ESTIMATION OF GENERAL PARAMETERS USING
PROGRESSIVELY TRUNCATED U-STATISTICS
by
Elizabeth R. DeLong
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1266
FEBRUARY 1980
·
.....:.
...
ESTIMATION OF'GENERAL'PARAMETERS USING PROGRESSIVELY
TRUNCATEDU-STATISTICS
by
Elizabeth' R.
DeLo~g
A dissertation submitted to the faculty of The
University .of North Carolina at Chapel Hill in
partial fulfillment of the requirements for the
, degree of Doctor of Philosophy in the
Department of Biostatistics
Chapel Hill
1979
Approved by:
r--'-").
.'
'. .' \.'.:....._.....J"'\~/
..,;/ .
"' .. ;~
:..~ ;-"
.....~.,.--~ .....",..",..
Advi,ser
ABSTRACT
ELIZABETH R. DELONG.
Estimation of General Parameters
Using Progressively Truncated U-Statistics.
(Under the
direction of P. K. Sen).
A method is developed for truncating generalized twosample u-Statistics progressively in time.
Such statis-
tics and functions of them form estimators at individual
time points and can also be viewed as stochastic processes •
•
The asymptotic distributional properties of these estimators and processes are studied and conditions are imposed
which guarantee large-sample weak convergence to Gaussian
I
pJ::'ocesses.
As an application of this theory, a truncated Wilcoxon statistic is defined.
This statistic and a function
of it are compared in terms of feasibility for modeling
the true parameter curve and predicting its asymptotic
value.
"Another application is explored in an extended version
of the above statistic.
The extended two-sample statis-
tic arises in the case when random variables distributed
according to known functions of the underlying distribution functions, F and G, arc observed, rather than variahIes from F and Gdirectly.
Restrictions on these known
.:,.
functions are developed in order to yield asymptotic
normality and weak convergence to Gaussian processes ..
....
ACKNOWLEDGEMENTS
I first wish to express sincere gratitude, appreciation and admiration for my advisor, Dr. P. K. Sen.
A full
discourse on the attributes of Dr. Sen would ·comprise more
text than the bodyrif this dissertation.
I would also
like to thank the members of my committee, Dr. James E.
.
'
Grizzle, Dr. C. Ed Davis, Dr. O. Dale Williams and Dr •
",
George S. Fishman, for their interest, encouragement and
time.
In particUlar, Dr. Grizzle's country logic has
helped me over some low spots.
My deepest appreciation also goes to my husband,
David, who has been \-Tilling'to arrange our lives around my
schedule and who has patiently let me interrupt his work
to' listen to my ideas.
I am also indebted to a very good
friend, James Lancaster, for constant prodding and encouragement.
Finally, I would like to thank Ann Davis for her
exce.llent typing job, undertaken at very short ·notice.
Financial assistance for my graduate study and research has been provided by the National Heart, Lung and
Blood Institute; under contract Number 1~HV-l~2243~L.
ERD
.:..
TABLE OF CONTENTS
. Chapter
I. .
INTRODUCTION AND LITERATURE REVIEW
1.1
Introduction. • • • • • • • • • • • • • • •
1.2
Some properties of Generalized
1
U-Statistics. • •• • • • • • • • • • • •
1.3
The Mann-Whitney U-Statistic. • • • • • • •
1.4
Truncation and
the Mann-Whitney
"
Statistic ••
10
•
14
~
24
• • • • • • • • • • • • • •
27
2.2
The One-Sample Case. • .. • • • • • • • • ••
28
2.3
Definitions and Notation for the two-
1.5
II.
3
Proposal.
..
•
...•
eo
•
•
• • •
•
•
. -.
•
•
•
•
• • • •
•
•
..
DISTRIBUTIONAL PROPERTIES OF PROGRESSIVELY
TRUNCATED TWO-SAMPLE GENERALIZED U-STATISTICS
2.1 Introduction. •
Sample Case. • • • • • • • • • • • • •
30
2.4
Lehmann's Two-Sample Scale Statistic • • • •
32
2.5
Covariance Structure at a Fixed Time
2 .. 6
33
...
34
U-Statistics as processes •
2.6.1
2·.7
Point. • •.• • • • • • • • • • • • • •
...
• •
The function spaces. • • • •
•
35
2" 6 • 2 .The Domain. • • • • • • • • • • • ••
35
2.6. 3
37
T~ghtness
•
e·
•
.. .. • .. .. • .. .. • .. •• • •
Tightness of u-Statistics.
.......
. ..
38
vi
.:,.
: 2.8
Weak Conve:rgence .of a Vector.-Va1ued
Process of U-Statistics. • • • • • • •
: 2.9
*
The Process BN.'
~
~
• •
~.
47
48
• • • • • • • •
III., THE PROGRESSIVELY TRUNCATED WILCOXON STATISTIC
..•
3.1
Introduction. • • •
•
56
3.2
Distribution of UN(T). • • • • • • • • • • •
58
3.3
The Conditional Estimator and PJ;'ocess. • • •
63
3.4
Tne Functions
3.4.1
• • • • • • • •
'
1P (T) and
peT) .' • • • • • • •• 73
1P (T) and peT) for the L~gistic
Distribution. • • • • • • • • • • • •
.
'
3.4.2
.....
1P (T) arid peT) for the Normal
Distribution. • • • • • • • • • •
•
3.5 Predicting P(F,O). • • • • • • • • • • • • •
THE TWO-SAMPLE EXTENDED CASE
•
IV.
•
t.
76
82
'4.1
Introduction. • • • • • • • • • • • • • • •
98
4.2
Lails Approach ••
99
4.3
The Pyke-Shorack Approach. • • • • • • • •• 102
4.4
Weak Convergence to a Gaussian Process. • • 121
4.5
V.
75
•
•
•
•
•
·e
o
•
•
•
. . -
•
•
4.4.1
The Individual u-Statistics • • • • • 122
4.4.2
* • • • • • •• 129
Weak Conve7t'gence of TN.
General Confidence Bounds. • • • • • •
SUMr~RY
,AND
S~,GGESTIONS
....
130
FOR FURTHER RESEARCH
5.1
Sununary.-. • • • • • • • • • • • • • • • • •• 134
5.2
S~9gestions
APPENDIX •
REFERENCES
for Further Research. • • • • • 135
....• • •
• • . . . . .
. . . . . . • • . . ••
. . . . . . . . . . · ..
• • •
137
142
e
.I.'
CHAPTER I
INTRODUCTION AND LITERATURE REVIEW
, '1'.:1:.' . :Il'it'roQ'uc'tion
The situation in which an experiment
i~ des~9ned
on
the basis of independent observations, but for which the
data arrives as a function of some
.'
presents a wide field
f~r
orderi~gsuch
analysis.
as time,
This dissertation
addresses the problem of estimation in such cases when it
is impractical or unethical to wait fcfr the full set of
data.
For example, in some clinical trials where the
variable of interest is time to failure or endpoint, the
trial is designed to terminate after a certain number of
years.
It is likely that not all of the survival times
will have been observed by the time the trial is terminated.
When this occurs, the data are collected in the form
of order statistics and therefore are not independent.
However, statistics which take into account this loss of
independence can be constructed from the data as it is
observed and thus provide methods for
meters without
waiti~9
estimati~9
para-
for the full sample.
We will deal here with the classical design in which
subjects are all assumed to enter the stUdy at the same
time.
Truncation will occur from the right in the sense
':"
that at any point T,
~stimation
will be based on all ob-
servations of magnitude less than T.
The problem con-
fronted will be that of pr!>gressively
estimati~g
sample
r~gular
which unbiased
thro~gh
two-·
parameters, i.e., those parameters for
estimators'exist~Theapproach
will be
the use of,generalized V-statistics, which lend
themselves readily to
pr~gressive
truncation and evalua-
tion.
Throughout the text we will let
.
'
Y , ••• 'Y
1
n
Xl, ••• '''m and
be independe'nt random samples from continuous
.....
.
distribution functions F and G respect1vely, with N=n+m
and "N==m/N.
,we,w~ll
.
also refer to the,empirical distribu-
,
tion functions ofF and G, denoted by Fm and Gn ' respectively. Convetgence in distribution will be denoted by
the symbol
f)
+
and we will use 0pCl) for convergence in
probability to a constant, while opel) will imply convergence in probabi Ii ty to zero.
We ~ill use ]RR. for
f.-dimensional real space and the symbol - to mean 'is distributed as'. '
The next section lays the foundation for the use of
generalized V-statistics in what follows.
Section 1.3
forms the background for the use of the Mann-Whitney statistic and then section 1.4 gives an overview of the work
that has been done in
truncated data.
usi~9
the Mann-Whitney statistic on
~
·.,
3
The theory of U-statistics was covered .comprehens~vely by
Hoeffdi~g
in 1948 for the one-sample case.
This
.work has been extended or elaborated by Lehmann (1951),
Sen (1960 and
1964), Bhapkar (1961), and Miller and Sen
(1972) •. This section of the dissertation presents for
further reference the notation, definitions:and properties
,"
A parameter8(F,G) is said to be regular over the
product space of continuous distribution functions,
. {FX G:
F
£
f, G £ G} if there exists a statistic
,(Xl' • • • '~1 ' y l' • ..', yx ) with Xl ~ 1 and X2 ~ 1, which is
2
symmetric in the X's and in the Y's, such that
E'(Xl'···'~ ,Y1 , ••• ,Yx
1
for all (F, G)
£ (F x
G).
2
) = 8(F,G)
Expressed in
int~gral
(1.2.1)
form, this
is
dF(~
I
If
)dG(Yl)." (dG(yx )=e(F,G).
2 .
(1~·2.2)
Xl and X2 are the smallest sample sizes for which such
·"
:.'
a ,statistic exists,..then the statistic
,yK' ), denoted simply by ,p, is called
KI 'YI' ••• ,
2
the Kernel for e(F,G) of d~gree vector (K1 ,K 2 ).
~(XI'·.· ,x
.
For example, P (F, G) =Pr (Y < X) is a
r~gu1ar
para-
we have
.'
co
00
•
.1_00 1_00 ,p (x,y) dF (x) DG (Y)
= 1_
00
Now,
00
G(x)
correspondi~g
U-statistic
UN
dF (x) =
(1.2.4)
P (F, G)
to a Kernel
,p, the. generalized
is defined as
m -1 n -1
UN = ( K) (K) E' ,p (Xa ., ••• , Xa ' Yf3 ,,; ... , Yf3 )
1
2
1
Kl
1
K2
where
t' denotes swnmation over all
1 ~ a l < a 2 , ••• , < a K < in, I ~ 13 1 < 13 2 , • • ., < 13K < n.
12-
uniformly minimum variance unbiased estimator of
(cf. Puri and Sen, 1971).
UN
is the
e (F,G)
It is unique when F and G are
5
:.'
both equal to .the class of all continuous distribution
functions (cf. Lehmann, .1951).
The
followi~g
notation and definitions will be em-
ployed throughout the text:
For
0 ~c~m, O~d~Ji, define
(1.2.• 6)
and
"",
(1.2.7)
In the case of several.
let UNl , ••• , UNR.
be 1 two-sample U-statistics defined on the same sample
U-statistic~,
of size N with, for each i and j
a.
E UNi = 6 (F,G), denoted by 6 i
i
(1.2.8)
and
b.
The Kernel for 6i is 4>i (Xl' • • • , Xx
1
' Yl' • • • , YK
2
)
6
.:,'
[4!J'(X1 , • •• ,Xc'~
. x...-.
1i+c' ., •• '-""K . +K. ,-c
11
Y1'''·'Yd'Y.-K
'
1J
,···,yX 21.+K 2J.-d)
2i+d'
-
9 (F, G)
J },
(1.2.9)
X1 ' ••• 'Xc and Y1 ' ••• 'Yd are the only subscripts
common to both Kernels.
where
"
Now, assume that
cl:
lim. AN ~ A
0<).< 1
N~CD
(1.2.10)
\
when
c2:
mIN
= AN·
(Sen, 1974):
(1.2.11)
i.e.,
(F,G) is stationary of order
zero.
c3:
Then the
fol1owi~g
have been derived:
(1.2.12)
properties of, generalized V-statistics
."
7
J..
. P2:
, 1
fUN -
iii
m,
. 1, n
'
~:l"'lO(Xa)- Ii B:1"'01 CY B) .,.
e(F,G)
I
=Op(N- 1 )
(1.2.14)
(1.2.15)
.'
and'the convergence can be shown to be uniform over
(-00,00)
P4
and Cf
x G)
under certain
•
r~gularity
conditions.
(Bhapkar,196l):
, ~N = (m(Uhl-e l ), IN'cuN2 -e 2 )·· .,1HuN-e»
(1.2.16)
is asymptotically normal with mean Q and covariance matrix
~
=
(Oij)' where
_ K1iKlj
a ij , A
PS
(1.2.17)
(Sen, 1960):
UN
"'O(F,G)
w.p. 1,
'that is, UN is a strongly consistent estimator for 8(F,G).
For 'this property, condition C3 can be relaxed somewhat.
8
J.'
P6: .Letc [0,.1] be the space of all continuous. ;functions
on the interval' '[0 ~1].
.{z; (t),
0
Then the process
~ t ~ I} defined belowconve~ges weakly in
the uniform
topol~9Y
onC[O,l] to a standard Brownian
motion as N approaches infinity
the, greatest
int~ger
II
Letti~g'
[p] denote
less' than or equal to p and
letti~g ~
and n x denote individual sample sizes when
the total sample size is X, with, Um..
as the
~'A,nK
.
U-statistic based on this sample, then
when [Nt,]). [Nt] < Xl
."
".
or
[Nt](l~A[Nt])
-8CF,G)}
.,.K{Unx,~····
.. ·
=
2
(1.2.18)
(1.2.19)
2
" l(:ttlO 'Xi,tOl 1/2
m
+
n }
N{
and
(1.2.20)
..when
KIN < t ~
K+l
--w'
'
K=O, 1 , ••• ,
N-l ..
Sen (1974) generalized this concept to a two-dimen-'
sional time parameter and demonstrated weak conver,gence to a two-dimensional Gaussian process.
9
'.
. A consistent estimator for the variance
~f.
UN has
been demonstrated by Seri' s .. (1960) property of structural
conve~gence.
We "will call upon this property in Chapter
Let
III.
and
"
m -1 n-l -1 .....
(Xl)
(X -1)
E2j'(X~1'''·'X~ 'Yj'YB1'···'YBx -1)
2
'.
· 1 2
•
E1j is summation taken over all
where
. {l ~ a
(1.2.22)
1
< a , ... , < a - < m,' (xi
2
K 1
1 <13 1 < 13 2 , ••• , < BX2'~ n1 and
'F.
j
for any i},
I 2j is taken over all
1 ~ a 1 , ••• , < ax
any i}.
~ mt ' {1< 13 1 < 13 2 , ••• , < Bx -1 ~ n, B 'F j for
i
1
2
And define
(1.2.23)
(1.2.24)
Then
(1.2.25)
10
and we have the next property:
P7:
,-
2·
2
Bvel ) and BV(2) are consistent estimators. of . t lO
and tal respectively and
The Mann-Whitney statistic (cf. Mann and Whitney,
.
"\
1947), designed to test
•
(1.3.1)
against
Hl :
X is stochastically smaller than· Y
(or, F(z»Gez) for all
z),
(1.3.2)
is defined as the total number of pairs eX"Y,) for which
.
the ~ preceded the X.
When
~
J
n=m, ~w is. equivalent to the
statistic proposed by Wilcoxon (1945), which is the sum of
the combined sample ranks of the Y's.
Mann and Whitney derived a recurrence relationship
for computi~g the distribution of
u:w under HO and have
tabulated the distribution for sample sizes up to m=n=8.
11
.:,.
Lehmann (1951) $howed that the
.t~st,
:which· rejects H for
O
small values of the statistic, .is consistent ~gainst all
a1tematives of the form Hl •. Van Dantz~g (l95l) further
showed that the test is. consistent ~gainst any alternative
. such that Prey :< xl .(
Mann
limiti~g
j .
and Whitney (1947) showed that, under HO ' the
distribution of
(U - ~ mn)/«m+n+1)/l2mn)1/2
.'
is standard normal,
.....
r~gardless
of the relative sample
sizes, and that, for sample sizes as small
as
#I
distribution qiffers only
sl~ghtly
m=n=8, the
from the normal.
Now
can be expressed as a generalized two-sample U-statistic
with Kemel
cfl(X, y) =
1
Y<X
o
otherwise.
(1.3.4)
As such, it inherits the properties discussed in the previous section.
In particular, when m and nappropch in-
fini ty· at the same rate and we assume
~ll <
OJ)
t Ol
> 0,
we have, even hi the non-null case, an
t lO
>0
and
12
asymptotically normal distribution:
'
.
f)"
,tic ' "~oi
P(F,(;» +N(O, ~ + (l:-~:».
. Lehmann (1975) has shown that
~lO >
0
and
tOl > 0
assumi~g
(1.3.5)
the conditions
implies the assumption that
P(F,G) = Pr(Y:< X) is not equal to
0,
or 1.
Work has been done by Van Dantzig(1951) and Birnbaum
and Klose '(1957) on bounds for the variance of
..
u:w
by
obtaining bounds on
.,
'
and
(1.3.7)
Thus, using the normal approximation and these bounds,
confidence intervals for P(F,G) can be obtained.
~owever,
tighter confidence bounds are produced by using Sen's
(1960) structural components and the resulting estimate
for 'the variance of
P(F,G) (cf. Sen, 1967), which will be
closer to the actual variance for large m and n.
Another method for constructing confidence bands was
explored by Birnbaum (1956) and Birnbaum and McCarty (1958)
for the case when the normal theory is not justified.
13
They used the
fo11owi~9:
,.
00
P (F, G) =" .1-00 Gno (x) (iFIIi (x)
(1.3.8)
'and
P (F ,G) =
I':: G(xldF(x) ,
(1.3.9)
where Fm and Gnare the empirical distribution functions
.
"
~1' •••
associated with
Then
e
P(F ,G) -P(~,G) ==
'Xm and Yl' ••• 'Yn
"
' respectively •
•
I:' G(t) d [F(t) -Fm (t) ]+ ':00 [G(t) -Gn (t) ldFm (t)
(1.3.10)
which, after integrating the first integral by parts,
yields
Ip(F,G)-P(F,G) 1=1/00
-00 (Fm(x)-F(x»dG(x)+
.
~
sup
=
n; + n:
-co <x<co
where D;
and D~
statistics.
(Fm(X)-F(X»+ sup
100CG(x)-G
-co
n (x)dFIn (.x)
-co <x<co
(G(x)-Gn(x)
.
(1.3.11)
are the familiar Kolm~gorov-Smirnov
The exact distributions of these statistics
are known and depend only on the sample sizes and not on
the underlying distributions, F and G.
I
Then, recalling
14
J.'
that
n: 'and D~ have identical distributions,
and.
usi~g
(1.3.12)
·a one-sided confidence interval for P(F,G) is obtained
thro~gh the distribution of
mIN.
n: + D:
in terms of the ratio
However, this approach is only good for la:rge sample
sizes and, unless P(F,G) is close to 0 or l,.the assumptionof nor.mality yields better confidence
.
'
bou~ds •
......
1.4
Trunea't'i'On' 'and' :th'e' Ma'nn-Wh'i'tne::y' S·t'a·t·i·s'tic
Several extensions ·of the Mann-Wnitney stat~stic to
the truncated problem have been proposed, some of which
are specific to a particular truncation or censoring
scheme.
We will say that an observation has been censored
if because of trlJIlcation, we do not have the full value.
Halperin (1960) considered the case in which all
subjects enter the study at the same time and both samples
are truncated at time T.
Letti,ng r m and r n denote the
number of censored observations from the m X's and nY's
respectively, Halperin's test statistic for 'testing
H : F = G.
O
is de fined as ,
It
15
.(1. 4 .1)
where.·Ur ,r
.
is the 'usual Mann-Whitney statistic based on
l 2
sample sizes of r 1 ,r 2 •· Uc is essentially the ordinary
'two-sample Uncensored statistic with an adjustment for
ties.
Halperin calculated the exact probability distribution Uc under HO ' conditional on r, the total number of
c~nsored observations.
He also found the unconditional
.'
mean and variance of Uc under HO and showed that Uc has
asymptotically a normal~distribution, the approximation
being acceptable for up to 75% truncation.
The test is
ff
shown to be consistent against all alternatives of the
form
H1 : F(x)/F(T) >G(x)/G(T), F(T) >G(T), -oo<x<T, (1.4.2)
which implies F (x) > G(x) for all -oo<x<T.
Sugiura (1963) proposed a. generalized U-statistic
for dealing with the situation 'in which truncation occurs
from the left, i.e., at the origin and when
F
takes the
form
F(X)
= pr(X
<x)
and likewise for G.
=
o
x <0
Px + 'Oxf(t)dt
(1.4.3)
x> 0
The test statistic employs the
16
.co.ncept .of midrank:; a.nd is defined to have kernel
m(X,Y} =
1
1
x>Y
X=Y
0
x<y
'2"
.(1.4·.4)
•
Then
Us
"1 . m n
E
.E m(x].. ,Y·.}
i=l J= 1.
J
= -mn
. (1.4.5)
is an unbiased estimator for
.'
"-
PriX > Y > o}
+' ~ PriX
= Y = O}.
(1.4.6)
•
The test based on Us is shown to be unbiased and consistent against alternatives of the form:
HI:
F
is stochastically
la~ger
than G,
(1.4.7)
assuming m and n approach infinity at the same rate.
When Halperin's statistic is reversed so that censoring occurs at .the origin and is standardized into the
form of a U-statistic then
U
c
where
'I
= ron
m n
E
E c(x]..,y.)~
i=l j=l
J
(1.4.8)
·e
17
. c(X,Y).'
=
1
,(1.4.9)
o
y~~>o,
and Uc is related to Us by
. r-.r
.IU n··
= .Us -..".-;....,;.;;,
, c
,·2 mn
(1.4.10)
U
where r m and r n are the number of zeros appearing in the
X's and the Y's respectively.
Sugiera calculated the ARE of the U~ test relative
.'
to the U·c conditional
t~st
at
r
=
(n+m)px
when testing
•
(1.4.11)
against
(1.4.12)
(1.4.13)
, G(x) = Py + It"f(t-' ) dt
0
Py· = 1-co f(t-S)dt
The ARE turns out to be 1 when
smaller or larger depending on
'm
1 and can be
N~ n
• m·
1J.m Ii ·
N-+co
lim -
=
A sequential procedure for testing
H : F = G
O
18
. against
HI: F (x)
.?: C:Hx) for all x'
0 and F ~ G
was proposed by Alli~g (1963) and uses the following relationship between
v
Um-r ,n-r and Um,'n=
m
n
+ (n-rn )r"m< Um,n< Um-r ,n-r + nrm =
. . m-r
. m,n-rn
m
n
= U
W(1~4,.15)
.
Alling then suggested
the one-sided test which
....
a. rejects HO when W< b ,
.
b.
c.
,
,
accepts HO when V> b , and
- a
continues for another observation when
neither of the above applies,
where
b a is
th~
a-level critical value of the ordinary
Mann-Whitney statistic.
This test thus provides for what
is called an early decision, while reaching the'same conelusion as would be reached if all the data were' available.
Its power and consistency are also identical to
that of the two-sample Mann-Whitney test statistic.
Alling demonstrated an
anal~gous
"0: F = G
two-sided test for
(1.4.16)
e
against
"0: F ~ G
(1.,4.17)
19
which. also preserves ,the ,power and consis,tency of the
two-sided Mann-Whitney test. . In e'ach 'case,' ratios of
aver~ge
samp1e size of
Mann~Whitney
Alli~9's
test and of the complete
test were calculated and indicate a
savi~9s
of more than 50% in small samples I with the ratio 'increasi~g
with
,N =
,m+n.
A method of
also devised so that the
st~9gered
iecordi~g
deaths was
entry problem in which
subjects enter the trial randomly rather than at the same
time could be handled by the same test.
.'
'e
In comparison'to Halperin's test, in which the time
'" random
T is fixed and rn.m
, r are
' l Alli~g's test is based
on fixed number of observations with the time of trunca•
tion being
ra~dom.
,has more power.
Due to this difference, Allings test
However, neither lends itself to the
problem of estimation.
Halperin and Ware (1974) further investigated early
decision whereby a trial is terminated only when it is
clear that continuation cannot reverse the decision.
Their idea follows Allings, with the exception that they
propose to terminate either according to the early stoppi~g
,rule or after a specified proportion of the observa-
tions have been collected.
Another approach was sU9gested by Gehan (1965), whose
two-sample test applies in the case of arbitrary
r~ght
censoring and random entry with the same entry distribution for both samples.
The test is conditional on the
20
pattern .of observations and tiea are al.lowed.
Xl , ....'Xm- r
.
. m
and
Letti~g
Yl' •.• ~ ~ Y:n - r be ·survival times from F
n
and G respectively
and X'm-r. +1' •••
.
. ,X'm , and
m
Y~-r+1' ••• 'Y~
be ·the :correspondi~g censored observan
..
tions, ~han's statistic W is the number of pairs in which
• ,.; ... _.
'.
an'
uncensored. Y .precedes any X, minus the number of
pairs in which an .,', uncensored X precedes
~y Y.
This is
ron times the U-statistic which has kernel
-1
Xi <~j
0
Xi =Y j
Xi >Y j
0'
Uij =
1
"
or
X. < Y~
J. -
J
pr (Xi,Yj) or X! < Y. or Xi >Yj
J.
J
or X! :> Y. •
(1.4.18)
J. -
J
"
e
When no observations are censored, this is equivalent to
the ordinary Mann-Whitney statistic and when censoring
occurs at a fixed time T and all censored observations
have. the same value, W is related to Halperin's Uby
c
(1.4.19)
The normal approximation for W is shown to be good for
sample sizes as small as
m = n = S,as long as not more
than 60% of the observations are involved in ties or
censori~g
and at least S distinct failure points are ex-
hibited.
The test based on W is consistent ~gainst all
alternatives of the form
e
21
(1.4.20)
Another statistic which employs a fixed number of
observations and random time 'T was introduced by Sobel
. (1957) and examined by Basu (1967).
Letting m, and n.J.
~.
represent the number of X and Y failures
amo~g
the first
i observations, the statistic is defined as
~,n
R
"
=
(1.4.21)
"
where R is the total number of observations at time T.
Basu established the null and non-null asymptotic normality of an equivalent statistic to ~,n and also showed it
R
to be a consistent test of
(1.4.22)
against one-sided alternatives of the form
HI:' G(x) = F{x-8)
(l.4.23)
8 > O.
Efron (1967) considered the completely general situation where entry distributions are not necessarily equal
and are possibly unknown.
Letting
Xl' ••• 'Xm and
'
t 0 fal.
' l ure, X"l' • •• , X"m an d Y"l' • • ., Y"
Yl' • • • , Y.n d ena t e t l.mes
n
denote times to truncation, then for
i=l, ••• ,m and
22
."
'-1., .,•• ,n,'
)-
Y! = miJi{Y. ,yin
)'
J
J
.
(1.4.25)
Information about the nature of each observation is
carried by the random variables
~l' ••• '~m
and
El, ••• ,en ,
given by
.'
X!1. = Xi
......
X!1. = X~'
1.
1
~i =
0
(1.4.26)
and
Ej =
1
Y'j = Y.
0
Y'j = Y'! •
J
J
(1.4.27)
X.1
- F,
Y. - G, X!,1. - H and YJ~ - I. In
.J
this setting, Ge~an's statistic is only asymptotically
It is assumed that
nonparametric whenH=I since the variance of W depends on
the relationship between H and 'I and F(=G under
H~).
Also,
when the null hypothesis is not true, the expectation of
W depends on H and I.
Thus W cannot be used as an estima-
tor.
. .
,..
Efron has proposed a stat1.st1.c W which is independent
of H and I but which relies on his self-consistent estimaA
A
tes F' and G' of
F'
~·l.d
G', the right sided cumulative
23
;.'
distribution .function$.
A self-consistent estimate of
F' is one which 'satisfies
(1.4.28)
where Nx(S) is. the number of
x: > s.
1-
lo' is defined'
.
iteratively by
1
.'
,.
. , NxC:sJ.,: : ' : .
F' (s) =
m-
XK_l<s~'1<
k-~ .<.~-.cL)
1':
... '1
•
f1
o
s>x
m
and has the usual properties of
cumulative distribution functions.
A,
(1.4.29)
2<K<m
i=l F"', ( Xi )
,."
'
•
discrete rightsided
Further, Efron showed
_
•
that F' and G' are the unrestr1cted max1mum likelihood
estimates of F' and G'.
,.
Then W is defined as
n,.
1: 0 (X. , Y .)
ron i=l j=l
1
J
,.
1
W = -
m
1:
(1.4.30)
with
O(x.,Y.)
1
J
(1.4.31)
24
.:.,'
.It is pr,oved' that
,.
OC)
,..
,A'
W = .-./-00: .F' (s)'dG' (8)
(1.4.32)
,.
and W is the maximwn likelihood estimate for' Pr (Y ~ X) •
Also
N1 / 2 (w_pr(Y < X)
.
.'
'
~N(O' 'Ak 12
+~2
~-A~2
)",
(1.4.33)
where, under HO '
(1.4.34)
Efficiency calculations are made in the paper and it
A
is demonstrated that W is more efficient than Gehan's W
providing not more than 66% of either sample is censored.
In the following chapters, the properties of progressively truncated. generalized U-statistics, viewed both
as point estimators and as processes over time, are examined.
In Chapter II the method of truncation is defined
such that the
resulti~g
statistics remain U-statistics.
Existing theory can thus be applied to yield the asymptotic distribution theory for the point estimates.
The
stochastic processes are then examined with the result
25
/'
that,. ,under certain $moothness .conditions on the first,
second and fourth moments of .the Kernel, .suchprocesses
conve~ge
weakly to Gaussian"' processes.
shows that processes derived as
A further result
well-behav~d
functions of
these progressively truncatedU-statistics are also
asymptotically Gaussian.
~he
truncation techriique developed in Chapter II is
then applied to the
Mann~Whitney u-statistic
unbiased estimator for
.'
for
WeT)
= Prey < X< T).
to form an
An
estimator
P (T) = Pr (Y < X IX, Y < T) is then const.ructed from a
"
function of this statistic and the cumulative distribution
functions Fm(T) and Gn{T).
This latber estimator is shown
to be unstable for T close to zero, so its domain of
application is restricted to T. greater than some positive
TO.
The stochastic processes induced by both of these
statistics are then shown to be asymptotically Gaussian.
Consistent estimators of the variances and covariances of these statistics at various time points can be
computed through the use of structural compponents and
then weighted' least squares can be used to fit functions
to the values of these estimators evaluated at several time
points.
In this way a prediction of the true asymptotic
value of the particular parameter is produced.
This pro-
cedure is incorporated on randomly generated exponential
data and the prediction produced by modeling
compared against that obtained by. modeling
"-
WeT) is.
"
peT)
with the
...
.. ....
26
;"
reault.that
"
A
",(T.)
produces a more .accurate estimate in
.this particular. case.
In Chapter IV the "notion
of"pr~gressive
truncation
is extended to a modification of the Wilcoxon statistic
.
.
for the case'when the random variables observed come from
distributions which are known functions of the distributions F and G.
Under certain integrability conditions on
these known functions, this statistic is shown to be
asymptotically equivalent to the sum of three U-statistics
.'
whose Kernels each obey the smoothness criteria set in
......
Chapter II for weak
conve~gence
to a Gaussian process.
Thus the conclusion: follows for the .-extended two-sample
wilcoxon statistic.
"e
...
.:"
CHAPTER II
DISTRIBUTIONAL PROPERTIES OF PROGRESSIVELY TRUNCATED
TWO-SAMPLE GENERALIZED 'U-STATISTICS
2.1 ' 'IIltr'oduc'tion '
By
truncati~g
the
~ernel
of a, generalized two-sample
U-statistic at a time point, T,we create another U......
statistic which is now a function of time as well as
sample size.
We can then consider
e~timators
that arise
as functions of such U-statisticsand estimate certain
parameters which vary with time.
These estimators then
represent random processes which exhibit certain behaviors.
The study of such estimators as processes is motivated by the situation in which it is either infeasible,
unpractical, or impossible to collect all of'the data
and the data becomes available as a function of an ordering, such as time.
For example, in an intervention trial
comparing a drug and placebo, it may become necessary to
terminate the trial before all of the data
ar~
collected.
\
When this happens, parameters representing the full effect
of the drug on survival time are still of interest and
must be estimated from the data at hand.
We can, at anyone time point, form the truncated
28
estima.te .of .the paraJQeter
~finterest usi~9'
,qathered thus far. 'However,
~itis
.th.e data
usually the case that
our main interest lies in the 'full-sample parameter,
i.e.; .the parameteresttmated by the full sample, and we
wish 'to quantify it on the basis of an
initi~l
portion of
the data.
Suppose that the parameter of interest is such that,
as a function of time, it ·traces a continuous path and
approaches an asympto.te as time. gets large.
Then it is
possible to model this path by means of progressively
truncated estimators, s1.nce at each time point representing the value of a data point, the truncated estimate can
•
be evaluated. ·We thus obtain a set of points on which we
can fit a polynomial, the degree of which depends on the
parameter being estimated.
In this way we develop a pre-
diction estimator for the full sample parameter, using
the progressively truncated process.
It is the purpose of this chapter to investigate the
properties of U-statistics truncated at a single time
point and also as random processes when they are pro-'
gressively truncated.
We first consider progressively truncated U-statistics
from a single ,opulation.
Let Xl,.··,Xm be independent
and identically distributed (i.i.d) positive random
tt
29
1,.
. . variab.les from a population with continuous distribution
.function F.
Then let
(2.2.1)
be a one-sample U-statistic of
d~gree
K with Kernel ;,
t· (X , ••• ,X~) indicates the sum over all
al
1 ~ a l < ••• < a K ~ m. . We define the. truncated estimate at a
point.T to be
where
."
(2.2.2)
·e
where
; (Xl'···
,Xx)
o
Xl, •••
,Xx < T
(2.2.3)
otherwise
For example, consider the empirical distribution
function Fm(T) given by
= 1
m
C(u)
=
m
.
t C(T-X.),
i=l
1
1
u>O
o
u <0 •
(2.2.4)
(2.2.5) .
We can alternatively express this as
(2.2.6)
30
:,.,
and see that Fm(T} represents a truncated V-statistic
with
d~9'ree
1 and Kernel
~T(X)
=
1
X <T
(2.2.7)
otherwise'
Note here that the full-sample parameter which Fm(T)
estimates as
T.9'ets1a~ge
is the number 1,
r~gardless
of
the distribution function F.
We will henceforth consider one-sample truncated
.'
V-statistics as a special case of two-sample V-statistics
as is 'mentioned below ......
•
The notation used here will be consistent with that
of the previous chapter.
and Yl'Y2' ••• ,Ynbe i.i.d.
co~tin~ous
X~,'.X2'··,·'Xm ~~_,i.i.d.(~l:
We let
with
(G)
dist"ibUtion functions"\
F,
and
·We
G
both·
d~o·.l~t·
N
.
I
. I
...
be the total sample size (N=n+m), AN=m/N, and assume AN
,
approaches some fixed
finity.
Also let
'
A, 0 < A < l, as N approaches in•
V~(T)
be the truncated version of'the generalized U-statistic
(2.3.2)
I
31
>.
with
.d~9'ree
a (F.,~) •
·vector
(Kl'~2)'
t'
Recall that
.Kernel
a
and expected value
(X~ , •.•• i Xa
1
cates the ·sumover· all
~
. Xl
'~e'
r
1 ~ a ••• < aK.
l
ax·
1
..• 'Ye )
indi-
K2
"IIi;
1< l < ••• <
< n.· .In the. case of a one-s·ample U-sta-.
2tistic, the d~gree ·vectorbecomes (Xl·' 0) or (0,X ) depen...
2
di~9'
on the scupple.
The truncation is performed by de-
fini~g
.-
=
.
Denote by
~T(Xl' •••
.
(2.3.3)
aCT) the expected value of
'Xx
,Yl,···,YK )·
· 1 2
We will abbreviate this to ~T
when possible.
Now
let
(2.3.4)
where
9'
is a differentiable function from R1
to Rand
U (T), 1 < j 5...1, are all two-sample U-statistics. from the
Nj
same sample, with expected values e.J (T) and degree
vector
.
Let
(2.3.5)
32
We will then examine .the properties of the
resulti~g
sta-
tistic,
(2.3.6)
under suitable restrictions on, 9 and on the individual
U-statistics.
.'
As an. example, consider Lehmann' s two-sample scale
"-
statistic, ~ , which has Kernel of degree (2,2). given by
•
(2.4.1)
otherwise
Then
(2.4.2)
It can be shown that, for the truncated statistic,
OCT)
= E~T(Xl,X2'Yl'Y2)
~ ~{F2(T)G(T)+G2(~)F(T)-'
F
3
(T)
3
3
_' 'G (-T)1
3
_' 2 F 2 (T)G 2 (T) + I T [F(X)_G(x)]2 d{F(X')+GC-X)1.
3
0
.
2
(2.4 .• 3)
Notice that. aCT) is a. ocontinuous ofunction of Twhich
approaches
9(F,~)oas
T. goes to infinity.
Now recall that the empirical distribution Iunctions
Fni(T) and Gn(T) are °also U-statistics and can be con. sidered two-sample V-statistics with
and (0,1), respectively.
°d~gree
vectors (1,0)
Thus we can estimate the trun-
cated distance function
(2.4.4)
.' .
by
L
= g(UN(T),Fm(T),Gn(T»
L
1 2
2
. F~ (.T)
= UN(T)- 2{Fm(T)Gn {T)+Fm(T)Gn (T)- m3
o
0
0
-l F~{T)G~(T).
3
. oG (T)
- n3
}
(2.4.5)
However, Wegner (l956) has shown that certain additional
assumptions must be made on the functions F andG to ensure asymptotic nondegeneracy for the full-sample case •
•
Returning to the treatment of the V-statistics at a
fixed time point; we have the
eo
followi~g
direct result of
property P4 of Chapter I:
Theoore°I[{ :2 ."5'.1:
0
Aossumi~g
conditions Cl-C3 of Chapter I for
each of the V-statistics involved, the vector
•
34
.:"
(2.5.1)
has asymptotically a mUltivariate normal distribution with
mean gand covariance matrix
ET = C(oij (T») where
(2.5.2)
.'
'With the foundation built around the
individua~
u-
statistics and their relationship at a fixed time point,
we are now in a position to investigate these statistics
as processes in time, with the aim of eventually showing
that a process can be constructed from BN(T) which converges to a Gaussian process.
To achieve this result, we
will first show that the vector
WN(T), considered as a
process in time, converges weakly' to'a Gaussian process
in n1 [o,m), the l-dimensional 'space of functions on [O,m)
with only jump discontinuities.
Conclusions will then
follow about the derived process,
*.
A
BN = {JNCBN(T) -B (T) ); 0 <T <
OC)}. •
(2.6.1)
When referring to a process rather than a statistic at a
35
:,'
fixed time point 'We will use the, • .notation.
In. general, .in order to demonstrate weak
'conve~gence
of a sequence of probability measures on a separable
."
metric space,' :we mus,t show
a)
weak
conve~gence
'of the finite-dimensional
distributions
'and
b)
the property of t:f,ghtness, which is discussed
below.
Some remarks are
2.6.1
neces~ary
at this point
"
The Function Spaces
:.( .. " ,. We are dea1i~g wi th spaces of functions which are
not continuous, but which have only jump discontinuities.
In these spaces, the uniform metric
p(x,y)
= supfx(t)-y(t)
f
t
(2.6.2)
does not provide a complete separable metric space (see
Billingsley, 1968, Chapter 1, Section 6).
2.6.2
The Domain
"
.. The domain of the functions in these D-spaces is
[0,00) in our case and [0,00) is not compact.
then two ways of approaching this problem.
There are
Whitt (1970)
x:e1ated weak convergence of probability measures'{PN} on
C[O,m), the space of cC?ntinuousfunctions on (0,00), to
weak convergence of asaociated probability measures
36
.
{P~}: on c j
= .C[O,.j.J •.
He .defined the .sequence of metrics
PJo (x,y \) = ,sup ,Ix(t)-y(tll
'. ·.'t~j'
and constructe'd the
followi~g
(2.6.3)
metric on C [0 ,00)
:
(2.6.4) .
He then showed that P makes
.'
C(O,~)
a complete separable
,
metric
space and concluded that a necessary and sufficient
.
cond~tion
that'
{PN}
converge weakly to
P
on
C[O,~)
with
Pw' is that, for all j , ' {~} converge weakly to
on C[O,j] with respect to Pj.
respect to
pi
However, we are able to use a simpler approach, since
the functions we are dealing with can be assumed to have
unique limits as T goes to infinity.
We will define the
function value at infinity to be the limiting value.
example, in the case of the
lim
~~
INcu (T)-6CT»= '.Ncu
N
N
Mann~Whitney
statistic
(~}-6(~»
= /.N(UN - 6CF,G»,
where
For
(2.6.5)
UN(T) is the truncated two-sample Mann-Whitney
statistic, UN is the full-sample statistic and
6(F,G) is
the unconditional probability that it estimates.
We will then define a bicontinuous, one-to-one map
e·
37
,:.,
h;' .,[0,.1] +·[0,00J. such .that hlOl = .0 and hell =.00..
In this
way we'can translate the problem to D'rO ,.1] by working with
fO (f) = ,f 0 h (t) =.f (Tr,· .T
E:
ro,.~],
.t E: [0,1] f E: D [0 ,00] •
(2.6.6)
In some instances, :we will implicitly assume the
function h and we will work directly with T.
2.6.3
Tightness
The proper topology for D[O,l] is the Skorohod
."
topol~gy.
However, as''is shown in Billingsley (1968,
p. 111) tightness with respect to the uniform topology on
f1
DIO,l] implies tightness with respect to the SkorOhod
topology.
Also, the weak limit, if one exists, will be
concentrated on C[O,l] (see Billingsley, 1968, p. 110-111
and p. 127-128).
Thus tightness can be demonstrated for
a sequence of probability measures' {Pn } by showing that,
for T) >0, x£D[O,co],
a.
there exists an a such that
pri {x: Ix (0) I>a} ~ n,
(2.6.7)
n >1
and
b.
for £ > 0 there exists a
integer
cS, 0 < cS < 1, and an
nO such that
p~{x: wcS(x)~E:l ~
n,
.n~nO
(2.6.8)
where
wcS(x) =
sup
'x(T)-x(S)
Ih:-lCT)-h-lCS)I<cS
I.
(2.6.9)
\.
38
In what follows the objective is to demonstrate weak
converge'nce of a se'quence of probability measures' on
D[0 ,~] or DR. [0 ,1] to a 'Gaussian process, 'concentrated on
R.
.'
C[O,l] or C [0,11.
Usi~g theabQve 'remarks as justifica-
tion, we need only show that the finite-dimensional distributions converge 'to a finite-dimensional Gaussian
distribution and that conditions (2.6.7) and (2.6.8) hold.
Returning to the
~ctor-valued
o <T<
process
oo}
,(2.7.1)
We first deal with one individual U-statistic asa process
in time and consider conditions under which we are
guaranteed tightness.
We will use the following conven-
tions:
a.
'T(Xl, ••• ,Xx 'Yl' ••• 'YK ) will be abb'reviated
. 1
2
to 'T and we will define
'ST
=
['T- 6 (T)] - ['S-6(a)].
(2.7.2)
(2.7.3)
39
.:,.
where. h; ]0 ,~]
+
10 ,.~J.
wa~
introduced in
section 2·.·.6.
c.
: 'Th'eo'relU
Let
PK(S,~)
= .I'ST]K.
Assume the
~·.7.·1·:
(2.7.4)
followi~g:
1.
F and G are continuous distribution functions •
2.·
lim N =.).,
.m
.
(2.7.5)
0 <). < 1.
N+ClO
+T(xl ,. "'''K,Yl ,. "'Yx ) can be written
1
2
3.
(2.7.6)
.'
where
i =·1,2
and with probability 1,
+i(T)is
(2.7.7)
•
nondecreasi~g
in T for
i = 1,2.
4.
There exists a strictly increasing, continuous
function H on [0,1] such that
P2(8,T) ~ H(t) - H(s),
P4 (S,T)
~
(2.7.8)
(2.7.9)
B(t) - H(s),.
and
9:l(T) - 8 (8) ~H(t) - H(s)
i
i = 1,2.
(2.7.10)
Note th·at(2 .1. 9) implies boundedness of the fourth moment.·
Then the process defined as
·40
0 ( .) ,
UN * .=.{
: UN t
._.11
.0
~t ~
1·}·
{. 0 C'h' ~1..T
C ) . , 0 ! T < QCl. }
::; : UN:
= ·N
U*
is
·t~9ht
Proof:
in D[0 ,1] with respect to the uniform metric.
Condition (2.6.7) is trivial since,.withprobabi-
lity one, U;(O)
= O.
The proof will consist of finding, a
such that, for fixed
c5
(2.7.11)
£
and
n,
(2.7.12)
•
First write
(2.7.13)
where
(2.7.14)
and
(2~ 7.15)
Then the t~ghtness of
* ' i
of YN,i
c5
such that
= 1,2.
U~* will follow from the t~ghtness
Thus, for fixed
£
and
n, we must find
41
(2.7.16)
Toward this end, we
Let
.
Jn~e th~ .. followi~g
construction:
N' = min·([:..!!!..l,··[~i'}
K
. K
l
2
(2.7.17)
and define the· process
(2.7.18)
"
where
(2.7.19)
Note that ZN,(t) is an average of independent random
variables and, likewise, so is
ZN,(t) - ZN' (s).
IN'
*
*'
. * .
{N [YN,l(t)-YN,l(s)]=
E{ZN,(t)-ZN'(s)
Ic} ,
where C is the product
sample order statistics.
Also,
(2.7.20)
a-field. generated by the individual
Thus, by Jensen's inequality and.
the expression for the fourth moment of an average of independent random variables (cf. Cramer, p. 345) we have,
42
(2.7.21)
Now, since
.'
mIN"" )., 0 < ). < 1, for N sufficiently large and
and for K = max(K ,K ), and
2
l
N' ~). °N/2K. Thus
.
).0 = min().,l~)') we have
"
°
For convenience, let
C
=
3(2K/).0)3.
(2.7.22)
By assumption 3, we
thus have
'*
'*
4·
E[YN,l(t)-YN,l(s)]
<C{[H(t)-H(s)] 2 +
H
(ot) ON-H. ( s) }
(2.7.23)
Therefore, when
£
N
<oH(t) - H(s)
we have (assuming
£
< 1)
'* l{t)-Y * l(s)]'4'<..:'2C [H(t)-H(s)] 2 •
E[YN
N,
,
- £
(2.7.25)
e
.
43
,:.
.Now
.~s~UJne.
and let
p > q. such :t.hat
~
there is
H(t+p} - H(t} >
~/N.
. O! t~·l-p.
Ap =.
H(t+p)-H(t).
.(2.7.26)
0
sup
o ~t~l-p
(2.7.27)
Then consider the random variables
.',
*
'.
*
.
XN,i=YN,1(s+1P)-YN,1(s+(~-1)p), i=1,2, •••
II
where
s
is some element of [0,1] and
tive integer.
,m
(2.7~28)
m is some posi-
Let
. (2.7.29)
Then, since (s+ip) -
Isi.. .Sjl =
I~,i
E [x.._
x_~
-~,i--~,j
(s+jp) >'p
Ax
i~
j
XN,jl
and
2C
] 4 <. -e:o
[H (s+ip) -H ("s+.jp) ]
=. '2Ce:
where
for
= H(s+Xp)
(
i
1:
X=j+l
we have
2
2
AK ) ,
- H(s+(K-l)p), i=1,2, ... ,m.
(2.7.30)
(2.7.31)
44
Thus
i
~ ~l. < ','2C ' ( I A )2.
-~,1 -~,J - ' ~X4 K=J+1 K
pflx- .-x_
.1
By Theorem 12.2 of
Bi11i~gsley
(2.7.32)
(1968), this implies
\
* 1 (s+ip) -Y * , 1 (s) 1 > A'\} <'
p{~a<x.lYN,
N
1_m
'~'~4:,'2 ( ~~
, .2CK '· 2
..
AX) 2
u
K=l
2 2
., -< eX4 4 ' '. mA p ,
4, 2
where X
,
(2.7.33)
'"
depends only on the numbers 4 and 2.
Now, for , s ~ t ~ s+p , with' h (s+p) fI denoted byS+P,
m
n
m
n
= I' (4)1 S+P-4>l S)- (K ) (X ) (a 1 (s+p)-a 1 (s»
",
1
2
.
.'
< [I' (4)1 s+ -4>1 S) - (K ) (X ) (a 1 (S+P) -a 1 (S) ) ]
, p,
1
2
(2.7.34)
Thus, for
s
~ t~s+p
45
:,'
.(2.7.35)
Likewise
. . < ( m) ( n ) Ap
-
1(1
1(2
(2.7.36)
There~ore
.'
*
*..
*-',
*
IYN,l
(t)-YN,l (s) I ~ IYN,l (s+P)-YN,l(s) I
(2.7.37)
"
which implies
sup
+ fflAp,
* 1 (t) -YN* 1 (s) I ~ 3 I!laxlYN
* 1 (s+ip)
IYN
,
1<m'
s~t~s+mp'
* ,l(S}
- YN
I+
IN Ap.
(2.7.38)
Now, if
.' (2.7. 39)
We can apply (2.7.33) to obtain
46
* 'I (s+ip) -YN* 'I (8) 1 +' 'IN Ap ~ 4£}
" < ' P{3 max IYN
"i<m'
, .
,
,
" < . P{max
'i<m
IY,*N
-Y *
1 (s+ip)
1 (s)
N
" ,
I' ~' tl.
i. ",2CK'
' ,.::,.2.': m2 Ap2 •
e:
(2.7.40)
2CK ,2c5/e:~ < n.
4
Now choose c5 such that
Then it follows
that
,
p{
"
1Y * 1 (t) -Y * 1 (s) 1, ~ 4e:} ~ nc5
sup
N
N,
s ~ t ~s+c5'
(2.7.41)
.1
•
if there exists an
m Ap
= c5
m and a Ap such that
int~ger
with
e:
N
< Ap < :e: •
-IN
-
This first requires an
m
(2.7.42)
such that
,
(2.7.43)
or
(c5/e:) IN
<m~
(c5/e:)N,
which is true for all sUfficiently large N.
(2.7.44)
47
.:.,.
,
Havi~9'
found an
JO,
:then
Ap =
,<5/~
and
(2,.7.45)
. Since' His, cont.inuous and st.r.ict.ly
increasi~g,
p > O.
The rest. follows as in Billi~gsley (1968>', p. 199
* is t.:L9ht in D[O,l]
YN,l
and we conclude that. t.he process
with respect t.o the uniform metric.
The same conclusion
* and we have the desired result •
holds, for YN,2
.
...,
'
2. 8
Weak 'Conver'genca 0'£ 'a' Ve'c'tor-Va'lUed'Pro'ca's's'of
U-S't.atisti'cs
•
The above result puts in a position t.o stat.e the
next theorem.
-rheo'rem 2'. 8 .1.
Under the assumption of Theorem 2. 7.1, the
vector-valued process,
o < t < l}
- -
o '" T
< Go}
=
converges weakly in
Proof:
alo~g
Applyi~g
(2.8.1)
DL[O,l] to a Gaussian process.
property P4 of the previous chapter,
with the Cramer-Wald device, the weak convergence of
the finite dimensional'distributions is, guaranteed.
,,'
48
.:,'
,
T~9htness
,follows 'f:romthe t:!.ghtnesfl of the;ma:rginal dis-
tributions (see Whitt, ,1970, 'p. 941).
2.9 : tI'he: 'Proce's:s: B*
N
~Now
we are prepared to discuss the process
=' {1fl[g(UNl (T) , ••• ,UNR,(T» :--g(e (T) , ••• ,eR, (T» 1,0 < T < co}
l
(2.9.1)
"
"For the major result, we will need the following theorem
by Hoeffding and Robbins (l948)~
tI'heo'rem 2 _;9 .1-: 'L~t!
•
, ,
(1)
. (K)
'!S = (YN • • • YN , )
A
•
and
Y
... = (y(l) ,y(2) , ••• ,y(K~ be K-dimensiona1 random variables •
If
b)
{~}
is a sequence of non-zero numbers
convergi~g
to zero,
and
c)
h: JRK .... lR is a measurable function with a total
differential at (0,0, ••• ,0), with
Then
49·
. h/c.Lv·(l) "d·y(2).. .... d·,v.(K} \.. h'Q 0" ... O},'
. ""~"'N ::';-W N ~ ·':·'·:-:'·-W"'N :' ..(;- .\,. "';';.'-;-": . ; 1J
•,~
.'"
K
,.
(i)
t hi·.OY
~~.
'
•
(2.9.3)
We will also call upon the
. '.l'he·or'e~";2:~':9:.2 .•... Let
followi~g
",0*.=· {wO(t)
N
.
N
'
preliminary result •
0 _< t _< I}
(2.9.4)
be a random process in D[O,l] which converges weakly with
respect to the uniform metric to a Gaussian process with
.'
bounded variance at each point.
Then, for
11 > 0, there
exists a K>O such that
fl'
p{
,
Proof:
~ke
(2.9.5)
sup
IWNCT} I >K} < 11/2.
O<T<ClO
~
£,·11 > O.
Then, since
."
wN
converges weakly with respect to the uniform metric, there
exists a
arbitrary
such that
(2.9.6)
Now choose
m
grid points
, ,
I t i -ti~ 1 I < ~
p{I
for all
.max W;Ch(t.»-
'l~j<m'
Also,
J.
i.
.
t l ' t 2 ' ••• , t m £ [0,1] such tha t
Then we have
sup W;ChCt»
'O<t<l'
I
>
el
< 11/2.
(2.9.7)
50
:,'
p{ max
, '~<,J. :C:m'
--
* C.htt'J.) II' > itl ~ '..t pflwN* (h{t j » I >:K}
JWN
J=1:
'
(2.9~
8)
* , » = ,WN{T ) is ,asymPtoti~allY normal with
But each' wN(h(t
j
j
finite variance.' ThUs, for each 't. there 'exists a K. such
J
J
that
(2.9.9)
Take
'K
= max
'l~j~m
K..
J
Then
p{sup (WNCT)
"' > x} <
I
T
n.
(2.9.10)
•
'The'orem '2.9.3.
Let, g(xl ,x2 , ••• ,xR.): mR. .... m ' be a
measurable function with total differential at
o < T < co
-' -
39 (Xl ,x2 ' • • • ,xR.)
aXi
and assume
1=1,2, ••• ,R.
are continuous and bounded in a neighborhood of
"B;CT) ='{IN'(BN(T)-B(T), O~T< co}
=' {ffl[9(UN1 (T) ,UN2 (T),'''. 'UNR. (T»
-g(91(T),92(T), ... ,9R.CT»], O~T<~}
(2.9.11)
converges weakly in DrO,l] to a Gaussian process, B* under
the assumptions in Theorem 2.7.1 for the individual
S1
., 'U-sta.tistics involved.
'Proof:
We "use "the Cramer-Wald device with Theorem 2.9.
1 to show that the
ve~geto
'finite~dimensional distributions
a multi-normal.
con-
Let Tl ,T 2 , ••• ,Tx be X time
. points in [O,eo] and let
... IN(UNl (TX) -9 1 (TX) ) • •• IN (UNI. (TX) -9 t (TX) ) ] •
(2.9.12)
.'
Then
:N is an
"I.·X-dimensional vector of U-statistics
and, by property P4 of the previous cJtapter, we have condition (a) of Theorem 2.9.1:
'!l!
a)
!N
where Y is an I.. X-dimensional normal.
c)
For arbitrary constan~s
Also,
and
Then
°1 ,° 2 , . " '''Ie'
let
h: m1K _.... lR is a measurable function with total
differential at (O,O,.~.,O) and with
52
C2.9.13)
Then'we have the conclusion
.
..
0·' t .
"
•
'D l·K
....
CO)
t h! Y 1.
i=l 1.0
,
C2.9.14)
which is normally distributed since Y
... is multivariate
normal.
Using this technique we also obtain the covariance
*
structure of the' process B.
Letti~g
K=2, S < T,
Q
1 = 1
a 2 = 1, we have
:C2.9.l5)
53'
'.
~
where'
(2.9.16)
. where' Is and
IT
were 'defined in Theorem 2.5.1 and
is computed similarly
chapter.
us~g,
I ST
Property P4 of the previous
Then, accordi~g to the theorem by Hoeffding and
**
'.'
.,
.
Robbins, B C8) + BNCT) is asymptotically distributed as
N
21
21 21
I h' yC1') N(O I
o_I hi'OhJ!O cov(yCi) "y(j». C2.9.17)
i=l iO
......' i=~ J-l
'
0
Similarly, we have
Var B* (8) =
•
~
~h!oh!O
i=l j=l J.
cov(yCi) ,y(j»
J
(2.9.18)
and
.21
21
(")
()
Var B (1')=
E
I h!Oh~O cov(y J. ,y j ) .
i=1+l j=R.+l J. J
(2.9.19)
Now,
- Var B • (S) - Var B * (T)}
=
I.
21.
I,
E h! h! cov(y(i) y(j»
i=l j=R.'+l J.O JO
.'
•
For tightness, .we, have
(2.9.20)
54
:.'
o -< T -<'
~}
.
'1· .•
"1, •
=".! IN.£ 9 (8 ~ (T) +' '/iUN1,(T)
,8 2,(T) +, '-mUN2 (T) , ••• ,8 t
(T)
•
,
,+ "1'
.'mUNt(T»g(8 1 (T),8
2 (T), . . . ,8 t (T»], O<T<
~}.
(2.9.21)
.By
* (T) is equal to
the mean value theorem, B N
t
E
.'
~g(Xl,X2"·'··"XR.)
ax.
i=l
].
U
•
(T)
Ni
YN1(T)'YN2(T)' ••• ~YN(Tj
......
(2.9.22)
•
where
yNi(T) is between UNi(T) and
8i (T}.
demonstrate conditions (2.6.7) and (2.6.8).
Again,
•
(2.6.7) is trivial since we have
all i.
We wish to
UNi(O} = 0 w.p.l, for
Now,
ag (xl' • • • ,xR,)
*
UNi(T}
aXi
, YN 1 (T) , ••• 'YNt (T)
t
:i.;1
,a,g(x1 ,·,·· ,xR,}
aXi
,
I' *
. UNi (S)
YN1(S},···,yNt (S)
•
*
IUNi(T)-UNi(S)
I
N (T)
5S
~'
(2.9.23)
,-
By the assumptions on, 9 and the
dual processes
. {U*
t~ghtness
of the indivi-
'.f,
Ni (T)} i=l ' we can make the first summation above arbitrarily small fo~' Ih-l(T)-h-l(S) f < 6
with high probability.
For the second summation we use
Theorem 2.9.2 along with the continuity of the partials of
"
9 to give the needed result.
'.
CHAPTER XII
THE PROGRESSIVELY .TRUNCATED WILCOXON STATISTIC
3.1 ' 'XntrodU:c'ti'on '
In many experiments, i:he primary paramater of interest
is i:he probability that Y is less than X when X and Y come
from continuous distributions F(x) and
G(x)
respectively.
This parameter can be expressed as
•
00
P(F ,G) = '0 G(x) dF (x)
(3.1.1)
e
and can be estimated from a random sample of N elements
-/WO -.JIlrJ'Jf/Ui
(m from F and n from G) by the
U. - JlakAJc.,
Mann-Whitney~±ooHOn &ta-
t:ist1:C'
1
n
m
I
I
ron j=l i=l
+(X,Y) =
, (XJ.. , Y .) ,
J
1
Y<X
o
otherwise
(3.1.3).
As was noted in Chapter I, this
i8 a two-sample U-
statistic with degree Kernel (1,1) and can be truncated at
a time point T by setting
It
57
°e
with
0
yl(
1
X~
T
(3.1.5)
0
0
otherwise.
Then UN(T) is an unbiased estimator for
= Pr (Y
:c X < T) •
(3.1.6)
'''.
The empirical distribution functions can also be considered as progressively truncated U-statistics and can
be employed to form an estimator of the parameter
. peT) = Pr (Y < XIX, Y < T) ,
(3.1.7)
i.e., the conditional probability that Y is less than X,
given that both are less than T.
This is estimated by
the statistic which we will call
PN(T) given by
A
0
(3.1.8)
where Fm and Gn are the empirical distribution functions
of the X's and Y's respectively.
°
This chapter will deal with the properties of these
0
58
tw~~sti.mat~rs" ,UN(T.) and ~N<'T.l ,~n .terms of distribution
and also in terms of prediction for the parameter P(F,G).
3.2 .' 'Di=s:tr'ibu·t:ion: ·of
Since UN(T) is a
.oNeT) .
U~statistic,
the properties given
in Chapter I and the theory develoPed' in Chapter I I apply
directly and we have the
followi~g
. 'The'ore'nf '3, ..2'.1.' The statistic
result:
N (T)-1P(T» is asymptotically normally distributed with mean 0 and asymptotic
.'
variance
ffl(U
.....
, 1.00TG2 (x) elF C.x) ,-.1P: 2,<: 1:')
V:T
=
-:----~:------}.
T 2'
2
.
.2
1"00
F. .(x) .dG.(x).-F .·C,T).G C.T.) .+.2.1P. (.T.).F (,T) .-.1P.. (.T)
(I-X)
+
Proof:
(3.2.1)
The asymptotic normality follows from property
P3 of Chapter I.
For the variance we compute
':' I_~ [/;dF(X)] 2dG-21P (1') 1_:0 l;dF(X)dG(y)
2
+ 1P (T)
==
1:
00
I':' dG (y)
,
(3.2.2)
[F(T)-F(y) ]2 dG (y)=21P(1') I_Too [F(T)-F(y) ]dGCy)
+ 1P2 (T)
=
I_TClO F2 (y)dG(y)-F 2 (T)G(T)+ 21P(T)F(T)-1P2 (T)
4It
59
·e
L;lk.el(ise ,
Since ·the Kernel is of
(3.2.1).
d~9ree
(1,1) we obtain expression
Note that, as T approaches infinity, VT coin-
cides with the variance of the asymptotic distribution of
the full-sample Wilcoxon statistic.
We now form the process
·e
and compute its covariance structure.
For
S < T we have
.=
(3.2.3)
e'
1P (S)
when
a.
1/1 (T)
b.
,S
G2 (x) dF(x)
-00
i,&k, j'& 1,
when
i=k, j'& 1,
61
,
1
l'
"e
t
'v(mnl+ V
- V
at~'" S
. ST"
+'."lIt (:$:) :U~ C:Tl-:F ($:) ].
(l:--Xl
"+'
ji e.S)' ['lit e.'S')
'--r: (T:)" 'J
. ). (1:-->'
r,
(3,;2'.6)
for
S
T.
<:
utilizi~g
Now
citly
assumi~g
h: [0,1]
+ [O,m]
followi~g
a
the :results of Chapter II and impli-
one~to-one
bicontinuous map
we have the
result:
, ;rheorem: '3,..2".2.
Assume F and G are strictly increasing
continuous distribution functions.
.'
2~6,
as introduced in section
* :' . , is weakly
UN"
conve~gent
Then the process
in D[O,ll to a Gaussian process
with ,covariance structure given by VST •
...
, Pro'of. We need only satJ.sfy the assumptl.ons of Theorem
.
.
2.7.1.
Now, we can write
~ ~T(xJ..'YJ.)-~(T»
(3.2.7)
is positive with probability 1.
Thus assumption
=
where
~T
.
~
{N(JL
Inn
j=l i=l
3 is met by setting
~l,T = ~T
and
~2,T =
O.
Now, we
must find a strictly increasing, continuous function H
such that, for
S < T,
).12 (8 ,T) < H.(T} " - H" (5) ,
).14 (S,T)' -< H{T.) - H(S),
'
and
~(T)-lPC5) ~H(T)
- H(S) •
(3.2.8)
1
62
:.,'
FiX'st considex-
.\
~
(3,.2.9)
Now
1
,0 '
Thus, for any
K
~
y -:c X,
s <X<T
otherwise
,0 we have
(3.2.11) ,
and we obtain
(3.2.12)
But
(3.2.13)
so we conclude
(3.2.14)
Now
j
63
Zo'
..
=[f(Tl-.(S)J~4If{T)-f(S)j2+6[.{T)~f(sJ)3
which 'arrives from
.'
expandi~g.and usi~g
(3.2.11).
Thus
< lJ 2 (S··,T)
~
•
f(T)-f(S).
When F and G are strictly
increasi~g,
(3.2.16)
fis also strictly
increasing and is a continuous function.
is satisfied by
setti~g
H(x)
Thus, (3.2.8)
= f(x).
The estimator
A
.•.
PN(T)
(3.3.1)
= FM(T)G
(T)
. n
arises as a function
tis tics •
UN (T).
g: R3
For (xl' x2 ' x 3 )
£:
-+
3
R of three truncated U-sta-
R '. 9 is given by
64
Note, however, that this function is unbounded when
or x
3
is near O.
x
2
Examining the behavior of
take, for example, .a sample whose first six order statistics are
Y
Y
x
Y
x
Y
Then
". T < T 4
T 5. T <T
0
A
1
PN(T) =
4
3
(3.3.3)
S
T5~T<T6
"4
regardless of the total sample size, N.
the estimator is unstable for small T.
our purpose is to estimate
Thus we find that
However, since
Pr (Y < X IX, Y < T) as
T~
CQ
,
A
it will not be inconvenient to restrict the domain of P
N
to larger values by waiting until we have observed a certain percentage of the total sample; i.e., we might only
estimate peT) after we have seen (k,q) observations, where
k.?:. ~ £0
. and
q.?:. n £0'
TO
=
Note that, for T in this
£0 and Gn (T) ~ EO.
-1£
-1 £
min(F (0/2),G (0/2».
range, we have F1I\(T)
Now let
£0 > O.
~
By the
Glivenko-Cantelli theorem, we can choose nO arbitrarily
small and make N sufficiently
la~ge
such that
65
_(3.3.4)
and
.
-
.
£0 ~o
P{sup IG(T) -G{~) I > /2} < / 2 .
Tn'
(-3.3.5)
Thus, for T in the -ra~ge where Fm(T) > £0 andGn (T) > £0 '
(3.3.4) and (3.3.5) imply
P{F(T) >' £0/ 2 , G(T)- > £/2} > l-n
,-
......
O
•
(3.3.6)
the- process P * - we
N
will need to recall that the theory has already been deFor the major result
concerni~g
veloped for the empirical processes
= {IN'(F (T) - F(T» , O<T<ClO }
(3.3.7)
(3.3.• 8)
and we have the
followi~g
(cf. Puri and Sen, 1971, p. 36):
Let
h ( t) = T =F- l
(t),
t
£
10 ,1] •
- (3.3.9)
66
·
IiOtFmF- 1ttl
-tl, ...f) ·'Z * [tl '
,
where .' Z*
is the Brownian
t £. EO ,1.J
(,3. 3 .10)
on' [0,1,], i.e., :the
Br.i~ge
.
*
finite dimensional distributions of . Z
are Gaussian
with .
*
E Z (t) = 0
*
*
E[Z (t·)Z (8)
Is <,1:)
= ,sC'l-t)
(3.3.11)
[0,00)
(3.3 .. 12)
Thus
.'
......
m(F (T) -FeT»
m
where
"
.
Z'
T
£
is Gaussian and
E Z' (T) = 0
. Theorem 3.3.1.
JJ z ~T)
E{Z' (T) Z' (8)
Is <T}
. F(S')'['l~F('Tl]
A
(3.3 ..13)
=
The process
(3.3.14)
conv~rges
weakly to a Gaussian process
p*
, with
67
·e
*
*
_:
Y(S·)'Gun··
'.
. COY (p. (.5), P. (T)). -. ,,.{T.)(H.T.}" VaX" p
* ($)
. " . . : : : ; p ('5): . . . . ::. [F' ('1') ...·F (Sl -] .
.
+p{sJG{S.) F ('1') GeT)
(l;A)
(3.3.15)
for
S < T.
Proof:
The
conve~gence
of the finite-dimensional distri- .
butions follows from the proof of Theorem 2.9.2 and the
covariance structure is computed similarly.
Now, for tightness, consider the function
. g(x,y,z) -~'x
yz .
G('1'l+
JL
IN
Then ....
G;('1'»
- g(W(T),F('1'),G(T»)
(3.3.16)
By the mean value theorem, this is equal to
(3.3.17)
where
'YNl ('1') is between 1JI(T) and UN(T),
'Y N2 ('1') is between FeT) and Fm(T),
and
'YN3 ('1') is between GeT) and
G~(T).
68
We wish .to demonstrate ,conditipns ,C.2.6.7} and (2.6.8) of
s'eotion 2'.6. 'For condition
.(2~ 6. 7), ~e
need an a ,for
each 'n >' ,0, .s'uch :tha:t '
where x denotes a sample path.
Now
X(T ) is of the form
O
(3.3.19)
-".
which
conve~ges
variance.
to a normal distribution with finite
Thus condition (2.6.7) holds.
Now consider
b
'. < ' t A
- i=l i
, where
,e
(3.3.20)
69
·e
.:..
- . (3. 3.21)
(3.3.22)
(3.3.23)
"
and
We will deal with the above terms individually, and will
use the fact that all three processes are tight with respect-to the uniform metric.
First consider AI.
Given
and an Nl such that, for
Also"there exists an
T), e:, > 0
we can find a «5 1
N > Nl
Nit such that
1
2
P{YN2 (T) Y (T) ;. '£40l ~ I-T)/12 when N > Ni.
N3
Putti~g
these two probability statements
t~gether.
(3.3.28)
gives us
70
;.'
...
: : '1 : : : , , .
.(T)y ,(T)
,sup
N2
N3. lli-l,('l) _h""!l (S)
pry
.
~
*
*
·:luNCT.),-UN (8)I
I < <5 1
·,2
.,
. £ : '£,0, :.' 4 '}
> 6
,.2 .
£0 '
--T--
=
.' ...
pry
~,
: : :J.: : :.
(T)y (T)
N2
N3
. .
*
.,
Ih-
sup
1 (T)
1 (8)
_h-
I < <5 1
'n.
i
for
. £'
(3.3.29)
> 6 ' N > N1 •
<
*
IUN (T)-UN Cs)I<6}
N > N2 '
, (3.3.30)
N > N3 •
(3.3.31)
and
< n·
6 for
Now for
p{
A4, AS and A6, let
sup
s£[O,oo]
lu:CS) I ~K~}
K = max{Kl,K2,K~}, where
> 1-'
'it '
(3.3.32)
71
'e
p{
. '$&
sup· IF:CS)
[0 ,.~J.
li~} > ~-" ~"
(3.3.33)
'.
and
p { ,sup I~ (8) 1 < K~} > ~-' ~ '.
S£[O,ClO]
'(3.3.34)
K ,1<2 and K are. guaranteed by Theorem 2.9.2•.
1
3
Dea1i~g
w·i th 'A4' we have
',' . . .'. , '1 . . . ',' . ; . . : : . :1 . . ,.
YN2(T)YN3(T)
"
.
"~
. r I'
-~YN2(S)YN3(S)
'YN 2 (S) YN3 (S)-'YN2 (T)·yN3 (.T)
KIYN2(T)YN3(T)YN2(S)YN3(S)I
*
1
UN(S)
" 'n
.w.p. > 1-"j"b. (3.3.35)
This is in turn
Y
(8) Y ,(S) -F.(S.)G(S)
N2
N3
< KIYN2(T)YN3(T)YN2(S)YN3(S)
l
+ K
1
F (T) G(.T) -YN2 (.T) 'Y
N3 C.T).
YN2(T)YN3(T)YN2(S)YN3(S)
We can find an N such that, for
4
r.
(3.3.36)
N > N4 , T £ [TO ,ClO] ,
p{ 1YN2 (T) YN3 (T) -F(T) G(T) r <'i~k
£4
..
.'.Jl > 1-' '31
(3.3.37)
72
and
'(3.3.38)
These two probability statements "imply
. . 'Y
(S}:Y
(S.) .-.F (S) .G (,S.) . . ..
.' .
N2
N3 ,
, ..: 1< e'} >l-..!L·
YN2 (S) Y (S) y
{T)·y ,(T)
18
18
N3
N2
N3
p{xl
(3.3.39)
and
"
'YN2' (.T) 'Y
(.T.) ..;;.F (,T.) ,G (.T,).
NJ
p X YN2 (8) Y (S) Y (T) Y (T)
N3
N2
N3
'{ ..I
..
.,.,',. "
I
. .
el
. ..
n .
< I8 > 1- I8
(3.3.40)
•
Now, the function F (T) G(T), T e [O,co], is equal to
F(h(t»G(h(t», t e [0,1), which is uniformly continuous
on [0,11.
Thus there exists a
Ih-1 (S) _h- l (T)
I < ~4
~4
such that, for
'
.
,
IF(S) G(S) - F(T) G(T)
"e
. 4
eO
I < 18K·I6 •
'(3.3.41)
This, combined with the above, gives us
'n
P{A4 > ~} < -
6
(3.3.42)
when' Ih-1CS)-h-1CT) I < ~4 ' N > ~4·
~gain, we can find a ~s' NS '
~6' and N6 such that
73
and
(3.3.44)
. ' { ~i"}6
Now let u~ = m1n
i=l
and
1 1=
IP;CT)-P;CS)
sup
p{
.
"6
N = max{N,}, 1.•
Ih-1 CS) _h- 1 (T) ,I ~ ~
"
1 < E}
6
> p{
sup
Ih- 1 CS)-h- 1 CT)
-
Then
1<
t A. < e:} •
~ i=l 1
"
sup
A 3 < ~}
1
1
Ih- CT)-h- CS) I
"
>
1-n.
(3.3.45)
So we have condition C2.6.8):
IP;CT) -P;CS)
p{
Ih- 1 CT)-h- 1 CS) l<c5
I > e:}
< n.
C3.3.46)
f
3.4 ' 'The" Fu"nc"tic>n's' " "lfCT) , 'a"nd poeT)
In order to be able to estimate
WCT) and peT) in
74
..
~
. ,terms ,of possible prediction ,fo;(. ,the parameter
p,{F,G), ,
we 'should have 'a feeli~g for the behavior of these parameters as functions of, time and, in particular, ,should '
know their shapes under, different distribution functions
F and G.
Toward this end we look at
1f1(T) when F and G
have continuous, bounded densities f and, g, respectively.
For the purPoses of prediction, we 'need not assume that
the random variables X and Yare positive.
Now
.'
C3.4.1)
fI'
which is continuous and strictly
F or G is strictly increasing.
increasi~g
whenever
Also,
00
'
lim 1f1CT) = pCF,G) = 1_00 G(x)dF (x) < 1.
T..-oo
'
C3.4.2)
Thus the function has well-behaved asymptotic behavior.
Also note that the function is differentiable with derivative
",'
1f1'Cx) = G(x)f(x)
C3.4.3)
Now consider the function peT), which has the same
asymptote as
at 0, for
1f1(T).
T > TO > 0
Altho~gh
peT) is not differentiable
it has a bounded and continuous
"e
75
;.,
der,ivative which can be e.xpre ssed as
P'
(x) --'
'f(x) " '"["1- {")"J _', 'g("x.}. (") ".
p x
- G (x p x . "
(3-.4.4)
"F (x)'·
Thus we see that peT) is not necessarily monotnic and that
it could have "several local maxima and minima •
.. .... ... . ....
3~4.1 ' :..p'(T)' :and P{T); ¥ori '!'he' 'Logistic Distribution
. -.
I.
In the case when F and G are both
l~gistic
distribu-
tion functions, we can simplify the expression for
."
..peT).'
We let
F(x)
"", ~ , , : '1"" ; " , "
= 1+e-(x-~1)7S1 '
_00
< Sc <
00
(3.4.5)
_00
<x <
00
(3.4.6)
and
G(x)
." ..1""·-
Then we have
,
(3.4.7)
which gives us
..p (T)
(3~4. 8)
76
1P (ttl, .deJ?ends on ;F(T), .the .ratio of the
.. ..
scale parameters, .and (J!2~}1il/$'2'.· . Without loss of infor-
,'rhus, ,we see ,that
mation, we: 'can assume' s2 ,= ,1 and
P2
,0.
=:'
Then
1P(T)
and
s
't' ~
F( 'T)' , .....
f..'
. , .. ,.
" ... , ".
'dt .
s1" 'l-p .
s,
.
, . , , , .'
, . 't··' ..e,·,·
'-1..' ' , ,1 ('I' 't')· ..... . . , , ,
- , .," , ',.
o. .,
peT)
=
.
...
s
F(T) [ '"
's' ,-p"',. . 's ] •
[F(T) l"l+e 1 [l-F(T) ] 1
',;, , .. ' . '[F(T)'l ,1 , , , ....
"
We can evaluate the above in
int~gral
(3.4.10)
for'various
values of T to determine the shape of the function.
Figures 3.1-3.4 show the results of
and then peT) against F(T) for
PI
= -2,-1,0,1,2.
s1
plotti~g
= 1,2
first
1P (T)
and
Note that, except in the case when
1 = 2 and P 2 = --2, tlle asymptotic behavior of the curve
is mostly determined by the time F(T) reaches .75. This
8
indicates that, when F and G are not vastly different in
terms of scale and location, prediction
~ght
?e feasible'
when the underlying distribution is logistic.
3.4.2
1P(T) and peT) For The Normal Distribution
In the case when the random variables are normally
distributed, we can also derive expressions for
peT) which can be evaluated numerically.
For
1P(T) and
77
:,.
VeT) FOR LOGISTIC DISTRIBUTION
. . . . . - ----------------------s
Fiaure 3.1:
'!' (T).
Ratio of Scale Parameters = 1
~
.'
•
.00
.25
.50
1
.75
F(T)
Fiqure 3.2r. Ratio of Scale Parameters
=2
...T----------------------.. . •
v (T)
lI)
•
•00
.25 .
.50
F(T)
.75
1
78
peT) FOR LOGISTIC DISTRIBUTION
Figure 3.3:
Ratio of
Scal~
Parameters = l'
O-+---t"'---t---.......
--"i---~--+---...----I
•
.00
.25
.50
.75
1
F ('1')
Figure 3.4: Ratio of Scale Parameters = 2
. . -r--------.-----------------
P ('1')
II)
•
~-t----r---r--...,..---t---..,...----t--. . . . " t----4
.00
.25
.50
FCT)
.75
1
79
;.'
'.
,.' ' . 2'
T .' .-,l.x~,lJ'11 "
F ('1'1 =
. ' . '" e·
:dx
0' 1 /2 11" - 0 0 , 20'~ ,
, . , :1
b
•
,
'
,
.'(3.4.11),
,
and
""
)2 '
. .-. C~-}12' , .
, . . <t ' " ',T
GCT) =
,
'
0'2 /2 11'
we let
s
= 0'1/0'2
Ie,·, 2
dx
'-00
C3.4.12)
20'2
,
lJ = ClJ2-lJl) /0'2 and
'
.
T' = CT-lJ 1 ) /0'1 '
with
.'
C3.4.13)
•
We can assume' s > 1
and
lJ > O.
Then, from the calculations in Appendix Awe have
111
(T)
~
1
-}
2
t ('1" )
1
o
2
-~'-- t(I2T' - x/l2)dx
2fi
lJ2
+.,i.. / ' e:I 1+.,2 { .
12Tt
-I.
1.1 e -1/4, x
,(1+z2)
'ZII
(1+z2) 1/2
.. -~(A+z2T·-h:i2
e
.
"
}dz
( 3.4~14)
12n
Figures 3.5-3.8 were thus produced similarly to
.e
Figures 3.1-3.4.
All of the, graphs are smooth and well-
behaved once
is beyond .10 or .15.
F(T)
However, we see
80
~(T)
FOR NORMAL DISTRIBUTION
Figure 3.5: Ratio of Scale Parameters = 1
.........-..-....._----------------------,
~(T)~
.
.00
.50
.25
.75
1
F(T)
...
l' (T)
Figure 3.6:
Ratio of Scale Parameters = 2
11)
•
.00
.. 25 .
.50
F(T)
.75
1
e"
81
•
peT) FOR NORMAL DISTRIBUTION
...
Figure 3.7:
Ratio of Scale Parameters
=1
"
11,-2~v
P (T)
Il,A/
.....
II)
•
JJ... 0
.AI, =- ;
.......
-...
N, :_~
,"
~
"'.
•
•
•
.00
.25
P (T)
I
* •
.50
•
I
.75
FeT)
...---:_---_._---Figure 3. 8:
"
•
I
Ratio of Scale Parameters
1
=2
..
II)
}ol,'O
u,s.-/
Itjhl.
•
0
•
.00
.25
.50
F(T) "
.75
1
82
.
that in this
would
. .
cas~, pX"edJ"ct~on (In
r~quirewaiting for
'"
the basis of PN{T)
a large peicentageof the
•
sample.
3.5 ' p're'dicthig' P(F,:G)' .
At various time points Tl ,T2 , ••• ,T , we not only
k
haveestimates UN(T.),
F m(T.)
andGn (Ti ), ·i=1,2, ••• ,k,
~
~
but we also have consistent estimates of the covariances
of these 3k statistics
thro~gh
structural components.
the use of Sen's (1960)
Furthermore, Sen (1977) has ex...•
tended work by Arveson (1969) to show that jacknifing
functions of generalized U-statistics.provides .strongly
consistent estimates of variance of the asymptotic distributions of the functions.
..
Thus we are in a position to
,use weighted least squares to model the estimates of
~(T)
and peT) as functions of F(T).
In particular, since
these functions are in general
well-behaved and approach
asymptotes as F(T) approaches 1, we can model them as
polynomials or sqme.· smooth functions in F(T).
We have
the following situation when modeling polynomials and.
using the method of components:
•
•
•
•
•
,
•
(3.5.1)
•
•
83
X
w
FJtilT1 } • •• IFJIl.IT.1 }]r
FJIi (T 2 ). • •• [ FJIl CT 2 )]r
1
.:'
=
:1
1
•
•
1
•
1
•
•
(3.5.2)
r
F JIl ('l'k) • •• IF.. (Tk ) ) ..
1
"'"
,
.
(3.5.3)
'
•
..
We a!!sume
•
'
(3.5.4)
and
A
!:N = X
r+
~2
(3.5.5)
with
(3.5.6)
Now, E1 is the dispersion matrix of the asymptotic distri-
•
A
bution of
~N
and we can estimate its (r-s)the1ement by
84
. : 1:
. v rS
= m(m-1)
•
.
-1/1N (TS )]
., '1'
'l
+
' .. n
E
[n(n-1)
m
E
j=1 m i=1
~T
..
'1
m
(x.,Y·)-1/IN(T )] [- E ~T (X.,Y.)
r 1 J
r
m i=l S 1 J
(3.5.7)
Likewise,
~2
is the dispersion matrix of the asymp-
totic distribution of
.
~N
and we can obtain, by means of
Taylor Series expansion, an estimate of its (r-s)th.
element.
When
T
•
r < TS we have
n'lm
E [- l.
j=l m i=l
'T
r
(X.,Y.)
1
J
•
85
A
--"'N(T ) ] [C(Tr-Y ) -G (T )]S
j
n
r
(3.5.8)
where we recall the function
,-
C(x) =.
1
x>O
-,
0
(3.5.9)
'-
x<O •
Alternatively, we can use the ja~knife technique to
produce an estimate of
~2.
(It should be noted here
that jacknifing is identical to using the method of components in obtaining an estimate for ~l.) We let
g: R3 +R be the function g(x,y,z) = x/(yz) and define
the vectors
O·
.
.
'
~N = {g ( UN (T1) , F.'J1l(T1) , Gn (T1) ) , • • • , 9 ( UN (Tk l ,~1Q (Tk ' Gn (Tk ) ) } ,
(3.5.10)
~~~:~{g(U~~i(T l ) ,F~_l (T l ) , Gn(T1 ) ) ,. • • _,g (U~~i (T~)
,F;_l (Tk ) ,
(3.5.11)
and
86
(3.5.12)
where
1°
. UN~11(T)
=·"I.:I.'m
-1 t
m- . n k=1
n
E ~T(X,.y.)
j=l:'
-It J
. (3.5.13)
k;!!
(3.5.14)
i
F _ (T) =
m 1
. '1
m=r
n . .,
t
C(T-~),
(3.5.15)
k=l
k~'~
and
. 1
.n
t C(T-Yk ) •
- n=r k=l
Gnj_·l(T) -
(3.5.16)
k~j
Next let
(3.5.17)
and
e• J. =
n
eO -N
(n-l)
ei
.2-n-l
•
(3.5.18)
Then the jacknife est'imator for e(T) = (p(T!) !=I, ... ,k)is
e.iA
\
87
.~.
eN'
~ ,(.,f1'e.,...... +, 'f
... .
. 1
Jf "
,
'1-
~-
,
,
.
~
.
e)'.l/N
1".
(3,.5.19)
and Arveson (1969) has proved that
Nl / 2
[e...N - p.(~»)
...
(3.5.20)
is asymptotically normal with mean
matrix
.'
e.
...1.
0...
and covariance
~2.
If we denote by Dl the sample covariance matrix of
and by D2 the sample
covariance matrix of .... J
then
,
e.,
Sen (1977) concludes that, under the condition that g have
bounded second partial derivatives in-a'
(1f'(Ti ), F(Ti ), G(Ti ) for each
ne~ghborhood of
i=1,2, ••• ,k,
(3.5.21)
Thus, letting
D
...
= m...
1 01
+ 1 D2
n...'
(3.5.22)
we can model
(3.5.23),
with
~3 -N(O'~2)'
,.
since we are assuming that ~N is co!"-
puted at time points for which 9 has bounded second partial
88
.:.'
.' ,de;J;j.vat1ves.
,Dependi~g
on th.eindividual, V-statistics
involved. and th.efunction,g, :th.is' method could, in
,general, provide 'a reduction in ,computation.
Now
,we~ghted
usi~g
these es'timates',
~e
can do a standard
,least squares analysis ,to arrive at ,the
predic~
tor equations.
Since we can assume 'a known covariance matrix in all
three of the above cases, we also have a.goodness-of-fit
test"given by
"
(3.5.24)
, or
(3.5.25)
or
(3.5.26)
Note, however, that these tests only tell how 'well the K
data points we have chosen fit the lines modeled to them
in the region for which we have data and not in the extrapolati~g r~gion.
3.5.1
Modeling In The Exponential Case
For the purpose of
compari~g ,the
curves obtained by
89
'",
extrapolati~g
from
A
1/J·(T.l and from .J? (T), .random variables
were. :geIierated from .th.e·
:follow~g
eXponential distribu-
tion's: .
F(x) = l-e~x
o <x <
co
and
(3.5.27)
G(x)=.l-·e -x12
The true curves,
'0 ~ x <
co
1/J(T) and P(T),are easily obtained as
(3.5.28)
•
and
,I.'
peT) = 1/3-e- T+ 2/3e-3/2T)/1_e-T..;e-T/2+ e~3T/2).
(3.5.29)
Note that
p(F,G)
= lim
1/J(T)
T-+oo
= 1/3
(3.5.30)
.. Figure 3.9 shows the true curve plotted along with
'"
estimates, 1/J(T
), obtained from independent samples with
i
A = 1/2 of sizes N
=
200 .(connected line), N
= 50
.(/1's) and
N = 20 (X's).
F~gure
3.10
simil~rly
shows peT) plotted along with
estimates from these samples.
The erratic behavior is
90
evident at the
b~ginnj.!lg,. ,b.ut.
for .the sample si.ze$ 50 and
200 these :estimatesseem to be 'closer to the true curve
as F. (T) , gets' close :to 1.
For the purposes' 'of prediction, a random sample of
size
'N =
.2000 with " ). = :1/2. was. generated and six time
points
Tl ,T2 , ••• ,T6 .were 'arbi"trarily chosen on which to
base the modeli!lg. A polynomial of d~gree two was fit to
the six points.
F~gures
3.11 and 3.12 show the results
of using the. method of components and
"
choos~g
six time
points, the maximum of which falls about the point
'"
F(x)
=
.70.
Figures 3.13 and 3.14 arise similarly, but
based on estimates at six other time points, the maximum
bei~g
about F(T) = .87.
W~
see that, in this case, the
predicted equations are much closer and, in fact, the
estimator
,.,
~(T)
quite accurately predicts P(F,G).
How-
ever, peT) does not seem to be a reliable predict~r.
The jacknifing technique for modeling peT) produced,
as expected, very similar estimates.
the results of all three trials.
Table3~5
presents
e
e·
e
,
-FIGURE 3. 9 :
PROGRESSIVELY TRUNCATED WILCOXON STATISTIC AND ITS MEAN CURVE
';
FOR EQUAL SAMPLE SIZES'a: 10 25, RND. 100
WHEN f(X)-1-EXP(-X) AND GeX)-i-(1-FCX) ••• 5
'..-l
i
,
.-
"(T)
LI')
o•
o
1l'I lef . .
'.00'
.25
.50
F(T)
.75
1
\0
I-'
92
Lf)
........
•
-"
anrz.
E-t
O
x
x
•
x
<J
•
<3<3
x
•
M
f===t==~f:====t=====J~o
t
S.
0·- •
-0
E-t
Qt
-
e
e
FIGURE 3.11:
FITTED CURVE VERSUS MEAN CURVE VERSUS DATA POINTS
PROGRESSIVELY TRUNCATED WILCOXON STATISTIC
WHEN FCX)t=1-EXP(-X) AND G(X}e::1-(1-f(X))4$.5
.-I
,
,
....
'Y(T) LJ')
•
..
=J~i
•
.00
p£'+
.25
I
A,?: ,
.50
F (T)
.75
I
I
1
\0
W
FIGURE 3 .12 :
FITTED CURVE VERSUS MEAN CURVE VERSUS DATA POINTS
,.
CONDITIONAL PROGRESSIVELY TRUNCATEDWILCOXCN STATISTIC
WHEN fCX)r=1-EXP(-X) AND GCX)1:I1-(1-fCX)).".S
...-c
.P(T)
i
,
~1
==
ts:
8:
lSI
lS
l
ZS;
-
C)
•
I
.00
I
I
.25
t
I
.50
I
I
.75
I
I
1
e
e
\0
~
F (T)
e
e
e
tit
."
F.IGURE 3.13:
':
FITTED CURVE VERSUS .MEAN CURVE VERSUS DATA POINTS
PROGRESSIVELY TRUNCATED WILCOXON STATISTIC
WHEN f(X)=1-EXP(-X) AND G(X)=1-(1-FeX»)••• 5
.....
i
•
'f(T)LI)
•
•
oj:
•
. 00
=
i
I
.25
lit I
I
.50
F(T)
7
' I
I
.75
I
I
1
\0
VI
'.
FIGURE 3.14:
FITTED CURVE VERSUS MEAN CURVE VERSUS DATA POINTS
I:
WILCOX~ STATISTIC
G(X)~1-(1-FCX» •••5
CONDITIONAL PROGRESSIVELY TRUNCATED
WHEN f(X)a1-EXP(-X) AND
~
•.
i
.. ,
P('1')~f
..
is
t
~
ts:
A
AA
==
I
..
_.....
97
.:.'
,Res'u1ts of'
Mode1i~g
. ,E,s.t.i~a,t.o,r:,' ',Fii:(.T,6,>, .I~.t~,r.c,e~,t: ,.S.l,o~.e .. 'oO~,dfrf~ ~~,c.. "t' "p.r':d{'Fic'Gte)·.~ ,G!flF
. , . ' . ' , . " , " " . " , .. ",. '. 'oe' . ·J.cJ.en'
,"$:..
,.
:,'
.
' ..... 035
.337
.305
1.32
.574
-.446
.,342
.470
.849
.692
.574
-.444
.341
.471
.844
1P
~866
.023
. -.130
.435
.328
.040
P
.866
.512
-.149
.052
.415
4.76
.866
.512
-'.149
.052
.415
4.75
1P
'.692
'.0026
,p
.692
,.
,.
e
,.
,.
,.
e
'"
,
Reca11i~g
that the true parameter, P(F,G). is 1/3, we
•
find that, usingpabout 86% of the F sample, we can produce
a good prediction estimate by mode'ling
~N(T).
;.'
CHAPTER IV
THE TWO-SAMPLE EXTENDED CASE
A natural extension of the statistic
PN(T)
= '-looTGn (x)dFill (x)
(4.1.1)
.....
is
s~9gested
by the experiment in which we wish to esti-
matepr(Y < X), but we' do not observe"the samples
.' {X1'}~
1 and '{Y.}~ 1 directly.
1=
J )=
*n 1
and '{Y.}.
) J=
where
Rather, we observe' {X~}~
1
1 1=
X,* - F * , Y*' - G* , and
1·
J
(4.1.2)
with
01 and
02
known.
some life-testing trial
For instance, suppose that in
X'l
and
1
'
X'1 2 are times to failure
of independent components, each with distribution F, and
the
~ntire
has failed.
and
system fails when either one of the components
Then, in this case,
...
....
99
.:"
,(4.1..3)
Sen (1979) propos.ed a linear rank statistic for this.
type 'of situation and developed the asymptotic theory for
both 'the 'null
(F=~)
and the non-null cases.
What we pro-
pose here is a ,somewhat different approach.
We can view O;l~:(X)' as an approximation to
O;l~*
F.
.'
= G and
likewise
OilF:(x) as an approximation to
The purpose of this chapter is to
invest~gate
the
properties of the statistic
.....
T -1 * -1 *
= JOG
dO F
2 n 1 "
(4.1.4)
_CD
. and, in particular, to examine the conditions on the
functions 01 and 02 which guarantee
la~ge
sampl.e normality.
4 .2 ' La'i 's' Approach
Let
. -I'
J(x,y) = 02-1 (x)Ol
(y).
Then the statistic'TN(T)
can be expressed approximately as
(4.2.1)
and as such, the untruncated version"
treated by Lai
(1975)~
TN(CD), has been
In his theorem 2 he gives several
100
px-operti,es ~f ~N und~r: fairly ;r;lgid asswnpti~ns. . .That
which 'is ,pertinent is p,res'ented below.
o
1J'h'e'oX'eiD1
·(Lai) .. ' Ass'ume .
J: [O,ll x [0 ~1] ... R
1.
is ,twice continuously differen-
tiable 'except possibly at the points (0,0) and (1,1).
There exist constants
2.'
N
0
~O
~
and
such' that, for'
= n+m,
om
lim N
N-+oo
=..~
1
o<~·<
, 0 _. ~<
-
~
~O
<1.
(4.2.2)
There exists a constantK such that
3.
J(2) (x',y) ~K,
.a.
0 < x,y < 1
(4.2.3)
•
or
b.
There exists a constant K such that
J(2) (x,y) ~ K[(max(x,y) }-5/2+6
+ max(l-~, l_y)}-5/2+6]~
!
-~
o < x,y < 1
(4.2.4)
where
•
(4.2.5)
Now write TN as
TN
= If»
*
*
*. 1
J(F (x),G (x»dF
(x)+ m
_01)
, 1 . n
*
*
,1: (~ (y),)-EtP (y),»
+
where
n )=1
m
1: (1J'(XJ.,)-EtP(XJ.'»
'I
1= .
+ R
n
,( 4.2.6)
101
1P *.(u) =
U .
/u:'
'oJ
ay
,*
*
*
(4.2.7)
(F(t), G (t)) dF (t).
o
Define
(4.2.8)
Then, when
"
0 < d <1/2 and
0 < )J < 1/2 +
1.
. 2.
o <y <
Setting
WN = N(TN -
00
f_e»
~
, we have
(1/2+~)-)J
•
J(F,G) dF), the process
converges to the standard Wiener process, where
(4.2.10)
In the case when 01 and 02 have bounded second derivatives, Assumption 3a. applies and the theorem holds.
How-
ever, when this is not the case, we must look to assumptions'1 and 3a, which impose
met in.many applications.
condi~ionstoo r~gid
For instance, suppose
to be
102
* ':::; min{X·l,x.
.
'
x.1
2}
.'
~'~.
*
.
"
Y.) .
'=.min{.Y )
'l"Y
- J'2}.
Then'oil(~)
= ,l-(l-x) 1/2,
"O;l(y) = l-Cl-y) 1/2 and neither
function is twice differentiable "at the point 1.
assumption 1 is violated.
Thus
Furthermore, note that a
stro~g
relationship .between F* and G* is required in order that
Assumption 3b. hold within its domain of application.
"
4.3 ' The ?yke Sh()r:ack APt>'ro'ach
Another approach uses the' hotion"S
o~
weak convergence
and follows along the lines of Pyke and Shorack (1968).
We will refer to the empirical processes
* = . {m'
1/2 [F,ur
* *-1 (t) -t] ,
m
U
0< t < l}
- -
(4.3.1)
* conve~ges weakly to a tiedand will use the fact that Um
* ='.{Uo(t): 0 ~t~l} where
down Wiener process Uo
o < S < t < 1.
C4.3.2)
As in Pyke and Shorack (1968) we denote by
, {V0 (t):
0~t
~
i}
a second tied-down Wiener process,
103
'* :andwill say , v~ (t) "conve;rges weakly
Uo
independent of
to ",V'*,., '
O
We 'will also rely on several preliminary lemmas', for
'followi~g
which we introduce the
• 'De'f1"·.....·•
·t'1·'0'n", '4' .'.'
'3' 1 '.'
'
,u;1"
families of functions:
Let Q be the set of continuous non-
negative functions g(t), defined on "[0,1] which, for some
y , are non-decreasing on [O,y],
non-increasi~g on
[1-y,l]
and are bounded away from 0 on (y,l-y).,
, 'De'finition' '4" ..3".2.
.'
Q~~~b) be the set of functions in
Let
Q whose reciprocals are
'(r)
o [a,b] =. { g
Note that, if
1"
g (t>'
£
b'
a
Q: I
r-int~grable
"1"
,
'q(t)
,r
(r > 0) on' [a,b]; i.e.,
·
'dt < oo}.
(4.3.3)
g £ OCr) [O,T] for some T, then
= 0, (t- 1 / r )
r
as
t -...
0"
(4.3.4)
This implies
H(X) =
x
'1
x-..O.
10 get) dt
(4.3.5)
We will refer to the general empi:r;ica1 process
o< t
< I}
(4.3.6)
where H is a distribution function ,and Hn is the empirical
104
· distribution function based on n observations •
. The first lemma is 'proved in Pyke and Shorack "s
paper.
· LeIriina" :3..4'.,1
(Pyke and Shorack)'.
· be 'such 'that q is
increasi~g
Let
non-decreasi~g on
on [l-y,l].
Then for
to
qe:Q~~~l] and let
y
[0, y] and non~ y ,
'(4.3.7)
and
......
(4.3.8)
For
(2)
qe:O[O,ll
(4.3.9)
Proof:
For y as defined in Lemma 3.4.1,
'
I
~
sup
o < t:
W*
n
.Ct>t....<
q ( t)
1
+
sup
-. 0 < t
~
*
'IWnJt)
Y
q (t)
I+
*
·I W... (.t~)·1
sup_n~":"'T"
y < t.:: 1- y q (t)
, :w*,C.t)
sup
1-y ~ t.::l
.
n·
q(t)
(4.3.10)
Now apply Lemma 3.4.1 to the two tails and use the fact
that, for
, . '1
t e: [y, 1-y), q (t)' . is bounded and
105
*
,IWnft),1
. s.up
Y~ t
~1-y
. R
,Op.(l) •
~"":1., (;t.)"
sup
,--n
< t < 1- £
t
£
I
-1:
0
,Fl , P
(1)
(4.3.12)
By Theorem 4 of C 'SO~90 and Revesz (1974) we have
Proof.
for
=.
,
_' '('lo'g 'n) 4
£n n
£
n
'Iii lHnH~lJt).~t),
sup
<t< 1-£
n
... :
(t(~~t)1~9 1~gn) 1/2
(4.3.13)
•
Then this can be written
£
n
sup
< t < 1-£
But, for
I
0p(l).
n
.(4 •. 3'~·14)
t £ [£,1-£)
(4.3.15)
Thus, for n
£
1a~ge,
sup
< t < 1-£
. 'De'firii't'iOn' '3 ..4'.,3.,
£n < £ and
(4.3.16)
For, a proces's '{x (t): 0 ~ t ~ i}, define
106
If
q
£ Q.(2).
10, ~]
Proof.
-
t. <cS}
+ sup{fx(t)-x(s)
I:
'Y ~s< t~s+cS ~l-y}
"'.
+ sup{lx(t)
I:
l-y~t~i}}
(4.3.19)
•
Now, Lennna 3.4.1 can be applied to Wn* (t)/qf(t)
the first
.
and last terms and we have.conve?=gence in probability to
. o.
For the middle term,
(4.3.2.0)
The function
l/q(t) is continuous on [y,l-y]
K such that, for
t £ [y, l-y], l/q (t) < K.
{:JO
.thexe
Therefore,
~s A
107
.(4,.3.21)
We use' 'Lemma 3.4.2"and the 'fact that q is- uniformly continuous on [y,':t-,'y] 'to' make "the 'first term. go to' 0 with
- The~9htness of 'Wn* erisuresthat the second term, go to
. Iienuna: :3';4'.,5., Suppose 'f has a ,derivative 'at a and
with
~
•
o•
Zn +p O.
Proof.
This is Homework problem number 1, p. 34, of
Billingsley (1968).
* ••• 'Yn* be the
Yl'
independent observable random variables 'from distributions
Theorem 4.3.1.
Let
* ••• ,Xm
* and
xl,
-.
F* and G* respectl.vely,
where
F* (x) = 01(F{x»
with
01 and
and G* (~) = 02(G(x»
02 known.
(Henceforth we will write com-
posi~ion
of functions in product form, i.e.,
01 (F(x,»
= 0lF(X».
1.
(4.3.23)
Assume the followi-!1g:
F and G are both continuous and differentiable
108
b .'
, c.
= '0,, " 'no,
(Il
1
Q
(01
, 1.
0
'-1
;:: ,I
,~r.i)
*
'.
qiOi (tl, £OIO'«'i J ' ,ri >2, , «1 ;::,F ,(Tl,
a 2 ;::' ,G* C,T.) with' qiOil non-decreasi~gin''IO~YiJ,
non-increa:~i~g in'
[1-yi ,1J Il [0 ,ai J and bounded
away from 0 'in' [Yi,.i:-y~n[O'~iJ
3•
N
= n+m
· 10 . m· '\
J.m N ;::. ,/\ •
, N-t<lO
and .
Define
IT
0'
-
G(t) dF(t)
(4.3.24)
IX)
•
Then
1
-1'*
+ --G(T)Ol
F (T) [F * (T)-F * (T)J
n:
+.
<>pel).
'
--J'g
(4.3.25)
.
109
e.
'.
,.
Integr.ating
,the second .ter.m by parts yields
,
,
,
"
~
, -Y1"4
T~ [01
-1F * (x) -01
' -1F *'(x») d0-1 G* (x)
'-
2
1Il
....
"
(4.3.27)
Deali~g
where,
with the first term, we rewrite it as
1etti~g
'" .rZ ; .'
S =.2(r - 1)
and
£
>
0,
(4.3.29)
2
(4.3.30)
(4.3.31)
A
'/,G*. rm}
7
= ·1- ~e: q2
vn ( t)
Q
;1
1 * * 1
dQ-l F G - (t).
(4.3.36)
(t)
* ';1'). < 1- e:. We
Note that A and A are not present if G(
6
7
wish to show that each of the ~ converges in probability
to O.
Toward this end, pick
£
> 0 with
e: < 'Y 2 •
Then
there exists N such that
(log N)
4
N
<.!.. < e:
NS
and
.G* (T)
.
1
< 1- - S •
Now, restricting attention to
:5./~
-5
IiiQ;lG~G·*-l
(4.3.37)
n > N, we consider each
of the terms individually:
fAll
•
N
(t) dQilF;G*-l (t)
. _so
. + I. n IiiO- 1 Ct) dO-lp*G*-l (t)
o
2
1
111
,
.
" ~. 'InO~~G~G*~l (n-Sl +
n~;l (Ii- S )
Recalling that 0-2 1 (t) =
..... O'p(·t2~~.lr and
+
00 •
S
. :'. :X':2' ;....
='2 (r -1)
2
, we
1 2S
p (t / ) and the second term above. goes
Furthermore, we have
see that O;l(t) =
toO as n
.
(4.3.38)
p
as
t oJ. O.
(4.3.39)
"
Since 02' is continuous and strictly ~notonic, we can say
-1
02 (02(t»
= t,
which implies
(4.3.40)
and this implies that
~~2(tl ]
+0
as
t '" O.
(4.3.41)
How, using Chebychev's inequality and the fact that O2 is
increasi~g, for . n > 0
we have
(4.3.42)
t
112
". '2S
. .... ( lii:l
nl
"'ng
..'
n
.
. Us
....
Q
}
2
eli/ ln
2Q
".C!
•
=' ' 'n ~n .~.
:'. '.' .. 0 as
. Qi(.-n1Iiif ·
."
r:: -1 * *-1 ' -S
rn02 Gn~
(n)
n +
co ,
we. ·.get
= optl) •
'(4.3.43)
Now for
~
~n
I
(4.3.44)
~ **-1
..r.D:(.GI1G
... (.t) ~.t I
r
sup
n-S < t <
=
where
,sup
Iiilo;lG=G*-l(t) - O;l{t)
n -S < t < £ "\
Iq20;l( ~n)
£
I •
is random and lies in the iriterva1 between
* *-1
GnG
(t) and
t; i.e.,
(4.3.45)
But, as
such that
~
.
n gets close to 0, . there exists a constant K
.
l"·········· k'"
.
IA21
.'
1
(
l / r ' so
Iq20;(~n) I
.I~nl
2
I
< sup
n- S < t < £
.
< .{ sup
,O<.t< e:
..
.
·r-:!"(G*G ir '!""l.(....,. )
rn.. ~" " ... ''-l.-.t
K
. u·
'lir"
~n2
r. * *-1· ..
,fli ,%.~
:: J.t:> .-.t)
l/r
K.
t
2
.' I.'
(4.3.46)
113
1.'
Now .
. ··t··· 1/·r2
. 1'1
sup
n -8<.t<
£
.
~
=..
,-'"
sup
.
·t··.. ··
.
"'*.*
..
1/r·
2
. -8'
b t+(l-en) n
G G -let)
n ..<t <£. in
.
= n~~~t<ere~;{i~~nl.~:~Ll(~l
2
(4.3.47)
1/r
t
By Lemma 3.4.3,
co
I· .f\;G,*:l:l,t.l./
in probabi1i tyfor
n -8 < t <
is bounded away from 0 and
Thus
£.
"
n-
sup.
8 <t< £
(4.3.48)
Now, since the function
get)
=
t
'1
l/r
is an element of
2
0(2) , we can apply the Pyke-Shorack lemma to the rest of
the expression.
i?{
sup
For
0 > 0 'we have
~
*G.*-1
yn(.G
. .(.t) -.t)
n
1/r2
O<t<£
, <
o}
> 1 -
o.
(4.3.49)
t
Thus, the whole expression is
Opel).
Likewise,
= 0.(1).
'p
~
(4.3.50)
Furthermore, the Pyke-Shorack lemma applies directly to
A and A to yield convergence in probability to O.
3
7
114
..T~ .d~al with. ~4 .' ~e chooseK such. that {;K+l} E: >.~"'e:.
, Then
=
*
_.,.· . , Vn,(t)
1 * * 1
1 * * 1
~
'd[Q-1 F~,G - (t)-O-1 F G - (t)].
lee: 'q a-let)
~"
.
2 2
+ c·1
Now, for
.
'(4.3.51)
he: < t < (h+1) e:, express'
"\
(4.3.52)
e
where
•
(4.3.53)
Using this,
(4.3.54)
e
115
~
(where ,(K+l)£ is replaced byl-£ as an upper boundary).
The first term can be rewritten as
" < ," {sup
IV** (t) 'I}{ I c(h+1)£ d
0 <t <1
n
~£
lQ- l F*G*-l Ct)
1
;
')9
.-:1 ,
......
~
sup
Iv**(t) j}{IOil,*G*-1«h+1)£)
'O<t<l
n
~JJI
< "{
"
"
•
(4.3.55)
Using the continuity of oil and the almost sure convergence
* to F * , we can say for
of Fm
t £ [0,1]
We now apply Lemma 3.4.2 to" {sup ,,/'* (t)
O<t<l n
obtain, with probability
K
t
'I}
and thus
> 1-£,
(h+l) £ **
-1 **-1
-1 * *-1
.
'h£
V
(he:)d[Q1 FmG
(t)-01 F G ( t ) ]
n
h=l
<
£
. {sup!'",**.
(t) I}.
t
n
(4.3.57)
116
.
We .must now e1iIrlinate in probability the second ,term
.' ..
,
in ·14.3.'54).' We have'
+ W£(V**)
n
K
}; 'c Ch+l ) £dO-lF*G*-l(t)
K=l ~£.
1
(4.3.58)
Lemma 3.4.4 gives us the result.
,All that i~ left is to s'how that
note that, for
entiable.
bounded.
£
•
.•
' -1
*'G'*1
- 1.Gn:
, ,Q'2'
' :.....; :'(tr~Q"
: . : ; :2 ;'Ct)
~ ,
GnG *-1~ (t),-t
where
=
~ Cl).
First
~ t~l-£, O;~!~~ is continuously differ-'-1
Thus, in thJ.s regJ.on O2 (t,)
"
Also
*
lAS I
=' , .... '1'-1"
"
is
q2 Q 2 Ct)
,(4.3.59)
117
t
'ion
0 < e < 1.
.an t + o(l.. eon-lG*oG*:,""l(tl,
. n·
n
:::::I
:(4.3.60)
-1'
Now, 02
Ct) is uniformly continuous in: [e: ,I-e) and
0
(4.3.61)
Thus
. (4.3.62)
Therefore
•
-1 * *-1
-1
. 0° 2 oG oG .. C.t.) :-.02' .Cot)
1
= I -e:{
~ *
0
n
0
-1 * *-1
o 02 GnoG.
<
;::
sup
e: < t< 1-e:
cp(1).
We now turn to
•
G G -l Ct )-t
e:
{
* *-1
G G
n
-1
.(ot) ~.0·2 .(t)
(t)-t
(4.3.63)
118
'- "IN
-m
.'
·e
(4.3.64)
Except for the first term above, each of the terms has its
analog in the previous discussion and . can be treated
.
simi;ta.rly.
~ow,
for
3.4.5 with
f =
'*
~
,.~G(T)
°
IFm(T) -
-1
1 •
F
-1 ~.n.l(T)-01
'*
-1*
[01
F (T)], we apply Lemma
.
Then, S1nce
'* (T) I
=
ep (1) ,
(4.3.65)
119
.:"
we have
'*
* (T)]
+ Zm[Fm(T)-F
"
e
(4.3.66)
~
the result follows and the proof is complete.
'Noteth'at we can now write
/.N(To(T)-PT) as
/.NCUN,l (T) + UN,2(T) + UN,3(T) + 0p'Cl) where UN,l (T),
UN,2(T) and UN,3(~) are one-sample U-statistics. We employ the function
C(x)
=
o
x <0
1
x> 0 •
(4.3.68)
Then
. ,UN , 1 (T)
(4.3.69)
120
.(4.• 3.70)
and"
, 'Oo'r'o'l'l'a'
.
.
' '!ry:'
Define
*
= loG. (T) loX y(l-x) Oil' (x) Oil' (y) dFG*-l (y)dFG*-l(x)
,(4.3.72)
=1:*
Then,' if
'.
,
(T) loX y('l-x) oil' (x) oil' (y) dGF*-l(y) dGF*-l(x)
....
(4.3.73)
2
aT(l) <
Q)
and
<
Q)
,
the asymptotic dis-
•
tribution of
is normal with mean 0 and variance
- 2G(T) oil' F* (T) [l-F* (T) ]
I:*
(T) x oil' (x) dGF*-l (x) •
(4.3.74)
Proof:
Since
'(4.3.75)
121
we
·canuse
·thetheo~ .o~
·U-atatistics.
, UN,l, isindepen:d~nt of. UN'·
We note .that ,
,2'
and· UN3
, . ' and each has mean
Property> P3 of Chapter 1 gives· :thedesired result.
, 4. 4 '
We'a~
COllve'r'g'erice;
'l"o'
,
o.
A 'Gau'sslai'l' P'ro'ce's·s
We may now apply the X'esults of Chapter 2 to the
process
whereTO is some specified upper limit of application.
.
In
some 'cases we can let T range
over the entire real line,
.
but in other applications we cannot satisfy the conditions
of Theorem 4.3.1 without specifying a TO.
For example, ,
consider
(4.4.1)
. where
, {y j
'{XiR.;
R. =1, ••• ,Pl} are i.i.d. (F) and
A: R. = 1,2, ••• , P2 } are i. i • d. (G).
Then
122
and
. "1'
'1-'-
*'
P1
= ~P1 (I-F)
= P (1-G * )
, '1' .
'1- -
P2
(4.4.3)
·2
No t e tha t b 0 th ' q10-11 and
q2 0-2
1
" f""t
approach 1n
1n1 y, as F*
and G* approach 1 and neither be1o~gs to OCr) [0,1] for
r >2
if P1 and P2 are. greater than 1.
pick some
stoppi~g
However, if we
point TO and then choose
a
1
and
a
2
such that
(4.4.4)
a = min(a 1 ,a 2 ) we have qiO~1(t) in OCr) [O,a]
i=1,2 and all r > O. In fact, in this case,
then, for
for
-1'
0i
(t), i=1,2
is well behaved near the origin and the
proof of Theorem
4.4.1
4.3~1
can be simplified.
The Individual U-Statistics
, In order to derive the weak
conve~gence
of the process
TN* we must first satisfy the assumptions of Theorem
2.7.1 for the individual u-statistics.
.
"
, ,...
. -1'-1'
,
, 'The'orem :4'.:4'.,1., Assume 0 1 (x) and 0 (2) (x) 0 ~ x ~ a
continuous and strictl~ increasing.
Then under the
are
~
123
assumptions of Theorem 4.3.1,
both satisfy the conditions of Theorem 2.7.1.
Proof:
* ,l ' we have
Dealing with UN
(4.4.6)
where
and
'T(Yj) =
I;* (T)C (G*-l (t) -Yj) Q;l'
(t) dFG*-l (t).
(4.4.7)
Now, since the Kernel is increasing in T, we can satisfy
condition 3, i.e., (2.7.6) by setting
'2,T = O.
'l,T=
T and
To satisfy condition 4, we must find a strictly
increasing, continuous function H such that (2.7.8),
(2.7.• 9) and (2.7.10) hold:
lJ4(S,T) ~H(T) - H(S)
lJ (S,T) <R(T) - H(S)
2
124
<
(4.4.7)
where the first
".
ineq~a1ity
arises because
IC(G*-l Ct ) - Y ~ -t
J
I
< 1 •
(4.4.8)
-
Now, expanding [F(T)-F(S)]4 shows that (4.4.7) is bounded
by
", <
28 [Q;l' (TO)] 4 [F(T) -F(S)].
Now consider
P2(S,T) =
E{(~T-~S)
-
CeCT)-e(s»}2
. < to- 1 ' (T )]2 [p2(T)-F 2 CS)]
2
0
,(4,.4.9)
125
(4·.4.10)
. Finally,
.. ~ . [0;1 (TO) ] [F.(T) -FCS) ] •
1
Thus,
. (4.4.11)
setti~g
.....\
1
.m = maX{28[0;1'CT )]4, 2[0;1 CT )]2, 0;1' (T )}C4.4.12)
O
O
O
•
Condition 4 of Theorem 2.7.1 is satisfied by
H (x) = mF (x), 0 < x < TO •
* ,2 is analogous.
UN
Under the assumptions for Theorem 4.4.1,
The proof for
~heorero
4.4.2
the process
. (4.4.13)
satisfies the conditions of Theorem 2.7.1.
Proof:
where
We write
126
.(4.4.15)
~gain
we have a positive Kernel which 'is
with 'probability 1.
.'
Now :for
increasi~g
in T
114 (S.,T) we have
+ [C(S-Xi )-F(S)]f01-I' F * (T)G(T)-Ol-I' F * (S)G(S)]} 4
",
(4.4.15)
where the first inequality above comes from an application
of
~e
cr-inequality (cf. Puri and Sen, p. 11).
Qil'~*(T)G*(T)
is
increasi~g in.T,
Since
the second half of the
last expression can be handled by expanding the fourth
power and it becomes less than
(4.4.16)
127
..NQ\'{,.,fo;rthe
fix-s.t pa,rt ofth.e
.
.
=.
exp~ession,
,we have
(F(T)-F(S)J-4(F(T)-F(S»2+6 (F(T)-F(S»3
3(F{T)-F{S»4.
(4.4.17)
This, in turn, is less than or equal to (F(T)-F(S»
o < F(T) -F(S)
<1
and, for
since
o ~x~l,
"
3x4 < x
3
2
x - 4x + 6x
e
is equivalent to
•
(4.4.18)
2
3
1 - 4x + .6x - 3x < 1.
This is equivalent to the expression
and is true if
the case.
4 - 6x + 3x2
has no real roots, which is
Th us
P4 (8 ,T) ~
+
. -1' F *.(T) G(T» 4 - (01
-1' F * (S)G(S» 4 }
56. {Cal
. -1'
8[01
*
F (TO>']
4
[F(T)-F(S)]
128
'. + 8 [oil 1 F*.CT )] 4 [F(T) -F(S) ]
O
(4.4.19)
Now consider
......
"
+ 4[Oi 1 'F*CTO)] [Oi 1 'F*(T)GCT)-Oi1 'F*(S)GCS»).
C4.4.20)
Finally
* CT) GCT) [F(T) -F(S)]
=. 0 -I'
F
1
'
-1' F(S)G(S)]
*
+ F(S) [01-1' F *
(T}G{T)-Ql
~
*
-I' *
0 -1'
1 F (TO) [F(T)-F(S)]+[01 F (T)G(T)
-I' F * (S)G(S»).
- 01
129
NQ~
.1et
and
O{
-1' *
4
-1' *
2
-1' *
m2 =ma?t 8 [01 F (TO») , 2 [01 F .(TO)] , 01 F (TO)}.
(4.4.22)
Then the conclusion follows by setting
(4.4.23)
H(x)
4.4.2 . We-ak ·conve:rgen·ce·.·Of TN*
Referring to Theorem 4.3.1 we ha"f'e
and, having chosen a TO ' the proof of Theorem 4.3.1 gives
(4.4.24) uniformly in T,
a ~T ~TO.
Thus, we can employ
Theorem 4.1 of Billingsley (1968, p. 25) to conclude that,
* + U* + U* converges weakly to some process
UNl
N3
N2
*
*
* This provides
U , then TN also conve:rges weakly to U.
if
the_p~sis
for the next result •
. 1J'h"e'orem '4 ..4'.3.Theorem 4.4.1,
Under the assumptions of Theorem 4.3.1 and
130
weakly to a. GauSs.;i.an p;r:ocess
.· .conve~ges
. .
· variance structure ~for
·+
.
{I:* .
. U* which. has co-
*
"
S ~ T ~ TO ' ..cov.[U (T),' U (S.)] =
8
(8) tOil (t) dGF*-l (t) -G.(s) oil' F* (8) F* (s) }.
-I' F * (T) [l-F " (T)}}
- G(T)Ol
+
(4.4.26)
Then g satisfies the conditions of Theorem 2.9.2 and the
result fo110l"1s immediately.
The covariance structure is
computed in the standard manner.
Although, in some cases, we cannot meet the assump-'
tion~
of Theorems 4.3.1 and 4.4.1, we can sti11 use the
statistic
131
.(4".4.27)
and can employ the technique 'used by Birnbaum and McCarty
(1958) to compute "confidence bounds •
. 'ThebreIn ~:"5·.·1·.·
Assume 'for
j=~,2
there. exist functions
Rj such that
IOj1 (a+b) -ojl (a) I ~ Rj
"
(b)
0
~ a, b ~ 1,
(4.5.28)
Then, for
(4.5.29)
we have
(4.5.30)
where HK is the distribution of the two-sided, one-sample
Ko1mogorov statistic DK based ~n a sample of size K.
Proof:
We write
which, upon
integrati~g
by parts, yields
132
: <
, -cb
rO;IG: -O;1G*,
sup
-
< '1' <
~.
('1')
('1')
I
Clo
(4.5.32)
Thus, by assumption,
. I~R
IAN
2
.'
(
IG* ('1') -G * ('1') I)
sup
n
-00<'1'< 00
"
+ 2 ,R1 (
sup
"
,
-co<T<oo
IF* (T)-F * ('1') 1)=
.=~*
m
•
R (D ).
2 n
+ 2 R1(Dm)
;,.
(4.5.33)
Thus, we have
(4.5.34)
which concl ude.s the proof •
.
'A
133
* ma.x{X'l'''.
.
X,':::;
~ ., X,
} and
1
,.
~'
),P1
.Y. ':= max{X'l' •• ~,',X, } we have .
J'
J'
'JP
, 2
.No.te .that when
'* ..
:..
F * (x)
= .01(F(X» =
*
G (x) = ,0 (G(x»
2
[F(x)] PI
'
= .IG(x)]
(4.5.35)
P2
and
O}1 (y)
= y
llPl
'-.
0; 1 (y)
=y
lip2
•
Neither of these functions satisfies ,the conditions of
Theorem 4.3.1 because of the behaviors of the origin.
However, applying the Cr -Inequa1ity (see Puri and Sen, p.
11) we have
(4.5.36)
Thus Theorem 4.5.1 is applicable in this case.
.CHAPTER V
SUMMARY AND SUGGESTIONS FOR FURTHER RESEARCH
5.1 . S'u'm'Iriary
This dissertation has addressed the problem of estimation based on progressively truncated U-statistics.
Chapters II and IV were theoretical in nature and laid
,"
the foundation for the
samples.
~se
of such statistics with
la~ge
Chapter III dealt with applications in the par-
ticular case of the Mann-Whitneystatfstic.
Chapter II introduced the progressive truncation
scheme and then studied the asymptotic properties of the
resulting stochastic process.
In particular, conditions
were found under which these stochastic processes converge
weakly to Gaussian Processes.
Chapter III built on this foundation as applied to
the Mann-Whitney statistic.
The emphasis was on modeling
the true curve estimated by the stochastic process, with
the aim of predicting the parameter P (F , G) = Pr (Y < X) •
Sample curves were exhibited for the
l~gistic
and normal
cases and then a linear models approach was taken for the
purposes of prediction on randomly. generated exponentially
distributed data.
In Chapter IV the properties of an extended
135
Mann-Whitney $tatistic were
under fairly
explo~ed
with the result that,
rest~ictive· ·conditions, ~eak
conve?=,gerice to
a Gaussian process is attained •
.
The theory developed in Chapter II can possibly be
extended to provide a basis for
deter:mini~g
uniform con-
fidence bands for the parameter of interest by utilizing
the properties of Gaussian processes.
Some staridardiza-
tion and modification might be determined to adapt the
process which arise to ·conform to standard Brownian
motion.
Also, since properly normalized U-statistics
•
are endowed with asymptotic behavior of standard Brownian
motion when viewed as functions of sample size, it is
possible to study the progressively truncatedU-statistics
as functions also of sample size and to try to develop
weak convergence to multi-parameter Gaussian processes.
Other possibilities for modification include methods
of taking into account the information which is available
in the data which has not yet been observed, as was done
in some of the test statistics introduced in· Chapter I.
Similar statistics might also be derived from other
estimators .of the distribution functions F and G: for
example, Efron's (1967) self-consistent estimator.
ever,
retaini~g
challe~ged.
the properties of U-statistics
How-
~ght
be
136
Chapter
research.'
I~Ileaves
Means of
open a wide field for future
opti:mizi~gthe spaci~g
and number of
time points m:tght be .:developed, .based on ass'umptions about
the data.
. a means of
A cost .function
relati~9
m~ght
also be 'implemented as
sample 'size and maximum time.point.
Applications of the theory of Chapter 'IV
~ght
also
be made, similarly to what'was done in Chapter III.
Different functional forms for 01 and 02 might be tested
to determine which types of functions yield acceptable
predictors. '
~9ain,
the properties of Gaussian processes
m:tght also be exploi tect.
•
137
APPENDIX'
We want
1/1 CT)
=/:00 GCx) dF(x)
-~ , ,1' "
'l' ' .'
_00 'c',,'
, 'IT
/2tr 01
"./2tr
.I t00 e
°2 , -
dt
(A.l)
Let
ICx)
~"2tr
,2-'1"
-
2
e -t
/2 dt, , epCx 1:= t' (x) ,
CA.2)
CA.3)
(A.4)
138
and
.
;=
·T.~J!i
.
(J~.5)
~.
'0
·1
Then
2
. , '1, .. T'
. . -y /2
1P(T) = .F(s,J.l,T') =':;211" I_co t(sY-J.l)e
dy
(A.6)
and
·'aF .
as
=
'1
~
v2'JT
T'
I_co
t (sY-J.l) Y e
""
-Y
2
/2
dy
~(S'Y"'.J.l) 2
., ;2
e
/211'
dy
1 2 222
1
T'
-"2"{Y +s Y -2 sYJ.l+J.l }
= 211' I_coY e
dy
"~'"
2
'_1{h+s 2 Y _ '/1+s , }2 .
=
Ye 2
dy.
CA.7)
Now let
(A.8)
139
and
z '=
h+s2,y
_. ; :S:lJ.'~ , " "
..1:+S
(A.9)
2, '
'2
'''s·2,
-.,....:[1- 1+s 2 ] , * '.;.1,' z2
..
aF
...:"
..
~
,
IT e '2'
'(: :,:Z: '
as - e , '. '" ,. '
r--,.
2~
~l+s. '].1
-00
+"
'S'Jl
1+s
:dz
)::
2
r--'J:
~l+s-
,, 2
1 *
-2 T 2 ,
<
*
~---'
~ (-e
)+
s~u
t(T )}
2 ~
1211' (1+s 2 )
(1+s 2) 3/2
,e
=
'_1 ( Jl 2>'
2, l+s
"'1"
{"-.. .,. . ._ _
fI,
(A.10)
Also,
aF(1,lJ,T' )
aJl
1
= -1211'
I
T'
- 4>(Y-Jl)e
1_ 00 - e
, . 1,
211'
= ,-
dy
-CIO
, . '1 ' , T'
=211'
_y2/2
,
_(Y-Jl') 2
,2
_y 2 /2
e
dy
'1 2 2
'
2}
T' -'2'{y +Y -2YJl+J.1
e
dy
'-00
, , l'
T"
: . 2/ } "2/
= -; --..:.'
e ~ {2
Y -Y)J+lJ 4 e -J.1 4
211'
L
CIO
.
dy
140
,. 2
-lL
... ,',4
'
= -:e " 't{I2(T'-p/2)} •
(A.Il)
2fi
.'
We also have
.....
F(I,O,T')
'1
= --12Tr
2
T' -x /2
I e
-00
~(x)dv
..
= ( : ~ (x) d~(x)
(A.12)
•
Now
F (1, lJ, T' ) - F ( 1,0, T' )
__ /,)J
0
'3F('l,'x,T')
3x
dx
and
F(s,lJ,T') - F(1,lJ,T')
Thus,
F(s,lJ,T')
__ ,.s' '3F(Y','lJ',T' ).
J.
--~-~~-~~
,1
3y
dYe
(A.13)
141
. x2
'::: t .2 er '1 .
.
2
_
.,Jl '~:~T ~&rz<'~'-X/2}
0 .. 21i
.
}dx
. l' : :~2
--(-).
. : "'~.'
. .,.: .lS .e.
~+;y22 t
1· {2tr (1+y)
. ylJ
~{h+y2
2 172
{l+y}
T' _. :Y;ll·
.}
r--"2
.
. "l+y-
(A.14)
........
•
142
REFERENCES
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