = (WO(ll), ... ,WO(lr)) has a (non-singular) r-variate normal
distribution with density
set
A,
I
¢(Xl, ... ,X ).
r
For any Lebesgue-measurable
80
p+l
pdu. - v. + u. - ... +(-1)
1
=
1
1
1
2
3
J••• J Pr{U.1
V
P
E
A}
-V. +U. - ... +(-1)
1
1
1
2
p+l
P
We complete the proof by showing that if
0, Pr{U.1 -V.1 +... +(-1) p+l V.1
1
2
p
Lemma 4.3.10:
0
1
t-c
WO(c ) +
l[Wo(c )-WO(c )]
c -c
2
1
l
2 l
=
~ C
l
A
E
A has Lebesgue measure
A/W ° (ll) = bl,···,W ° U r ) = br } = 0
E
~
Then for
Let
V
3
< C
2
and let
-
t-c l
_
c2 cl
Z(t) =
t-c
WO(c 2) + c -~ WO(c l )
2
< t < c '
WO(t) - Z(t) is a normal random variable and
2
l
is independent of WOes) for each s E [O,c l ] U [c 2 ,1].
(Note:
c
this result bears out the intuitive notion that in the plane,
the point
(t,E(WO(t)/WO(cl),WO(c2)))' for
line segment joining the points
cl
<
(cl,WO(c l )) and
t
<
c 2 ' lies on the
(c ,WO(c 2 )).)
2
Proof:
For
s
E
[O,c l ]
and
t
E
(c l ,c ),
2
E WOes)
and it is easy to check (using the relationship
t l (1-t 2)
for
0
WO(t) - Z(t)
~
tl
s
t2
~
1) that
=
E WO(t )WO(t )
l
2
E WO(s) (WO(t) - Z(t))
= O.
is a linear combination of normal random variables and
therefore normal, so
larly for
~
= 0 = E[WO(t)-Z(t)],
E
[cl,l].
WO(t) - Z(t)
is independent of
WOes).
Simi-
o
It follows that given
arc mutually independent.
U.
tribution of
of
1
WO (ll) , ... , WO (l ), U. , V. , U. , ... , V.
r
1
1
1
1
1
2
3
p
If we can show that the conditional dis-
is absolutely continuous, the absolute continuity
1
T(k) (WO) will follow.
P
Pr{
Lemma 4.3.11:
sup
WO(t)
~
A/WO(c ) = b , WO(c ) = b }
l
Z
Z
l
cl~t~c2
_ZA,(A'-b)
c2
e
for
c
l
h
, were
< t < c
WO (t) - Z(t)
, WO(t)
2
c
A'
is the sum of the normal random variable
t-c
cz-t
1
b b
2
c -c
cZ-c l l
2 l
and a constant
It is easy to compute that for
E(WO(s) - Z(s))(WO(t) - Z(t)) =
c
~
l
s
~
and writing
and
s' =
a
:5
c
2
Making the change
(c -c )
2 l
o
t' = _ _
1_
c -c
2 l
~
s'
t'
:5
~
s'
~
t'
~
1.
t'
~
1 ,
U.
1
sup
1
WO (t) - Z(t)
sup
O~t'~l
cl~t~c2
lIence
sup
cl~t~c2
Hence
in (c 2 -c l ) ~ times a "Brownian Bridge" random
1
process as defined in 1.4.1.
Pr{
~
Xes') = WO((c 2 -c l )s'+c l ) - zecc2-cl)s'+cl) , we have
EX(s')X(t') = (c -c )s' (l-t') , for 0
2 l
X(t'L
~
t
(s-c l ) (c 2 -t)
t-c
of variables
.
WO(t)
~
A/Wo (e )
l
= bl ,
WO(c }
2
= b2 }
X(t' )
1 -
82
=
Pr{ sup (X(t')
+
O~t~l
Pr{ sup
(X(t') + (b 2 -b 1 )t')
O~t'~l
=
Z((c 2 -c )t'+c 1 ))
1
Pr{ sup
(cWO(t')
t'b)
T
~
+
b1
~
A/WO(c 1)
~
A}
A'
A'-b
(W(t) - - t
c
(see 1.4.1).
A'
-
c
~
O} ,
where
W is the Wiener process
By a result in [10], this last probability is equal to
o
We next prove
Proposition 4.3.12.
Proof:
A(k,c)
in two parts.
II~
For each
1
c > 0, Pr(A(k,c))
k
as
1
~
~
00
is as defined in 4.3.3.
We will consider
Z?
b2}
=
-2 -A' 0A'-b
(_)
c
c
where
=
c = (c 2 -c ) ~ , b =
1
A'} , where
O~t'~l
1 - e
b , W'(c 2 )
1
=
WO(t)
=
Wet) - tW(l), 0
sup
IwO(s) - WO(t)
def i-I
i
-k- -<s<t<- -k
~
t
~
1.
Then
I
inf
(WO(t)
i-I
i
-k- --k
<t<i-I
i-I
suP. [(W(s) -W(T)) + (s - T)W(l)]
=.
1-1 < <~
k
-s-k
i-I
info [(W(t) -W(l())
.
1-1 <t<~
k
--k
+
i-I
(t - T)W(l)] .
,
83
If
Z.
1
=
def
sup IW(s)-W(t) I , then it is clear that
i-I < <t d
k -s--k
lz.1 - Z?!
~ -k2
1
W(l) ~
P
Hence it will suffice to show that
a
k ~
as
Pr{ max
Z.
1
l~i~k
By the reflection principle,
inf
i-I
W(i~l))
(W(t) i
00
•
~ c} ~
1
as
sup (W(s) _ W(i-l))
i-I
i
k
- - <s<k --k
k
~
00
•
and
are each distributed as twice a normal random
J ( <;t~
1
k
variable with mean zero and variance
Zi
~
Then
suP. (W(s) _W(i~l))1 + 2 . inf
21.
Ik
1-1 < <~
k
i-I
. (W(t)-W(-r-))
1-1 <t<~
_s-k
I
--k
is stochastically smaller than 8 times the absolute value of a normal
random variable with mean zero and variance
1
k
Since Zl"",Zk
are mutually independent,
max
pr{
l~i~k
We claim that
Z.
1
c} = (Pr{Z.
~
1
k
k
(~(Ck2)) ~ 1,
which will complete the proof.
Indeed,
it is well known that
k
~ (ck 2)
1
> 1 -
l' e
c2k~
- -2- ( lk)
(211") 2
so
(~(ck~))k
for
k
> 1 -
sufficiently large.
k~
k e
(211") 2
Hence
2 J/ck 2
ck 2
--2-
(~(ck~))k
~
1
as k
~
00
0
84
For
Proposition 4.3.13.
as
~
2,
c 4- O.
Proof:
€
p
> 0,
A(K) = {
Let
choose
max
l::;i::;Zp
For fixed
K} •
K sufficiently large that
"2€ .
Pr(A) > 1 -
Let o ::; tl::; ... ::;t _z ::; 1 be chosen so that for s = 0 or
p
s = 1, Tp_Z(WO) =(-l)S(WO(tl)-WO(tz)+···+(-l)P-lWo(tp_z))
this
WO(t)
can be done whenever
is a continuous function of
t, i. e. ,
almost surely.
For some
. I
t i' _l ::; Jz~
i
1
and j
I
1 ::; i
I
::;
p-l
I
Zp ,
::;
j =1, ... , Zp , contains points
WO (u) - WO(v) ,. c
WO(u l
)
>
and points
c.
u
to
= 0,
t p _l
=
1.
Indeed,
t.-t.
1 ~ ~l
1
1p-
i=l, ... ,p-l,
A occurs, then for
small, with probability greater than 1 -
-
1::; j
.I
We wish to show that when
WO(v l )
and
~::; ti ' where we define
<
p ~ Z, for some
if
I,
and
u l and
€
Z
sufficiently
c
i
each interval [ j-l
Zp' Zp]
v
such that
u
VI
such that
ul
<
v
<
and
VI
and
Since such points therefore can be found in
. 1_1
~;] defined in the last paragraph, then either
zp
(_l)s[(_l)i ' (WO(u) - WO(v))] > c or (_l)s[_l)i ' (WO(u l ) - WO(v l ) ) ] > c,
the interval [J
T (WO) - T Z(WO) > c.
p
pAssume for definiteness that
so
b
l
::;
b Z'
pr{
As in 4.3.11, for
WO(j'-1)
Zp
A~ b ,
Z
sup WO (t) ::; A/W o (j'-l) = b
2p
l'
j!-l<t<i'
Zp- -Zp
-.4p(A-b Z) (A-b l )
= 1 - e
'1
=
b
l
and WO(~)
Zp
=
b Z with
85
.1
J ) , if
WO (-2
Under the assumptions
sup
., I
·r
L-<t<-L
WO(t)
>
b
2
p
+ c, then points
u
and
v , u'
and
v'
as
2p- -2p
j'
j'-l
described may be found; indeed in the interval [-zp' 2p] , the process
starts at
b , goes up to
l
b 2+c
and back down to
b .
2
Then
•r
Pr{
sup
WO(t)
b l , WO(2~) = b 2 }
=
j!...l<t<i'
2p- -2p
-4p c(c+K)
-4 P c (c + b 2 - b I )
= I
- e
under the assumption that
~
A(K) occurs.
I - e
If
small, this value may be made smaller than
c
E:
2 .
is chosen sufficiently
o
We have now proved (I), (II), and (III), showing by 4.3.4
that
f(WO)
is absolutely continuous.
CHAPTER 5
Statistical Properties
5.1
Related test statistics.
From among the metrics which generate the topology of weak conver-
gence on
peS), we have elected to examine
dp
and
d
because they
L
are the ones which have appeared most often in the literature.
alternative, we may consider, for any
P(F) s Q(F s ) + bs;
all closed
d~b)(p,Q)
b > 0,
F~S} or dib)(p,Q)
P(-oo,x-s] - bs s Q(_oo,x] S P(_oo,x+s] + bs ; for all
are topologically equivalent to
dp
same uniformities.
b > 1,
(Indeed, if
and
=
As an
inf{s>O:
=
inf{s>O:
x
E
Ek } , which
d
and have respectively the
L
d(b) (P,Q)/b s d (P,Q) s
p
p
s bdL(P,Q) ; if
b
<
1, reverse
the inequalities.)
The two-sample test (for equal sample sizes) based on
seems to have advantages over that based on
d p (11
n
,'J
n
)
d(2) (11 'J)
P tAn' n
because of dif-
ferences in the null distribution for finite sample sizes.
J l1 ,'J ) s c} is constant in the intervals
Pr{d 1:'n n
1 s i s 2n, because
11
n
(F) -
points at which the atoms of
multiples of
points
1/2n.
'J
n
(F c )
11
n and
[i/2n, (i+l)/2n),
is always a mUltiple of
'J
n
lin, and the
are located are separated by
Thus, intuitively, there is more "going on" at the
2/2n, 4/2n, ... ,1
than at the points
1/2n, 3/2n, ... ,(2n-l)/2n.
87
For
(2)
dE) (11, v )
n
divisor
n
the interval between atoms is the same as the common
of the possible values of the test statistic and so there is no
such distinction between points of the form
(2i+l)/2n.
This difference between
tables in the appendix.
dp
2i/2n
and those of the form
d~2) can be seen in the
and
The result seems to be a faster, or at least more
d~2) (l1 n ,V n ) to its asymptotic distribution.
uniform convergence of
further difference is that given our restriction on the values of
A
z
as
described in Chapter 3, we can construct more extensive tables of the
distribution of
d~2) , using the same methods as for dp-
Indeed, in
place of 3.1.1, we have
(2)
Pr{ d p
(11, v )
n n
I
A(p,n,z)
O~p~z
The rate of convergence to the asymptotic distribution for the onesample
dE)
test statistic seems roughly on the order of that of the
Kolmogorov test judging
from empirical distribution functions based on
actual randomly-generated samples.
Our exact distributions show that the
convergence for the two-sample
test is slower than for the two-sample
Kolmogorov-Smirnov test.
dp
The proof of Proposition 4.1.1
may give some
explanation for this, but we have not been able to give a precise explanation.
The Monte Carlo one-sample distribution functions indicate that the
upper bound given by
Pr{T (WO)
l
~
2c,
T2 (WO)
~
3c} is fairly close to
the true asymptotic distribution and the lower bound,
Pr{T (WO)
2
~
2c}
is not as good.
In [26], Rosenblatt suggests a test based on the metric
d 2 (P,Q) =
I
sup
an interval
Ip(I) - QCI) I , stating that it has somewhat better
88
power than
d
against certain alternatives.
K
for this test statistic, multiplied by
generalization of
d
The asymptotic distribution
n ~ ,is
As a
' we suggest:
2
the supremum being taken over all
dk(P,Q) = suplp(F) - Q(F) I,
k
FEU F. ,where F. = {F E F: q(F) = i}, q(F) being
(1)
.11
1=
1
defined as in Chapter 4, and
dp(P,Q) = inf{E > 0:
(2)
all
*
dp
F
E
d*
p
Ip(F) - Q(F)
dp
and
dp
put a "penalty" of a factor of (k+l)
F under consideration.
~
tion).
c}
~
F
0 as k+oo (because WO(t)
F;
E
indeed,
a.s. has infinite total varia-
Intuition does not indicate why the factor should be k+l.
ever, a factor of
inequality,
k=1,2, ... ,
k
as
Ip(F) - Q(F) I ~ E for all
Ip(F) - Q(F)I ~ kE
perties of
dp
and
How-
would not be appropriate, since by the triangle
It should be noted that
on peS)
q(F)) allowed
Some such penalty is necessary if
the metric is to take into account all
Pr{Tk(WO)
~(k+l)€
except that the "end effects" for each inter-
on the number of component intervals (more precisely on
for the set
I
F}
is thus the same as
val are ignored;
for k=1,2, ... ,
dL ·
dp (relative to
section, would be shared by
for all
d;
F E F implies that for
l
F
E
F .
k
does not generate the same topology
However, we would expect that the power prod ) which will be described in the next
K
d*
p
5.2 Power of the test based on dp .
We first show that the tests based on
Let
U be the uniform measure on [0,1].
dp
Let
and
d
L
are consistent.
V E P([O,l]), V
~
U, and
89
let
V
be the empirical measure of a sample of size
n
test the hypothesis that
dL(Vn,U)
a.s.
or
and
V
=U
~
dL(V,U) a.s.
Since, if
dp(Vn,U)
F
1
{X.}
1
V.
We
~
dp(V,U)
>
0
is a continuous dis-
tribution function, {F(X.)} will be a sample from
sample
from
by using one of the test statistics
dp(Vn,U). By Proposition 1.3.5,
dL(Vn,U)
n
U if and only if the
F, our one-sample test is consistent
actually comes from
against all alternatives to a continuous null distribution.
To show consistency for the two-sample test, we use the following
result related to Proposition 1.3.12.
Proposition 5.2.1:
Let
P
{1/2n, 2/2n, ... ,1} and let
Then
dp(Pn,Qn)
Proof:
put mass lin on n of the points
n
~
2 dK(Pn,~)/3
If for some
x
E
~ ([X,X+E])
:0;
on the other
n
points.
- lin.
[0,1], Ip [O,x] - ~[O,x]1
n
2
Pn ([a,x])
?:
E
Qn([a,x] ) + E - lin
P ([O,x])
?:
Q ([x,l]E) + E - lin
n
n
Indeed,
put mass lin
(2nE+l)/n,
and
~([X-E,X])
:0;
3E
>
0, then either
or
(2nE+l)/n.
Hence
o
(By using a more laborious argument along the lines of 4.1.5,
can be replaced by
dK(P
n
,Qn )/2.)
dK(P
n
,Qn )/3
The consistency of the
two-sample tests now follows from the consistency of the KolmogorovSmirnov test.
Massey [20] has given an example to show that the KolmogorovSmirnov test is not unbiased.
test based on
dp
Similarly, we can show that the one-sample
is not unbiased.
Suppose we take a sample
{x.}
1
of
90
si ze
on
n
.
lc n
v
from the probability measure
-~
a
,1- c n
a
-~
) , where
test, and which places mass
(Assume that
c
is the critical value for the size
a
c n
-~
=
X!~l-cn-~.
a
1
c
a
1
Let
U*
n
sample from
U and let
sample from
U.
~
c n -~ .
X!
1
a
~
c n
1
-1:-
a
2,
E
(c n
-1:2
a
{X!}
1
.)
We may
from
1
c n
a
X. = 1 - c n
1
a
U
n
-1:2
-1:2
)
'
if
the the empirical measure of the original
I
U,
Indeed, for any closed set
a
c n
< 1 -
a'
and
a
-~
be the empirical measure of such a trimmed
{X!} from
For any sample
X!
if
n-~ if
c n
-~
a
and 1 - c n-~.
a
by "trimming" a sample
{X. }
U[O,l], i.e., by letting
X.1
c n-~
at the points
a
is sufficiently large that
n
generate such a sample
and letting
which equals Lebesgue measure
c n
*
c n
U (F a
U (F a
n
n
-1:-2
F;::[O,l],
]
)
so
c n-~]
sup
U(F) - U (F a
n
closed F;::[O,l]
) ~
sup
cZosed F;::[O, 1]
c n-~]
U(F) - U*(F a
).
n
With positive probability,
For example, if
has about
U
n
formly distributed on the interior of
remainder evenly divided between small intervals about
dp(Un,U)
val s about
{can
-4-
is close to
° and 1)
,I-Can
-~
34
can
-~
° and 1,
then
(worst set is the union of the small inter-
4 n -~ (worst set
and cL (U *,U) is close to -=c
~
n
5 a
}). Hence the one-sample test based on dp is not unbiased.
With additional complications, a similar example can be constructed
for the two-sample case.
91
To attempt to gain some insight into the power of the
dp
test
against various kinds of alternatives we have made some empirical comparisons of power of the one- and two-sample tests based on
that of the test of the same size based on
test the
dp
d
X
'
dp
with
For the one-sample
critical value was taken from the distribution functions
which were estimated in a separate Monte Carlo simulation.
For the
sake of convenience, for the two sample tests the size was chosen so
that a non-randomized test could be used for both
d
and
X
dp
Random samples were generated from several alternatives to the null
distribution.
The results of these comparisons are given in the
following table.
•
92
One-sample tests
Description
(sample
size)
U[O,l] vs. U[O,l]
U[O,l] vs. U2
a = .05
Number
Trials
Number
Rej ected
Proportion
Rejected
d
K
dp
dK
dp
(40)
4000
189
194
.047
.048
(40)
2000
1899
1900
.950
.950
2
U[O,l] vs. 2-sided U
(40)
2000
743
1445
.371
.722
U[O,l] vs. bimodal
(40)
2000
287
358
.143
.179
U[O,l] vs. trimoda1
(40)
2000
173
195
. 086
. 097
U[O,l] vs. U[O,l]
(100)
1000
47
53
.047
.053
2
U[O,l] vs. 2-sided U
(100)
1000
923
999
.923
.999
U[O,l] vs. bimodal
(100)
1000
293
672
.293
.672
U[O,l] vs. trimoda1
(100)
1000
150
239
.150
.239
Two-sample tests
a
= .25
N(O,l) vs. N(0,1)
(30)
200
46
48
.230
.240
N(O,l) vs. N(.5,1)
(30)
1000
703
694
.703
.694
N(O,l) vs. N(.75,1)
(30)
1000
914
913
.914
.913
N(O,l) vs. N(l,l)
(30)
1000
987
983
.987
.983
N(O,l) vs. N(0,2)
(30)
1000
369
438
.369
.438
N(0,1) vs. N(0,4)
(30)
1000
693
825
.693
.825
2
U[0,1] vs. U
U[O,l] vs. U3
(30)
1000
808
798
.808
.798
(30)
1000
988
988
.988
.988
2
U[O,l] vs. 2-sided U
(30)
1000
562
657
.562
.657
Explanation:
U[O,l] denotes the uniform distribution on [0,1].
denotes the normal distribution with mean
the "Lehmann" alternative generated as
~ and variance
0
max(U , ... ,U ), where
k
1
2
•
U "",U
1
k
93
2
"2-sided U " , "bimodal" and "trimodal"
are uniform random variables.
are random variables with respectively unimodal, bimodal, and trimodal
distribution functions generated by taking appropriate mixtures of
2
random variables wi th distribution u ; speci fically if
B
l
and
B
2
arc independent Bernoulli random variables mutually independent of
V, a random variable with distribution
(l-B ) (1-V/2)
1
2
U , then
VI
is our "2-sided" random variable, and
(1-B )(.S+V /2)
2
l
is our "bimodal" random variable.
Let
random deviates were generated as follows.
0,1, and 2 with equal probability.
random numbers were used in the
required to evaluate
¢
(X. )
1
Then
V
3
one-san~le
= BlV/i
+
V2 = B2V/2
+
The"trimoda1"
B take the values
= 3B
+
V1/3.
Only uniform
tests, because the time
for the normal distribution function
¢
threatened to become excessive for the large samples and large number
of trials involved.
The "2-sided", "bimodal" and "trimodal" a1terna-
tives were suggested by the examples given below, as being potentially
favorable to
dp , because
dp
can "look at" several intervals simul-
taneously, not just at one as does
d .
2
The evidence is not sufficient
to substantiate or refute such a conjecture.
It can be seen that in general
dp
performed better than
dK ,
except in the case of alternatives which were stochastically greater
than the null distribution, in which case the differences are not significant.
Rosenblatt [26] gives examples comparing
d
2
and
d
which are
K
useful for our purposes, shedding some light on the above results.
example in which
d
2
performs better is an alternative
An
V to U[O,l]
94
which has distribution function
where the sample size
ance region for the
and for
d
2
F(x)
=
.03,
0 :0: x :0: .06
F (x)
=
x,
x
1600
and
n
=
~
.06
a. = .05.
dK test is (using [29])
is (from Rosenblatt's results)
A sample from
V will always satisfy
On the other hand, the
dK(Vn,U) > 1.36/40
d (V ,U) > 1.75/40
2 n
d (V ,U)
2 n
no observations in (0,.06), and hence the
ility 1.
In this case the accept-
d
=
.034,
=
.044.
.06, since there are
~
test rejects with probab-
2
d test has power close to .05, since
K
it can be shown (for example as Rosenblatt suggests, using a result of
Uspensky cited in [25]) that
Pr{
sup IU[O,t] - Vn [O,t]
O:o::t:O::.06
I
>.034} is
small for this value of n.
The dramatic difference between
to the proper choice of n
superior to
d
at most .7/40
~
dp(Vn,U)
.02
2
=
d2
d
in this case is due
K
and
and, of course,
V.
dp
is also similarly
in this case; the critical value for the
dp
test is
.0175, based on our Monte Carlo tables, but
(consider the set (0,.06)), so the
with probability 1.
has poor power, while
However, if
d
2
dp
test rejects
F = .0225 on [0,.045], then
rejects a. s.
dp
also
(Rosenblatt also gives an
example,
F(x)
=
x;
0:0::
=
.5 - 0; .5 - cS::; x
= x; .5
for which
<5
and
d
is superior to d 2
K
+
X <
.5 - 0
:$;
.5 + 6
0 < x :0: 1,
and also to
dp ' for proper choice of
n.
As an example of the same type in which
dp
performs better than
95
ei ther
d
or
2
d , let
K
.01 at the points
0
n
1600, a
==
and .04
.05, and let
==
and mass .02 at the point .02, and be
Then if
equal to Lebesgue measure on (.04,1].
measure of a sample of size
the set
dK(V,U)
{0,.02}), so
.02
==
==
V place mass
n
from
V,
V
is the empirical
n
~
dp(Vn,U)
.02
(consider
rejects with probability 1.
dp
However,
d (V,U), while their critical values are .034 and .044
2
respecti vely, so as in the previous examples. it can be shown that the
d
and
K
d
2
tests have low power for the values of
n
and
a
chosen.
Massey [20] gives a lower bound for the power of the KolmogorovSmirnov test against a class of alternatives to the null distribution
P
of the form
{PI E peS): dK(P,P')
c}.
==
The result provides an
alternative proof of consistency of the test.
Let
Proposition 5.3.2:
~
and
Q be elements of
be the empirical measures of samples of size
respectively.
c}
P
We give a similar result
Let
c
for some
where
ility
Proof:
Y is a
>
Pddp(Pn' P)
c
1
B (n '2)
a
} ==
If
a
Pr{dp(Qn'P) ~ c a }
then
a
c
~
~
(i. e. , binomial based on
peS)
n
and
P
P
and
from
and
n
Q
Q E {P'EP(S): dp(P,P')
1
n
- Pr{Y- -n2
:0:
-n(c-c )},
a
trials with probab-
~ of success on each trial) random variable.
Since
dp(Q,P)
Pr{dp(Q, ,P)
n
==
c, there exists a closed set
F S S
O
such that
> C }
a
1 - Pr{~ (F)
c ]
:0:
peF a ) + c
a
c ]
a
~ 1 - Pr{~ (Fa):O: P(F O )
+
cal
all closed F
c
S}
96
n~(FO)
is a
B(n,Q(F O)) random variable.
Pr{n(~(FO)
- Q(F O))
-n(c-c )} is maximized for Q(F O) = } , since the binomial random
a
variable has maximum variance in this case.
~
o
APPENDIX
A.l
Tables of the one-sample distribution function.
h:
n 2d
The distribution function of
~
n = 5,10,20,40,60 and 80, where
~n
p
(~ .~)
n
has been estimated for
is uniform measure on [0,1]
is the empirical measure of a sample of size
n
from
procedure was to generate 4000 random samples of size
for each sample the value of
d
p
(~
n
,~)
and
~.
Our
n, to compute
and record the number of
samples for which the value fell in the interval [.OOli, .001(i+l))
for i=O, ... ,199 and [.2,(0).
[.002i, .002(i+l)).)
for values of
c
(For n = 5
h:
To get the tables which give Pr{n 2 d
p
which are multiples of
early between the values estimated for
and
h:
Pr{n 2 d
p
(~ ,~)
n
~
and n = 10, we used
h:
.001(i+l)n 2 }
The random samples from
~
(~
n
,~)
~
c}
.01, we interpolated linh:
Pr{n 2 d
p
(~
n
,~) ~
~
.001ion }
for i = 0, ... ,199.
were generated using a program from
the UNC Computation Center's "Scientific Subroutine Package."
The
program uses the "congruential" method, as explained in [Maclaren
and Marsaglia, J.A.C.M.
12~
83-84].
It is noted in this article
that the random numbers produced by such a method may not be mutually
independent, and a correction suggested in the article has been used:
100 numbers generated by this method are placed in a table and labeled
00, ... ,99; two digits from an additional random number are used to
determine which location in the table to take the next "observation"
from, and the observation used is replaced by a new random number.
98
The same procedure was used to generate the U[O,l] random sample
for the power comparisons of Chapter 5.
(see Bell C.A.C.M. 11, 498 and Knop
The Box-Mueller method
C.A.C.M. 12, 281) was used to
generate the normally distributed random numbers for the power
comparisons.
Several computer programs were written to compute
the methods of Chapter 2:
d
p
using
one to use the Hungarian algorithm in the
general two-sample case; one for the case of two discrete measures
1
1
on E , and one for the one-sample test statistic on E.
grams have been checked by using them to find
The pro-
in a number of
P
cases which can be computed by hand, involving various arrangements
of the points of the "worst set."
d
The two-sample programs provide
a check on each other.
The programs for
£1
take on the order of 75 seconds to perform
4000 evaluations of
for n = 80.
The program written for
the Hungarian algorithm is considerably slower, taking on the order
of 30 seconds to perform a single evaluation for n
required increases rapidly with
n.
=
30, and the time
The Hopcroft-Karp algorithm pre-
sumably will permit the evaluation for general metric spaces for
larger values of
n.
99
One-sample distribution:
x
.21
.22
.23
.24
.25
.26
.27
.28
.29
.30
.31
.32
.33
.34
.35
.36
.37
.38
.39
.40
.41
.42
.43
.44
.45
.46
.47
.48
.49
.50
.51
.52
.53
.54
.55
.56
.57
.58
.59
.60
n=5
.000
.000
.002
.007
.012
.021
.035
.052
.070
.095
.124
.156
.194
.236
.279
.322
.358
.400
.445
.487
.527
.560
.592
.628
.655
.684
.709
. 731
.753
.774
.794
.815
.833
.848
.863
.877
.889
.901
.913
.923
Pr{1n dp(~n'~) ~ x}
10
20
40
60
80
.000
.001
.002
.005
.010
.018
.028
.038
.056
.081
.103
.131
.158
.194
.235
.282
.318
.356
.392
.440
.476
.513
.552
.592
.627
.660
.688
. 712
.738
.764
.784
.804
.828
.841
.855
.869
.881
.894
.905
.915
.000
.001
.003
.004
.008
.015
.022
.033
.052
.069
.096
.120
.145
.184
.219
.262
.301
.336
.384
.421
.457
.501
.530
.568
.604
.630
.659
.687
.712
.739
.760
.784
.801
.818
.837
.850
.863
.877
.888
.899
.000
.000
.001
.003
.006
.011
.015
.021
.031
.054
.071
.098
.119
.140
.180
.208
.253
.286
.332
.368
.416
.448
.481
.526
.555
.598
.622
.660
.685
.706
.737
.756
.776
.795
.820
.834
.855
.867
.878
.893
.000
.000
.000
.002
.003
.008
.012
.017
.026
.037
.063
.083
.103
.141
.165
.199
.236
.264
.307
.360
.392
.431
.482
.509
.561
.589
.613
.655
.676
.710
.736
.754
.776
.804
.818
.843
.854
.866
.884
.892
.000
.000
.001
.001
.002
.005
.012
.017
.029
.046
.060
.078
.109
.131
.157
.209
.240
.271
.319
.362
.392
.444
.472
.501
.541
.590
.618
.641
.678
.701
.720
.758
.770
.785
.807
.828
.842
.863
.877
.884
100
x
.61
.62
.63
.64
.65
.66
.67
.68
.69
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
.80
.81
.82
.83
.84
.85
.86
.87
.88
.89
.90
.91
.92
.93
.94
.95
n=5
.933
.941
.947
.952
.958
.962
.967
.970
.973
.974
.978
.981
.983
.986
.988
.990
.991
.992
.994
.995
.996
.996
.996
.997
.998
.999
.999
.999
.999
.999
.999
.999
.999
.999
.999
10
20
40
60
80
.922
.930
.938
.944
.951
.958
.963
.967
.970
.973
.978
.980
.983
.984
.985
.988
.990
.991
.992
.994
.995
.996
.996
.997
.997
.997
.998
.999
.999
.999
.999
.999
.999
.999
.999
.907
.915
.925
.932
.940
.950
.954
.959
.963
.968
.972
.974
.976
.978
.980
.983
.986
.988
.989
.990
.992
.993
.994
.995
.996
.996
.997
.997
.998
.999
.999
.999
.999
.999
.999
.902
.912
.919
.927
.934
.941
.950
.956
.959
.963
.968
.970
.975
.978
.982
.984
.986
.988
.988
.989
.990
.991
.992
.993
.994
.994
.995
.996
.996
.996
.996
.997
.997
.998
.998
.899
.911
.920
.928
.935
.941
.948
.955
.960
.966
.968
.971
.974
.977
.980
.982
.983
.986
.987
.989
.990
.991
.993
.993
.994
.995
.996
.997
.997
.998
.998
.998
.998
.998
.999
.894
.909
.915
.926
.936
.942
.917
.954
.960
.964
.969
.973
.975
.979
.982
.984
.985
.987
.988
.990
.992
.992
.994
.995
.996
.996
.996
.997
.997
.998
.998
.999
.999
.999
.999
101
A.2 Tables of the two-sample distribution function.
The values of
z
a
K(i,n,p,s,t)
were computed for i = 0,1,2;
0,1, ... ,12; and n = 1,2, ... ,100, for each value of
~
p
~
a
z,
s
~
min(z,p), using equation 3.5.2 with the simplifi-
~
cations described in Section 3.6.
2
z
~
i=O s=o t=O
K(i,n,p,s,t) depends on
Section 3.1,
A(p,n,z)
L L L K(i,n,p,s,t),
then computed as
which is
Pr{d
p
(~
n
,v )
n
~
z/2n}
as defined in 3.1.1 is
where as described in
z.
In the following tables we have listed
which is
d,s, and t,
(2n) -1 L\'
A (p, n, z) ,
n 0~p~z/2
( 2n) -1
n
, and also
\'
A (p ,n, z) ,
L
O~p~z
Pr{d(2)(~
,v ) ~ z/2n} , where d(2) is defined as in Section
p
n n
p
5.1.
Because of the recursiveness in equations 3.5.2, there is reason to
suspect that rounding errors will affect the results.
the computations were repeated with values of
values of
n
To check for this,
up to 50 for larger
z, using "double precision" storage in the computer memory.
Storing the K(-,-,-,-,-)
values in double precision did not affect the
result while storing the D(-,-) values in double precision made a difference in the fourth decimal place of the computed probabilities.
(There was no marked effect for smaller values of
presented here were computed with the
sion
D. (-,-}
1
z.)
The tables
values in double preci-
Storage limitations preclude the use of double precision for all
the numbers for the full range of parameters.
The
values of
D.1 (Z,s)
~,
values can be checked by hand for moderately small
sand z , sufficient to test the operation of all parts
of this segment of the program.
The probabilities in the table have
102
been checked for n
= 6,
7 and 8 for several values of
z, by generating
all permutations of the two samples and using the two-sample program for
to find
for each permutation.
An
additional check on
the correctness of the programming (although not necessarily of the
z
theor~
is that for
z
large enough
~
L
p=O
A(p,n,z)
= (2n)
n
.
Pr{dp(~
n
z=l
-
2
3
4
5
n
6
,v )
n
~
7
z/2n}
8
9
10
11
12
1 1. 0000
2 · 6667 1. 0000
3 .4000 1.0000
4 .2286 .9429 1.0000
5 .1270 .8651 .9683 1.0000
6 .0693 .7705 .9199 1.0000
7 .0373 .6719 .8631 .9953 1.0000
8 .0199 .5747 .7995 .9849 .9975 1.0000
9 .0105 .4836 .7329 .9692 .9914 1. 0000
10 .0055 .4010 .6660 .9490 .9815 .9997 1. 0000
11 .0029 .3283 .6004 .9248 .9682 .9986 .9998 1.0000
12 .0015 .2658 .5376 .8971 .9519 .9964 .9992 1.0000
13 .0008 .2131 .4783 .8666 .9330 .9929 .9979 1.0000
14 .0004 .1692 .4232 .8338 .9117 .9880 .9957 .9999 1.0000
15 · 0002 .1333 .3725 .7993 .8886 .9818 .9926 .9996 .9999 1.0000
16 .0001 .1043 .3262 .7635 .8637 .9742 .9884 .9991 .9998 1.0000
17 · 0001 .0810 .2846 .7269 .8375 .9654 :9832 .9984 .9995 1.0000
18 · 0000 .0626 .2472 .6900 .8103 .9552 .9770 .9972 .9990 1.0000
19 .0000 .0481 .2139 .6530 .7823 .9439 .9698 .9956 .9983 :9999 1.0000
20 .0000 .0367 .1845 .6162 .7537 .9315 .9616 .9936 .9972 .9998 .9999 1.0000
21 · 0000 .0279 .1585 .5800 .7248 .9181 .9526 .9911 .9959 .9996 .9999 1.0000
22 .0000 .0217 .1359 .5445 .6957 .9037 .9427 .9881 .9941 .9993 .9998 1.0000
23 .0000 .0160 .1161 .5100 .6666 .8885 .9320 .9845 .9920 .9990 .9996 1.0000
24 · 0000 .0120 .0989 .4765 .6377 .8725 .9205 .9805 .9895 .9984 .9993 1.0000
25 .0000 .0090 .0841 .4442 .6091 .8558 .9084 .9759 .9866 .9977 .9990 .9999
26 .0000 .0067 .0713 .4133 .5809 .8386 .8957 .9709 .9833 .9969 .9985 .9998
27 .0000 .0050 .0603 .3837 .5531 .8208 .8824 .9654 .9796 .9958 .9979 .9998
28 .0000 .0037 .0509 .3555 .5260 .8025 .8686 .9593 .9754 .9946 .9972 .9996
29 .0000 .0027 .0429 .3288 .4996 .7839 .8543 .9529 .9709 .9931 .9963 .9994
30 .0000 .0020 .0361 .3035 .4738 .7649 .8396 .9459 .9660 .9915 .9953 .9992
"""
0
CoN
n
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1
.0000
· 0000
· 0000
· 0000
· 0000
.0000
· 0000
· 0000
.0000
.0000
.0000
2
.0015
· 0011
.0008
· 0006
· 0004
.0003
· 0002
.0002
.0001
.0001
.0001
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
· 0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
3
.0303
.0254
· 0212
· 0177
· 0148
.0123
.0102
.0085
.0071
.0059
.0048
.0040
.0033
.0027
.0023
.0019
.0015
.0013
.0010
.0008
.0007
.0006
· 0005
.0004
.0003
.0003
.0002
· 0002
.0001
.0001
· 0001
.0001
.0001
4
.2797
.2573
.2363
.2167
.1985
.1815
.1657
.1511
.1376
.1251
.1137
.1031
.0935
.0846
.0765
.0691
.0624
.0562
.0506
.0456
. 0410
.0368
.0330
.0296
.0265
.0237
.0212
.0190
.0170
.0151
.0135
.0120
. 0107
5
.4489
.4247
.4014
.3790
.3574
.3366
.3168
.2979
.2800
.2625
.2462
.2306
.2158
.2018
.1886
.1761
.1643
.1532
.1427
.1329
.1236
.1150
.1068
.0992
.0921
.0854
.0791
.0733
.0679
.0628
. 0581
.0537
.0496
6
.7457
.7264
.7069
.6873
.6677
.6481
.6286
.6092
.5899
.5708
.5520
.5334
.5150
.4970
.4793
.4618
.4448
.4281
.4118
.3958
.3803
.3652
.3504
.3361
.3222
.3087
.2956
.2829
.2707
.2588
.2473
.2363
.2256
7
.8246
.8092
.7936
.7777
.7617
.7455
.7291
.7128
.6963
.6799
.6634
.6471
.6307
.6145
.5984
.5824
.5665
.5509
.5354
.5201
.5050
.4901
.4755
.4611
.4470
.4331
.4195
.4061
.3930
.3802
.3677
.3555
.3435
8
.9386
.9308
.9226
.9141
.9052
.8959
.8864
.8765
.8664
.8560
.8454
.8345
.8235
.8122
.8008
.7893
.7776
.7658
.7539
.7419
.7299
.7177
.7056
.6934
.6812
.6689
.6567
.6445
.6324
.6202
.6082
.5961
.5842
9
.9607
.9550
.9490
.9427
.9360
.9290
.9217
.9141
.9062
.8981
.8898
.8817
.8724
.8633
.8541
.8448
.8352
.8255
.8157
.8057
.7957
.7855
.7752
.7649
.7545
.7440
.7335
.7229
.7123
.7017
.6911
.6805
.6698
10
11
12
.9896
.9875
.9851
.9825
.9797
.9766
.9733
.9698
.9661
.9621
.9579
.9535
.9489
.9440
.9390
.9339
.9284
.9229
.9170
.9111
.9050
.8987
.8923
.8858
.8791
.8722
.8653
.8582
.8510
.8437
.8363
.8288
.8212
.9941
.9927
.9911
.9894
.9874
.9852
.9829
.9804
.9777
.9748
.9717
.9684
.9649
.9612
.9574
.9534
.9492
.9448
.9403
.9356
.9308
.9258
.9207
.9154
.9100
.9045
.8988
.8930
.8871
.8811
.8750
.8687
.8624
.9989
.9985
.9981
.9976
.9970
.9963
.9955
.9946
.9935
.9924
.9912
.9898
.9883
.9867
.9850
.9832
.9812
.9791
.9769
.9745
.9721
.9695
.9668
.9640
.9611
.9580
.9548
.9516
.9482
.9447
.9411
.9374
.9336
I-'
0
~
n
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
83
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
3
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
· 0000
.0000
.0000
.0000
· 0000
.0000
.0000
.0000
.0000
.0000
· 0000
.0000
.0000
.0000
· 0000
.0000
.0000
· 0000
· 0000
.0000
.0000
4
.0095
.0085
.0076
.0067
.0060
.0053
.0047
.0042
.0037
.0033
.0029
.0026
.0023
.0020
.0018
.0016
.0014
.0012
.0011
.0009
.0008
.0007
.0006
.0006
.0005
.0004
.0004
. 0003
.0003
.0003
.0002
.0002
.0002
.0002
.0001
.0001
.0001
5
.0458
.0423
.0390
.0360
.0332
.0306
.0281
.0259
.0238
.0219
.0202
.0185
.0170
.0156
.0143
.0132
.0121
.0111
. 0101
.0093
.0085
.0078
.0071
.0065
.0060
.0055
.0050
.0046
.0042
.0038
.0035
.0032
.0029
.0026
.0024
.0022
.0020
6
.2153
.2054
.1958
.1867
.1778
.1694
.1612
.1534
.1459
.1388
.1319
.1253
.1190
.1130
.1072
.1017
.0965
.0915
.0867
.0814
.0778
.0737
.0697
.0660
.0624
.0590
.0558
.0527
.0498
.0471
.0444
.0420
.0396
.0374
.0353
.0332
.0313
7
8
9
10
.3319
.3205
.3094
.2986
.2880
.2778
.2678
.2581
.2487
.2396
.2307
.2221
.2137
.2056
.1978
.1902
.1828
.1757
.1688
.1621
.1557
.1495
.1435
.1377
.1321
.1267
.1215
.1164
.1116
.1069
.1024
.0981
.0940
.0900
.0861
.0824
.0788
.5722
.5604
.5487
.5371
.5256
.5141
.5028
.4916
.4806
.4696
.4588
.4481
.4376
.4272
.4170
.4069
.3969
.3871
.3775
.3680
.3587
.3495
.3405
.3317
.3230
.3144
.3061
.2979
.2898
.2820
.2742
.2667
.2593
.2520
.2449
.2380
.2312
.6592
.6486
.6380
.6275
.6169
.6065
.5960
.5857
.5753
.5651
.5549
.5448
.5348
.5248
.5149
.5051
.4955
.4859
.4764
.4670
.4577
.4485
.4394
.4304
.4215
.4128
.4041
.3956
.3872
.3789
.3707
.3627
.3547
.3469
.3392
.3316
.3242
.8135
.8057
.7979
.7900
.7820
.7740
. 7660
.7577
.7495
.7413
.7331
.7248
.7165
.7081
.6998
.6914
.6831
.6747
.6663
.6579
.6496
.621.12
.6329
.6245
.6162
.6079
.5997
.5914
.5832
.5751
.5669
.5588
.5508
.5428
.5348
.5269
.5190
11
.8560
.8495
.8429
.8362
.8295
.8227
.8158
.8088
.8018
.7948
.7877
.7805
.7733
.7660
.7588
.7514
.7441
.7367
.7293
.7219
.7145
.7070
.6996
.6921
.6847
.6772
.6697
.6623
.6548
.6473
.6399
.6325
.6251
.6176
.6103
.6029
.5956
12
.9297
.9257
.9216
.9174
.9131
.9087
.9042
.8997
.8950
.8903
.8855
.8807
.8757
.8707
.8657
.8605
.8553
.8501
.8448
.8394
.8340
.8285
.8229
.8174
.8117
.8061
.8004
.7946
.7888
.7830
.7772
.7713
.7654
.7595
.7535
.7475
.7415
f-'
0
(Jl
Pr{d (2) (ll ,v. )
p
n n
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
z =1
2
3
4
5
6
$
7
Z
/2n}
8
9
10
11
1.0000
1.0000
.9000 1.0000
.7429 1.0000
.5714 . 9841 1. 0000
.4156 .9546 1.0000
.2890 .9103 .9977 1. 0000
.1939 .8546 .9918 1. 0000
.1264 .7899 .9819 .9997 1.0000
.0804 .7194 .9678 .9986 1.0000
.0501 .6462 .9493 .9965 1.0000
.0307 .5728 .9266 .9930 .9998 1.0000
.0185 .5016 .9000 .9881 .9994 1.0000
.0110 .4343 .8698 .9816 .9986 1.0000 1.0000
.0065 .3721 .8366 .9733 .9973 .9999 1.0000
.0038 .3157 .8007 .9633 .9955 .9997 1.0000
.0022 .2654 .7627 .9516 .9931 .9994 1.0000
.0013 .2212 .7232 .9381 .9900 .9990 1.0000
.0007 .1830 .6826 .9230 .9862 .9983 .9999 1.0000
.0004 .1502 .6414 .9063 .9816 .9974 .9998 1.0000
.0002 .1225 .6002 .8881 .9761 .9962 .9996 1.0000
.0001 .0993 .5593 .8684 .9699 .9947 .9994 .9999 1.0000
.0001 .0800 .5191 .8475 .9628 .9929 .9990 .9999 1.0000
.0000 .0641 .4799 .8254 .9549 .9907 .9986 .9998 1.0000
.0000 .0511 .4420 .8022 .9461 .9881 .9980 .9997 1.0000
.0000 .0405 .4057 .7782 .9365 .9851 .9972 .9996 .9999 1. 0000
.0000 .0320 .3711 .7533 .9261 .9817 .9963 .9994 .9999 1.0000
.0000 .0251 .3383 .7278 .9149 .9778 .9952 .9992 .9999 1.0000
.0000 .0197 .3074 .7019 .9029 .9735 .9940 .9989 .9998 1.0000
....0
0\
n
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
1
2
3
4
5
6
7
8
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0153
.0119
.0092
.0071
.0055
.0042
.0032
.0024
.0019
.0014
.0011
.0008
.0006
.0005
.0003
.0003
.0002
.0001
.0001
.0001
.0001
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.2784
.2514
.2264
.2034
.1822
.1627
.1450
.1289
.1144
.1012
.0894
.0788
.0693
.0608
.0533
.0466
.0407
.0355
.0308
.0268
.0232
.0201
.0174
.0150
.0130
.0112
.0096
.0082
.0071
.0061
.0052
.6756
.6490
.6224
.5958
.5693
.5431
.5172
.4917
.4667
.4423
.4185
.3954
.3730
.3513
.3305
.3104
.2912
.2728
.2553
.2385
.2226
.2075
.1932
.1797
.1669
.1549
.1436
.1329
.1230
.1136
.1049
.8902
.8768
.8627
.8480
.8328
.8170
.8006
.7838
.7666
.7491
.7312
.7131
.6948
.6763
.6770
.6390
.6203
.6016
.5829
.5643
.5459
.5276
.5094
.4915
.4739
.4565
.4394
.4226
.4061
.3900
.3742
.9688
.9636
.9579
.9518
.9452
.9382
.9307
.9228
.9144
.9057
.8965
.8869
.8770
.8666
.8559
.8449
.8336
.8219
.8100
.7978
.7853
.7726
.7598
.7467
.7334
.7200
.7065
.6928
.6791
.6653
.6514
.9925
.9908
.9889
.9868
.9844
.9818
.9789
.9757
.9723
.9686
.9647
.9605
.9560
.9512
.9462
.9409
.9353
.9295
.9234
.9170
.9104
.9036
.8965
.8891
.8815
.8737
.8657
.8575
.8490
.8404
.8316
.9986
.9981
.9976
.9969
.9962
.9954
.9944
.9933
.9921
.9908
.9892
.9876
.9859
.9839
.9818
.9796
.9772
.9746
.9719
.9689
.9659
.9626
.9592
.9556
.9518
.9479
.9438
.9395
.9350
.9304
.9256
9
10
11
.9998 1.0000
.9997 1.0000
.9996 .9999 1.0000
.9994 .9999 1.0000
.9992 .9999 1.0000
.9990 .9998 1.0000
.9987 .9998 1.0000
.9984 .9997 .9999
.9980 .9996 .9999
.9976 .9995 .9999
.9971 .9994 .9999
.9966 .9992 .9998
.9960 .9990 .9998
.9953 .9988 .9997
.9945 .9985 .9997
.9936 .9983 .9996
.9927 .9979 .9995
.9917 .9976 .9994
.9906 .9972 .9992
.9894 .9967 .9991
.9881 .9962 .9990
.9867 .9957 .9987
.9852 .9951 .9985
.9836 .9945 .9983
.9819 .9938 .9980
.9801 .9930 .9977
.9782 .9922 .9974
.9761 .9913 .9971
.9740 .9904 .9967
.9718 .9894 .9963
.9694 .9883 .9959
......
o
'-l
n
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
1
2
3
4
5
6
7
8
9
10
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0044
.0038
.0032
.0028
.0023
.0020
.0017
.0014
.0012
.0010
.0009
.0007
.0006
.0005
.0004
.0004
.0003
.0003
.0002
.0002
.0002
.0001
.0001
.0001
.0001
.0001
.0001
.0000
.0000
.0967
.0891
.0820
.0755
.0693
.0636
.0584
.0535
.0490
.0448
.0410
.0374
.0342
.0312
.0284
.0289
.0236
.0214
.0195
.0177
.0161
.0146
.0132
.0120
.0108
.0098
.0089
.0080
.0072
.3588
.3438
.3292
.3150
.3012
.2878
.2748
.2622
.2501
.2383
.2270
.2161
.2055
.1954
.1856
.1763
.1673
.1587
.1502
.1425
.1350
.1277
.1208
.1142
.1079
.1019
.0962
.0908
.0856
.6375
.6238
.6097
.5958
.5819
.5681
.5543
.5406
.5270
.5134
.5000
.4868
.4736
.4606
.4478
.4351
.4226
.4103
.3982
.3863
.3745
.3630
.3517
.3406
.3297
.3191
.3086
.2984
.2884
.8226
.8134
.8040
.7945
.7849
.7751
.7652
.7551
.7450
.7347
.7244
.7140
.7035
.6929
.6823
.6716
.6609
.6501
.6394
.6286
.6178
.6070
.5962
.5854
.5747
.5640
.5533
.5427
.5321
.9207
.9156
.9103
.9048
.8992
.8935
.8876
.8816
.8754
.8690
.8626
.8560
.8492
.8424
.8354
.8283
.8211
.8137
.8063
.7988
.7912
.7834
.7756
.7677
.7598
.7517
.7436
.7354
.7272
.9669
.9643
.9617
.9588
.9559
.9529
.9497
.9465
.9431
.9396
.9360
.9323
.9284
.9245
.9204
.9163
.9120
.9076
.9032
.8986
.8939
.8891
.8842
.8792
.8741
.8690
.8637
.8583
.8529
.9872
.9860
.9847
.9834
.9819
.9805
.9789
.9773
.9756
.9738
.9719
.9700
.9680
.9659
.9637
.9615
.9591
.9567
.9542
.9517
.9490
.9463
.9435
.9406
.9377
.9346
.9315
.9283
.9250
11
.9954
.9949
.9943
.9937
.9931
.9924
.9917
.9909
.9901
.9892
.9884
.9874
.9864
.9854
.9843
.9832
.9820
.9807
.9794
.9781
.9767
.9752
.9740
.9721
.9705
.9688
.9671
.9653
.9635
I-'
a
00
n
90
91
92
93
94
95
96
97
98
99
100
1
2
3
4
5
6
7
8
9
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0065
.0059
.0053
.0048
.0043
.0039
.0035
.0032
.0028
.0025
.0023
.0807
.0761
.0716
.0674
.0634
.0597
.0561
.0527
.0495
.0465
.0436
.2787
.2691
.2599
.2508
.2420
.2333
.2250
.2168
.2089
.2012
.1937
.5216
.5111
.5008
.4904
.4802
.4700
.4600
.4500
.4402
.4304
.4207
.7189
.7105
.7021
.6937
.6852
.6767
.6682
.6596
.6510
.6424
.6338
.8474
.8417
.8360
.8303
.8244
.8185
.8125
.8064
.8002
.7940
.7878
10
.9217
.9183
.9148
.9112
.9075
.9038
.9000
.8962
.8922
.8882
.8842
11
.9616
.9596
.9576
.9556
.9534
.9512
.9490
.9467
.9444
.9420
.9395
....
o
<.D
110
A.3 Bounds on Pr{f(WO)
c}.
~
Proposition 4.1.6 gives as upper and lower bounds on
~
Pr{f(WO)
~
Pr{T (WO)
2
and
c} the values
2c}.
Pr{(mO,MO)
and
of
Pr{(mO,MO)
o~
x
and
0
and
A
the form
MO
~
~
B
~
0, y
~
~
x
and
0
B}
E
~
x
~
~
y
2c}
2c
and
and y-x
~
3c},
1.4.3(c) permits the evaluation
~
a, 0
~
y
mO
b}}
~
(The conditions
0
~
and MO
0.)
can be approximated arbitrarily closely by sets of
a, y
~
2n
~
b}.
For example
-(i-l)c
A ~ [ u {(x,y):
- {-2c
2n-I
ic
{(x,y):
--~x~­
i=l
{-
i
-c
n
~
x
~
x
0,
_00
~
~
x
~
_00
(i-I)
2c
~
c}]
n
< y ~
O} .
< y < 2c _ (i+l) c}]
n
n
-{-2c
Each set of the form
~
(i-I) c;
n
< y
_00
n
i=l
u
and
Pr{(mO,MO)
~
y-x
yare unnecessary since
{(x,y) = x
A? [
~
0, 0
are as in 1.4.3.
{(x,y) = 0
E
3c}
where
E A}
B = {(x,y): -2c
mO
~
2c, T (WO)
2
Equivalently, we may write
A = {(x,y): x
where
~
Pr{T (WO)
l
0,
c;
- (i-I)
n
_00
< y ~ O} •
_00
<
y
2c - (i-I)
n
<
c}
may be expressed as
{-
i
-c
n
~
x;
_00
<
y
<
2c - (i-I)
n
c} - {-
(i-I) c
n
x
~
~
0;
2c
c
- (i-I)
n
},
so that 1.4.3(c) can be used to compute its probability.
The set
B
_00
can be approximated in a similar manner.
<
y
<
III
We give the upper and lower bounds computed in this way for
selected values of
c.
The intervals between values of
care
chosen to be smaller where the distribution functions are increasing most rapidly.
table Al for n
Compare the upper bound for
= 40,
60 and 80.
Pr{f(WO)
~
c} with
112
c
Pr{T (WO) ::;2c}
2
(lower bound)
PdT (WO) ::; 2c, T (WO) ::;3c}
1
2
(upper bound)
.225
.001
.25
.006
.275
.022
.3
.054
.325
.000
.107
.35
.002
.178
.375
.009
.263
.4
.021
.356
.425
.043
.447
.45
.075
.536
.5
.178
.690
.55
.314
.802
.6
.465
.879
.65
.609
.929
.7
.728
.959
.75
.821
.977
.8
.889
.988
. 85
.934
.994
.9
.963
.997
.95
.979
.999
1.00
.989
.999
•
BIBLIOGRAPHY
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[3]
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114
[14]
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5
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c
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!
it