A CLASS OF RANK
BSTIMA10RS OF RELATIVE RISK
by
Janet M. Begun
lltiversity of North Carolina at Chapel Hill
Institute of Statistics Miaeo Series 11581
December 1981
-e
.
<
DEPARTMENT OF STATISTICS
Chapel Hill, North Carolina
A CLASS OF RM'K
ESTH-fATORS OF RELATIVE RISK
by
Janet M. Begtm
University of North Carolina
Short Title:
Rank Estimators of Relative Risk
S~NARY
It is frequently of interest to compare the survi,-al times of independent
samples from two populations.
~hen
the hazard functions corresponding to the
two populations are proportional, the constant of proportionality -- knohn as
relative risk--provides a descriptive summar)' of the survival experience of
one population relative to the other.
each population,
relative risk.
\~e
Using complete, uncensored samples from
construct a family of nonparametric rank estimators of
Based on their asymptotic behavior, we then identify a b'o-step
estimator which asymptotically achieves the minimum
this class of estimators.
as)~totic
variance for
Finally, the asymptotic behavior of the likelihood
ratio of the ranks under the proportional hazards model is used to
sh~\
that
the two-step estimator is asymptotically optimal (in the sense of minimum
L~)~totic
variance) among all regular rank estimators of relative risk.
Research was supported by NSF grant 81-00748.
FOOTNOTES
lAMS 1970 subject classifications.
2Key ~Tds and phrases.
PrimaT)' 62GOS; secondary 62FOS, 62F15.
Relative risk, proportional hazards model,
empirical df, Chernoff-Savage statistics, Lehmarm alternatives,
representation theorem.
~jek' s
3
1.
Introduction, sunvnarr and notation.
Let F and G denote continuous
distribution ftmctions (dfs) on :R+ :: [0,00) with survival flUlctions
F:: 1 - F
and G:: 1 - G respectively. By a proportional hazards model we mean that the
cumulative hazard flUlctions corresponding to the continuous dfs F and G are
proportional ,,1 th constant of proportionality 0
-
<
8
< 00:
-
-log G • -8 log F .
The ratio e of the cumulative hazard flUlctions is kno\\n as relative risk.
Equivalently, F and G satisfy the Lehmann alternative G• F 6 •
Suppose further that F and G are absolutel)" continuous
and g respectively.
~ith
densities f
Then the proportional hazards model is equivalent to the
proportionality of the hazard functions corresponding to F and Gwith the same
constant 8 of proportionality:
gIG" e • f/F. This t\\"o-sarnple model is a
special case of Cox's (19i2) general nonparametric regression model.
~hile
Cox's method of analysis of censored sun"h"al data has received considerable
attention (Cox (1972, 19(5); Efron (19i7); Kay (19ii)), technical difficulties
remain lUlreso1ved. Hence we focus attention on the special case of complete,
lUlcensored samples from t"o popUlations Kith proportional hazard flUlctions.
In Section 2 Ke exploit a simple identity for the relative risk e in terms
of the dfs F and G and a score flUlction J to construct a family
eJ
of
nonparametric rank estimators of 8. Ke discuss the strong consistency and
asymptotic nonnality of "6J lUlder some smoothness and grO\\"th conditions on J.
In Section 3 we minimize the asymptotic variance of the rank estimators "8J
over the appropriate class of score flUlctions.
involves the parameter
two-step estimator.
Since the optimal score flUlction
a, we discuss the asymptotic behavior of the appropriate
In Section 5 we use the asymptotic behavior of the
likelihood ratio of the ranks to show that the
th~-step
estimator is
4
asymptotically optimal (in the sense of mininun asYJl1'totic variance) amng all
regular rank estimators of
e.
Let Xl <... < x.n and Yl <... < Yn denote the order statistics of two
independent saJlt)les fran F and G respectively. Denote the corresponding
. ZeIt-continuous eJ11)irical dfs by Fm and Gn respectively. Let WI <... <l'1N
denote the order statistics of the combined sample of N • m + n observations
and denote the associated left-continuous enq>irica1 df
by~.
Thus, with
AN :: miN,
(1.1)
For A
£
Let
[0,1], set HA :: AF + (l-A)G.
~ <••• < R
m
set
~
!i ::
(Nl , .. ·, NN) by
::
(~, •••
denote the ordered ranks of the X' s in the combined samp Ie;
, Rm).
Define the risk vectors
~
:: (Ml' ••• '~) and
(l.l)
and
In Cox's (1972) tenninology,
~
of the X's and N of the V's are "at risk" at
k
time Wk.
We will use the following notatien:
Lebesgue measure en [0,1] by I.
over R+ :: .[0,00).
Unless noted otheNise, all integrals are
For functions f
ftmction g on R+, define
Denote the identity ftmction and
l
and f
l
on R+, and a nonnegative weight
5
For real numbers a and b, set a
2.
dG/dF •
1\
b :: min{a, b}.
G. Fe, then
After mul tiplying both sides by F and using G. Fe, the
Estimation of relative risk.
ef. 6-1.
If F is continuous and
Radon-Nikod)'JTl derivative may be re-expressed by the formula
(2.1)
(I-F)dG
= 6(1-G)dF
.
Hence a natural starting point for estimation of 6 in a proportional hazards
model is the identity
e=
(2.2)
J J (F ,G) (1- F) dG
J J(F,G)(l-G)dF
'
?
where J: [0,1)-'" lR is any score function for \\'hich
o < JIJ (F ,G) I(1- G) dF
(2.3)
• J~IJ(l-I, 1-1 6)11: dI
<
00
•
Then, for a suitable collection of score functions J (defined belo"), a family
"
{6J }
of nonparametric estimators of
e is
obtained by replacing the continuous
dfs F and G in (2.2) with the corresponding left-continuous empirical dfs Fm
and G .
n°
1\
6J •
R&WU\ 2.1.
f
f
The estimator
J(Fm,Gn) (l-Fm)dGn
J(Fm,Gn) (l-Gn)dFm
1\
eJ
may be expressed in tenns of the ordered
6
ranks of the observations from F and G.
Under the proportional hazards mdel,
the distribution of the rank vector, and hence of the rank estimator
depends on
e only
8J'
(See Lehmann (1953), p. 26.)
and not on the continuous df F.
It follows that, in our asymptotic results concerning ranks and rank estimators,
the convergence is 1.Dlifonn for all continuous dfs F.
Hereafter, we indicate
this by saying that the convergence holds '.'uniformly in F."
RfNARK 2.2.
With the ident i ty score f1.Dlct ion J E 1, (2. 2) becomes
!(l-F)dG
6 • /(l-G)dF •
where X - F is independent of Y - G.
P{Y<X}
P{X<Y}
,
The corresponding estimator
gJEl
is a
1\
simple function of the Mann-\\hitney-Wilcoxon statistic U • ! (l-Gn)dFm;
specifically, @JEl a.s. (1-0)/0.
Consequently, @JEl is an attractive estimator
since, regardless of whether the proportional hazards model holds, it aZuays
consistently estimates a natural quantity of interest.
an analogue of
0 for arbitrarily right
(Efron (1967) discusses
censored data.)
The purpose of the score f1.Dlction J in the nonparametric rank estimator
is to assign weights to the ranks, giving heaviest
are most sensitive to the value of 6.
\~ight
1\
eJ
to those ranks which
\\'hen the relative risk
e > 1,
the df F
is stochastically larger than the df G, implying that the X's tend to be larger
than the Y' s • This general tendency is reflected in the ordering of the
largest observations,
PROPOSITION 2.1.
~
and Yn , from the two samples.
Undel' a Pl'0p0l'tiona Z haza1'ds mode Z
as m,n
~
CD
as m,n
+
CD
e• 1
e>1
if e < 1
if
if
7
tit
uni fom Zy in F.
PROOF.
J Fn dFID •
n m
Note that Pe{Yn<Xm} .. J G dF • When e· 1, write Pe{Yn<Xm} •
miN. \\'hen e > 1, use the change of variables F· 1 - (x/m) and
apply the dominated convergence theorem.
\\'hen
e < 1,
write the proportional
hazards model as F • Go/e) and note that Ole) > 1.
Since the value of the parameter
e determines,
0
in probability, the
ordering of the largest observations from the two samples, the score function J
is allowed to gro" unbounded as both of its
coordinate~
approach one.
The
shape of the score function J is motivated by the locally most powerful rank
test (l)1PRT) of
e ..
e
eo versus
> eo'
Specifically, the L'IPRT is based on a
"generalized Savage statistic" T;\(eo )' \\'here
T~.(e)
(2.4)
.\
.. e J J*~.(F
.\ m,6n )(1-6n )dH~.
1\
and
To introduce a suitable collection of score functions, define the bounding
2
function R on [0,1] by
R(s,t) :: {(l-s)
DEFINITION 2.1.
Let
+
)(00)
(l-t)}-l for (s,t) ( [0,1]2 .
denote the collection of all real-valued
2
functions J defined on [0,1] , excluding the function J :: 0, which satisfy the
following conditions:
(Jl) (Differentiability):
J is differentiable on [0,1)2 with partial
8
derivatives
J 1 (s,t) :: a~ J(s~t) and J 2(s,t) ::
*
J(s,t) •
(J2) (Boundedness): There are constants M > 0 and a < ao for which
/J(s,t)1
:S
l-lRa (s,t)
for (s, t)
£
[0,1] 2 ,
IJ1 (s,t)1
:S
~lRa+1 (s, t) for (s, t)
£
[0,1) 2 ,
/J2(s,t)1
:S
~lRa+l(s,t)
£
[0,1) 2 •
for (s,t)
and
(J3) (Continuity of partial derivatives): The partial derivatives, J l and J 2 ,
are continuous on [0,1) 2 •
The power a of the bounding function R is called the growth rate of the
score function J.
If J is bounded by ~lRa, then J satisfies the integrability
requirement of (2.3) if and only if a
<
Our proof of strong consistency of
2.
1\
the estimator eJ requires J to satisfy the differentiability condition (Jl)
and the boundedness condition (J2) with ao • 2. To prove asymptotic nonnality,
we impose stronger smoothness and boundedness conditions by requiring J £
J(3/2). The strengthened boundedness requirements correspond to the usual
restriction of a finite variance for asymptotic nonnality, as opposed to just a
first moment for almost sure convergence.
TIiEORfM 2.1 (Strong Consistency Of @J).
Suppose the saore function J
satisfies conditions (J1) and (J2) ZJith groLJth rate a < 2.
proportional hazards
modeZ~ 8
J "a.s.
e
as m,n"
1\
TIiEORfM 2.2 (Asymptotia Normality of 8J) .
llIl
ThBn~
under a
unifomlll in F.
Suppose the saore function J is
9
in )(3/2) and suppose ).t\ -+ ). (" [O,lJ as m,n -+
00.
Then~
under the
proporti.onal haaards mode l ~
mn)~
['1\
t.
(eJ-€l) -+d r\(0
.,
o"(J)
(2.6)
To stabilize
as)~totic
variance, define e - log 6 and
cC'nd~t~ons
of'
T::e~rew:
=I
1\
eJ
1\
:: log EJ •
2. r~
-+d X(O ,c 2 (J)) as rn,n -+
The change of variables 1 - F
= o2(J,6,A)
o2(J)
cc
- {J J(F,G){l-F)dG} • {f J(F,G)(l-G)dF} .
mn)~ (5" -6)
("'!\
J
2.3.
as m,n -+
f J2 (F ,G) (I-F) (l-G)dH).
COROLLARY 2.1. Un.der tr.e
~~
,e 2 a 2 (J))
cc
in (2.6)
sho~s
does not involve the continuous df F. This observation,
combined with Remark 2.1, shows that both the finite sample and the
distributions of
that
1\
eJ
as)~totic
1\
and 6J are independent of F.
Crowley (1975) suggests the score function JC(F,G)
(1-A)(1-G)}-1. {I-H/.}-l for 0
= {)'(l-F)
+
A < 1. Clearly, J C satisfies the bO\mdedness
condition (J2) \doth gro\\"th rate a • 1 if and only if 0 < A < 1. Also,
<
Crowley's score function appears in the Savage statistic for the
versus
e > 1.
~WRT
of e • 1
(See (2.4) and (2.S).)
/I
Both the ntunerator and the denominator of 6
J are generalized Chernoff-
10
Savage statistics as defined by Lai ((1975), p. 834).
With the aim of
obtaining asyuptotic approximations for the stopping times of certain
sequential rank tests, Lai represents the generalized Chernoff-Savage statistic
as the
Sllll
of independent identically distributed random variables plus a
remainder term.
By analyzing the order of magnitude of the remainder term, Lai
obtains an invariance principle and a law of the iterated logari dun for
generalized Chernoff-Savage statistics.
Theorem 2.2 can be derived from Lai's (1975) work under the additional
hypothesis that the score function J has continuous second partial derivatives
which are each bounded in absolute value by MRQ +2 for some Q < 3/2. However,
we give a separate proof of asymptotic normality lv.ithout this additional
assumption.
The remainder of this section is devoted to proving Theorems 2.1 and 2.2
through a series of propositions and lenmas.
Each intennediary result holds
uniformly in F since either the statistic is a rank statistic or, with a simple
change of variables, the resul t does not involve F.
For the sake of brevity,
the assertion of uniformity is deleted from our statements throughout the
remainder of this section and in Section 4.
To decoIIq)ose ~J - e into manageable terms, we introduce the notation
(2.7)
KN =KNJ = f
J(Fm,Gn ) (l-Fm)dGn '
=KNJ = f
J(Fm,Gn)(l-Gn )dFm ,
KN
=f
J(F,G) (l-F)dG ,
= KJ = f
J(F,G) (l-G)dF •
K = KJ
K
Simple algebra yields the equation
and
11
(2.8)
With this decomposition, the proof of Theorem 2.1 on the strong consistency of
"e
J
follows directly from Proposition 2.2.
PROPOSITIO~
2.2.
K
Since
~
Under the
-+
a.s.
K
KK is synunetric to
cond~tione
and
K,. -+
,\ a.s.
0:
K
Tr.eor~ 2.=~
as m,n'"
oc
•
Kt\' only the arguments for the convergence of
K!\ \d 11 be presented.
Kext we decompose
N-
K
K
a
K~ - K
by adding and subtracting terms:
J{J(Fm,Gn ) - J(F,G)~(l-Gn)dFm
+
J J(F,G){l
+
J J(F,G) (l-G)d{Fm -
- G - (l-G)}dF
n
m
F} .
In the first term on the right-hand side, the integrator Fm essentially
restricts the range of integrators from R+ to [O,X ] "'nile the factor (l-G )
n
m
in the integrand restricts the range of integration to [O,Y ]. So the actual
n
range of integration is ~ = [0, X A Y ] • Hence, by the differentiability
m n
assl.UTlption (J1) and the mean value theorem, on the interval ~,
where !Pm is between Fm and F and \lin is between Gn and G.
It follows that
12
(2.10)
where
~
= f~IJ1(~m'~n)1
• IFm - FI(l-Gn)dFm '
bN = I~IJ2(~m'~n)1 • IGn - GI(l-Gn)dFm '
~
= fIJ(F,G)1
~
= II
• IGn - GldFm ,
and
J(F,G) (l-G)d{Fm - F}I .
Thus to prove Proposition 2.2, and hence Theorem 2.1, it suffices to verify the
following result.
PROPOSITION 2.3.
Under the aonditions of Theorem 2.1 1
-+
Oland et. -+
0 as m,n
N -+a.s. 01 b N -+a.s. 01 c..
~N a.s.
'-N a.s.
a
-+
Q;l
•
The proof of Proposition 2.3 is delayed tmtil Section 4.
To prove the asserted asymptotic normality of (mn/~)~
(6 J -6),
we begin by
multiplying both sides of (2.8) by (mn/N)~:
(2.11)
It follows immediately from Proposition Z.Z that, tmder the conditions of
Theorem Z. 2, KN -+P K as m,n
-+
Q;l •
Hence, to prove Theorem 2. Z, it remains only
to examine the joint asymptotic behavior of (mn/N)~ (KN-K) and (mn/N)~ (KN-K).
Since Xl' .•• '
x.n
are independent of YI' ••• , Yn' the empirical processes
m~ (Fm- F) and n~ (Gn - G) are independent for all m,n ~ 1.
\~e introduce
two
special independent tmiform e"1'irical processes U and Vn (discussed in the
m
13
filii
Appendix of Shorack (1972)), which converge to two special independent Bro\\nian
bridge processes U and V in the strong sense that
(2.12)
for all weight functions q in
Q(~),
the collection of all positive monotone
decreasing continuous functions on [0,1) for "'hich
J; q-2 dI
<
ClC.
The special
empirical processes Um(F) and \'n (G) have the same distributions as the
original empirical processes m~(Fm-F) and n~(Gn-G), respectively.
The
special independent Brohnian bridge processes are independent mean-zero
Gaussian processes on [0,1], with continuous sample paths and common co\ariance
structure
(2.13)
(ov(U(s) , U(t))
Ie
(o\(\"(s) ,r(t))
= s /\ t - st for 0
S
s,t s 1 .
The results obtained bel 0"· ',hich involve the stronger convergence ...p
apply only to the specially constructed processes.
But since the original
processes ha\e the same distributions as the special ones, all conclusions
concerning convergence in distribution also hold for the original processes.
~o",
after adding and subtracting terms using (2.9) and replacing the
original processes with the special ones, we conclude that
(2.14)
where
14
~ =f
SoN
J2(~m'~n)Vn(G)(1-Fm)dGn '
~
=f
J(F,G)Um(F)dGn '
~
=f
J(F,G) (l-F)dVn(G) ,
~
= f5.N
Jl(~m'~n)Um(F)(l-Gn)dFm '
~
= f5.N
J2(~m'~n)Vn(G)(1-Gn)dFm '
~-
=f
~
J(F,G)Vn (G)dFm ,
and
n_
= f J(F,G) (l-G)dUm(F) •
~
Define the following random variables:
(Z.15)
A=AJ = f
JI(F,G)U(F) (l-F)dG ,
B = BJ = f
JZ(F,G)V(G) (l-F)dG ,
C = CJ = f
J(F,G)U(F)dG ,
o= OJ = f V(G)d{J(F,G) (I-F)}
PROPOSITION 2.4.
,
A = AJ
=f
J1(F,G)U(F) (l-G)dF ,
B = BJ
=f
J 2(F,G)V(G) (l-G)dF ,
C =CJ
= f J(F,G)V(G)dF,
D = DJ
=f
and
U(F)d{J(F,G) (I-G)} •
If the score function J is in )(3/2)" then under the
proportionaZ 1u:u:ards mode Z~
15
PROPOSITIOt\ 2.5. Under the conditions of Theorem
((y)~ (~-j(), (f-)~ (K~-K))
"p
2.2~
((l-)')~A+)'~B- (l-)')~C -)'~D,
(l-)')~A+)'~B-)'~C-(l-)')~D)
as
Proposition 2.5 is a corollary of Proposition 2.4 and (2.14).
of Proposition 2.4 is delayed until Section 4.
From the basic identity (l-F)dG
-
and B - 6B = O.
= 6(1-G)dF,
m,n"
oo
•
The proof
-
it follows that A - eA • 0
Hence by (2.11) and Proposition 2.5, under the conditions of
Theorem 2.2,
where
(2.16)
The limiting random variable ZJ is normally distributed \\"ith mean zero,
and by the independence of the processes U and r,
(2.17)
since
-
Var ZJ. {(l-),) • \'ar(C-eD)
+ ). •
- -
Yar(€'C-D)} 6/K • K
e • K/K. Combining calculations (2), (4), and (5) of the Appendix, and
the basic identity (l-F)dG· 6(1-G)dF, yields
16
Var(C-8D) • Var C - 28 Cov(C,D) + 82 Var D
(2.18)
• 8
J J 2(F,G) (I-F) (l-G)dG
•
Since the proportional hazards rodel can be written as F. Gu/ e), then by
interchanging the roles of F and G and replacing e with l/e, it follo\'ls that
Var(eC - ih
(2.19)
• a2 Var(C - a-I 0)
• e2e- l
J J 2(F,G) (I-F) (l-G)dF
• a J J 2(F,G) (I-F) (l-G)dF .
From (2.17), (2.18) and (2.19), we have that
\>onere a2 (J) is given by (2.6) and H).
= ).F
+
(l-:\)G.
Hence, to prove Theorems
2.1 and 2.2, it remains only to verify Propositions 2.3 and 2.4.
3. An optimal rank estimator. By Theorem 2.2, the nonparametric rank
estimators gJ with J in )(3/2) are, under a proportional hazards model,
asymptotically normally distributed with variance e2a 2(J) defined in (2.6).
From Remark 2.3, oZ(J) is independent of the continuous df F. In fact, a2 (J) a 2(J,a,A) depends only on the score function J, the parameter a, and the
limiting proportion )..
PROPOSITION 3.1. Suppose G• Fe Lrith 0 < a <
o<
A< 1~
by J. where
0
2
CD
and F continuous.
FoZ'
(J) is minimized over Bcore functions J in J (3/2) uniformZy in F
17
J.(S,t) :: {),(l-s)
+
e(l-A)(l-t)}-l f01' (s,t) ( [0,1)2 •
l'urthe%'fTIo1'e~·
(3.1)
02 CJ .) ..
{J
J.(F,G) (l-F)dG}-l
.. {11[~
o
PROOF.
e(l-I,)
Since H). :: AF
+
+ e(l_~)I(f-l)/e]-l
dI}-l •
(l-A)G and G .. I - (l-F)e, then dH; .. 0.
+
(l-F)'?-?) dF so that
(I-F)dH~ .. J;l(F,G)dF .
(3.2)
Hence,
.,
JJ;(F,G) (I-F) (I-G)dH"
2
o (J*)
• {J J.(F,G)(l-F)dG}-l by (3.2)
.. {11[~
o
+
€(I_A)I(e-l)/9]-1 dI}-l
,,"here the last equality follows from the change of \'ariables 1 - G .. I and
regrouping of tenns.
Applying first (2.1), second the
Cauchy-Sch~arz
inequalit)·, and third both
(2.1) and (3.2), we have that, for all J in )(3/2),
{J
J(F,G) (l-F)dG) • {f J(F,G) (l-G)dF)
• e{f
s
J(F,G) (1-G)dF}2
e{J J 2(F,G) (l-G)J;l(F,G)dF} • {f J.(F,G) (l-G)dF}
.. {f J 2 (F,G) (l-F) (l-G)dH A} •
{J
J.(F,G) (l-F)dG}
18
or, after rearranging tenns, that a2 (J.) s a2 (J), with equality if and only if
o
J • cJ. for some constant c.
Since the optimal score ftmction J. involves the unknown parameter 6, a
two-step estimation procedure is required:
(consistent) estimator
go ; then
J.(s,t) - {A(l-s)
+
First obtain a preliminary
g
estimate 6 by
1\ ,
where
J.
goCI-A)CI-t)}-l for (s,t)
[0,1]2
!
Provided the initial estimator is consistent, the two-step estimator 6,...
"
asymptotically achieves the minimum asymptotic variance e2a 2 CJ.) for
t~ class
of estimators
gJ
indexed by J in J (3/2) .
MOIDI 3.1 (Asymptotic Normality 0;
estimator of 6 and AN" A
g,... ). Suppose
J*
(0,1) as m,n"
!
,...
6
0
is a consistent
Then, under tr.e proportional
(I) •
hazards mode l,
as m,n"
(I)
•
1\
Furthermore, if eo "p 6 uniformly in F, then the convergence in distribution
is aZso unif01'f1l in F.
PROOF.
Suppose the initial estimator ~o is consistent .. (If the
consistency is tmifom in F, then by Remark 2.1 the convergence asserted in
Theorem 3.1 will also be tmifom in F.)
0o(t,e) having P{Oe} > I sufficiently large.
E
on which
Given
Igo
-
E
eI
> 0, there is a subset no
< e/2 for all m and n
Also suppose that AN" A ! (0,1) as m,n ..
(I) •
=
19
•
But since
and by Theorem 2.2,
as nf',n"
cc ,
then· it suffices to sho" that
"
(f" - "
6J ).. 0
J",
'"
P
as m,n ..
ClC
•
l\lth the notation of (2.i), the spirit of (2.11), and the identity
(3.3}
"
"''here Jo(s,t) = -(l-i.)(l-t)J",(s,t)J",(s,t)
for (s,t)
£
~
[0,1]-,
we find that
(3.4)
- KN,J
By
~
assumption,
(ao -8)
•
0
p
(1).
By
o
•
nm)~
(1\
"
(8J", - 8)} /\.
Theorem 2.2, (nm/N)~
~
•
I"J",
(aJ '" -6)
• Opel).
It suffices to consider the behavior of the remaining terms in (3.4) on the
subset no'
Then, for m and n sufficiently large, J o is in )(3/2). Hence, by
20
Proposition 2.5,
mn)~ (leN
-....
J
(1f
J
-Ie
'00
) • 0pU) and
~ J
and by Proposition 2.2,
(mn)~
1f
(ICN J
-K
J
'00
) •
°
p
(1)
• Opel) and
• , 0
K
A
N,J Ir
•
(K
0 -
N,J Ir
leN J ) +
• ,
'Ie
K~
.,
J
#I
o
4.
Proofs.
Our proofs of Propositions 2.3 and 2.4, and hence of Theorems
2.1 and 2.2 respectively, rely on several
empirical df.
~ell-know
properties of the lDlifonn
For convenience, we reformulate the relevant results via the
inverse transformation (Shorack (1972), Proposition Al) in Lemmas A,
a,
and C.
Also, as discussed in Section 2, our proof of Proposition 2.4 uses the specially
The inequalities in (4.1) below are a direct
constructed processes.
consequence of the strong convergence property (2.12).
LBNA A (Shorack's (1972) linear bounds).
n = n£
< 1 and a subset
'1. = 0l,N,e:
Given
n(l-G(x))
$
> 0" there exists 0 <
having P{nl} > 1 - e: on t.ihich
1 - Fm(x) :s (l-F(x) )/n for x
n(l-F(x)) s 1 - Fm(x)
£
£
for x s
+
lR "
~
,
1 - G (x) :s (l-G(x) )/n for x
n
£
+
lR ,
for x
$
Y •
1 - Gn(x)
n
and
21
Wt-lA. B (We ZZner ' s (19 77b) "near'Ly 'Linear" bounds).
•
msts 0 <
properties:
n :: n(t)
For a'Ll
<
~
n2
~
2
n
0
0 0 0
fOl~
denote ti:e set
0.4"
[0,1]
r:h(F) (Fm,F) -+a.s. 0 ae m -+
<
Xm"
£
lR .-
+
a-..;,d
DC
H( ; )"
E:
•
Proposition 2.3 concerns the strong consistency of
function J is bounded by MR~ for some () < 2.
H( { )
"
a'L: nomlegat:ve deareasing .-lUnatione h
:-::eYl fOr' an2 h
£
lR
fOl' x :s Y •
n
nel-Gex))1 s 1 - Gn(x)
Lemna B, on Lenvna C \\i th h
+
£
X -
1 - Gn(x) s e(l-G(x) )/n)l/T for' x
Or!
:: no(w, '1")
i1"'/f'Ly
n(l-F(x))' s 1 - Fm(x)
Let H( ~)
> 1" there
tJi th P{nZ}. 1 having the fO HMng
1 - Fm(x) s ((l-F(x))/n)l/1 for x
TY.eorem) •
t
Po " there are integere m :: m (w, T) and n
W f
for tJhiah m ~ m a'1d n
o
and a subset
Given
/I
eJ
"..hen the score
Our proof relies heavily on
defined by
h :: (1- 1) 1- 0 on [0,1 ]
for some 6 > 0, and on Lemna 4.1 below.
Proposition 2.4 gives the asymptotic
nonnality of "
eJ \"hen J has a gl'O\\'th rate a < 3/2.
processes with the weight flDlction q
f
Our proof uses the special
Q(.j.) defined by
q :: (1-1)~ - 6 on [0,1]
22
for some 6 >
o.
With this choice of q e: QU), the following consequence of
(2.12) will be very useful (cf. Shorack (1972), Remark AS):
For almost every
sample path w, there is a constant M for which
w
IVI s
(4.1)
q and
IV w
ISM q for all n ~ 1 .
w
n
~I
Lenmas A and C and LeJmla 5.2 below are also used in our proof of Proposition
2.4 •. The loss of half a power in the growth rate of the score function from
a < 2 for strong consistency to a < 3/2 for asymptotic nonnality corresponds
to the half a power difference between the weight functions h and q.
Recall that the bounding function R is defined by R(s,t) ::
{(l-s) + (l-t)}-1 for CS,t)
IDN~.
For 0 <
4.1.
£
e<
(Xl
[0,1]2.
and a < 2" there exists aT> 1 and a
<5
> 0
T ~
1
8uoh that
(4.2)
J R~CF,G)h(G)dF
(4.3)
J Ra +1 (1_(I_F)T , 1-(l-G)T)h(FHI-G)l/T
dF <
ao"
(4.4)
J Ra+lCl-Cl-F)T, l-Cl-G)T)hCGHl-G)l/T
dF <
(Xl.
Moreover" if ex
and au'
<5 ~
~
2 and
e•
<
(Xl
"
and
1" then eaoh integraZ is infinite for aU
O.
IDMA 4.2.
For 0 <
e<
(Xl
and a < 3/2" there is a
(4.5)
J RaCF,G)qCG)dF
(4.6)
J Ra +1 (F,G)qCG)(1-F)dF
(4.7)
I
<
(Xl
"
<
RQ+lCF,G)q(G) (l-G)dF <
IlO "
IlO of
<5
> 0 8uoh that
23
(4.8)
I
RQ+1(F,G)q(G) (l-F)dG < ~ ~
(4.9)
I
Q
R +1 (F,G)q(G)h(G)dF
E'urtherrmol'e~
if
Q
~
3/2 and (, •
<
1~
~
and
•
then each integ1'aZ is infinite fo1' all
o > o.
~1~
4.1.
The second statement of Lemma 4.1 (Lemma 4.2) indicates that
the strong consistency (asymptotic normality) of
score function J has a gro',1:h rate ex
~
2 (\:1
~
e"J
at
e•
1 fails if the
3/2).
The proofs of Lemmas 4.1 and 4.2 rely on simple algebra and inequalities
such as Rex (F,G) ~ (I-F) -~(l"'-)
. for ex > o.
_
•
PROOF OF
PROPOSITIO~
~.3.
Assume G ..
_0
F~
ldth 0 < €I <
0:
and F continuous .
Assume the score function J satisfies the differentiability condition (Jl) and
the boundedness condition (J2) with grO\'1:h rate ex
<
z.
Condition (J2), the inequality 1 - G s h(G), (4.2) of Lemma 4.1, and the
Strong La',' of Large Nlunbers (SLL'\) imply that d:\ "a. s. a as m,n"
Lemma C, (4.2), (J2), and the SLL'\, lve conclude that c:\ "a.s.
To consider a:\ and b , fix
K
~
:: [0, Xm"Yn ] .
(4.10)
loU f
a
~.
From
as m,n ..
r/Z and restrict attention to the interval
By Lenma B,
~
m s max{Fm,F} ~ 1 - n(l-F)l
~. s max {G ,G} ~ 1 - n (1- G) 1
n
n
and
1 - G s ((l_G)/n)l/l •
n
e
Using condition (J2) and the fact that R increases in each coordinate, we
conclude that, for i .. 1,2,
ex>.
24
•
(4.11)
a+l
IJ i (~m''''n) I s
MR
S
(~m''''n)
MRa+l(l-nCl-F) T , l-nCl-G) T)
• Mn-(a+l) Ra +1 (1_(1_F)T ,l-(l-G)T) •
The last equality reflects the "reproducing increasing" character of the
bolDlding flDlction R (Shorack (1972), p. 426).
JsNIJ l C41m,'Jin)IIFm -
l(2Z aX :: l(2z
Nn-(a+l)-(l/T)
S
x
..
a.s.
p
h(F)
(F
m'
Fl (l-Gn )dFm
F)
J Ra +1 (l_(1_F)T, l-(l-G)T)h(F)(l-G)l/1
0 as m,n"
using Lemma C, (4.3), and the
SLL~.
dF
m
00
A similar argument with (4.3) replaced by
(4.4) sho\\s that 1,...~'2 b,."
1'4
a. s . 0 as m,n"
and b;\j "a. s. 0 as m,n ..
Hence,
00.
Since P{r.2} • 1, then a.,"
. .:'\ a. s . 0
0
00 •
PROOF OF PROPOSITION Z.4.
Assume
G· Fe
\dth 0 <
e<
00
and F continuous
and asstune J is in J (3/Z) • As explained in Section 2, we will use the special
processes which have the strong convergence properties (Z.12).
The random variables ~, ~, ~, and ~ are symnetric to ~,
~
respectively.
Also, the arguments for
~
and
~
reasons, we present only the details for ~, ~, and
By integration by parts,
~
--J
VnCG)diJ(F,G) (l-F)}
are similar.
ON.
B:-J,
S,
and
For these
25
•
Using the differentiability condition (Jl) , we have that ~ - ~l
and D· Dl + DZ
+
D3 ,
+
~2
+
~3
where
D~l
= J Jl(F,G)\~(G)(l-F)dF
,
~2
= J J 2(F,G)Vn (G) (l-F)dG
,
~3 = -J J(F,G)\~(G)dF ,
B1 = J J1(F,G)¥(G)(1-F~dF
°=J
2
,
J 2 (F,G)¥(G)(1-F)dG,
B3 = -J
and
J(F,G)¥(G)dF .
Using the boundedness condition (J2), the strong convergence property
~
given in (2.12), and (4.6) of Lemma 4.2, we have that
ID~l -
1\!
If
a
J 1 (F,G) (\'n(G)-V(G))(l-F)dF!
s ~1:'q(G)(rn(G), reG))
"'p 0 as m,n'"
00
J ROo + 1 (F,G)q(GHI-F)dF
•
Repeating this argument twice, replacing (4.6) first b)' (4.8) and then by
(4.5), shows that ~2"'P
rn,n ...
00 •
and ~3 "'p D3 as m,n'"
By adding and subtracting terms, we ha,'e that
Dz
QC.
5
I~ - BI s L
ial
e
where
~ . and
1
I~ -
.z
Cis
L
j-1
-
-
Hence, DN "'p D as
~.
J
,
26
•
Bm. ::
f~nsy IJ2 (4)m,Il.'n) - J 2 (F,G)1 • IVn(G)1 (l-Gn)dFm '
~2 ::
f
~3 ::
fI J 2 (F,G)1
• IVn(G) - V(G)ICl-Gn)dFm
~4 ::
fI J 2 (F,G)1
• IVCG)I • IGn - GldFm '
~s
::
c IJ2 (4)m'lPn ) - J 2 (F,G)1 • IVn(G)I (l-Gn)dFm '
~nSy
II J 2 CF,G)VCG)Cl-G)d(Fm-F)I ,
~l :: IIJ(F,G)I • IVnCG) - V(G)ldFm ,
C~2 ::
\\ith S
y
If
JCF ,G)V(G)d(Fm-F)
and
I ,
:: [0, F-I(l-y)] • [0, G- l Cl-y8)] for 0 < y «
1 and SC :: R\S •
y
y
To
complete the proof of Proposition 2.4, it suffices to show that ~i "'p 0 for
i • 1, •.• , 5 and that CNj "'p 0 for j • 1,2.
Consider
~S'
For all sample points w, the integrand J 2 CF,G)VCG)CI-G)
is a continuous function on ]R+.
If the score function J grows so slowly that
the integrand is bounded on R +, then by the weak convergence of Fm to F, \ie
have that B ... 0 as m'" co. Suppose, on the other hand, that the integrand
NS
of Ro is unbounded. In vie\i of (2.12), choose M so that IVCG) ISM q:CG)
~S
w
w
hnere q :: q(I_I)o/2 • CI-I)~-o/2 e: QU). Hence, by CJ2) and (4.1),
IJ (F,G)VCG) (l-G)/MRQ+lCF,G)q(G) (l-G)I s
2
Since (4.7) and the SLLN imply that lim
conclude that
~S -+
0 as m -+
co
f
Mw(1-G)O/2
-+
0 •
RQ+l(F,G)qCGHl-G)dF <
m
Ccf. Breiman (1968), p. 164).
argument may be used for ~2'
Applying (J2) and (4.1) gives the inequality
co,
we
A similar
27
•
"'a.s. 0 as m,n'"
co ,
where the convergence follows from Lemma C, (4.9), and the SL1..'\.
Using (J2), (Z.12), (4.7), and the
SLL~,
we have that, on the subset
r.1
of Lemrna A,
1" R'3 := 1r,
~"1 -~
~;1
fl J 2 (F,G)
... 0 as m,n ..
p
e
Since P{r?l} > 1 -
\dth
£
argument using (J2),
m,n ...
£
(~.12),
i:rn (G)
cc
- reG) I (l-G )dF
n
m
•
> 0 arbitrary, ~3 "p 0 as m,n'"
(4.5), and the
SLL~
shO\,'s that
cc.
C~l
A similar
...P 0 as
co •
~ext
inten"a1
consider
Sx
Bxz.
If \,'e fix :..
€
$11 and restrict our attention to the
:= [0 , X
m1\ Yn ], then by repeating the arguments behind (4.10) and
(4.11) 'Kith Lemma B replaced by Lemma A, we conclude that
> 0, there is a constant ~fe: and a subset ne: having
In vie\,' of (2.12) giYen
£
p(n£) > 1 - e: on \"hich
IVnl s ~fe: q for all n ~ 1.
Thus,
28
•
1n- nO ~2 :: 10 nO
-J. E
1 E
$
f
c IJ2C~m,1Vn) - J 2 CF,G)
~nSy
MM n- 1 {n- Ca+1 )
+ 1}
E
I SC
I•
IVnCG) I C1-Gn )dFm
Ra +1 CF,G)qCG)C1-G)dF
m
y
+ . l-M n- 1 {n- Ccx +1) + l}
a.s.
E
f
s~
Ra + 1 CF,G)qCG)(1-G)dF:: By as m +
~2
arbitrary, we conclude that
Since P{n n roe:} > 1 - 2e: \vith
l
+p 0 as m,n +
,
By the monotone
\mere the convergence follows from C4.7) and the SLL'J.
convergence theorem, By • 0 as y • O.
00
E
> 0
and y • O.
00
1
Finally, consider ~l' For x € SN n Sy \dth Sy
[0, F- (1-y)] •
-1
e
[O,G (l-y)] and 0 < y« 1, on the subset n of LellJTla A,
1
=
~mCx)
s max{Fm(x) , F(xH :s 1 - n(l-F(x»
:s 1 - ny
. ~nCx) s max{Gn(x) ,G(x)} s 1 - n(l-G(x»
:s 1 - n'(
By (J3) , the partial derivative J
2
and
e
•
is tmiformly continuous on [0, I-nY]
x
[0 , 1- ny e], and hence
1
n1
~l :: ln fSNnsy IJ2(~m,1tn) - J 2 CF,G) I
1
s
+
Mw n- 1
a.s.
•
In
sup
1 ~nSy
0 as m,n +
00
IJ2C~m'~n)
•
IVn(G) I (l-Gn)dFm
- JZ(F,G)
I
x
Is
y
q(G)(l-G)dFm
,
where the inequality follows from (4.1) and LellJTla A and the convergence from
the G1ivenko-Cantelli Theorem and the SLLN.
arbitrary, we have that, for fixed y > 0,
Since P{n1} > 1 -
~1
+p 0 as m,n +
E
00.
with
E
> 0
o
29
s.
e.
Regular rank estimators of
The two-step estimator of Section 3 is
asymptotically optimal (in the sense of minimum asymptotic variance) among the
class of nonparametric rank estimators
next question is
~nether
"eJ
with J in J (3/2) • Naturally, the
the two-step estimator is also asymptotically optimal
among all regular rank estimators of
e. This question leads us to consider the
.
Fisher information for estimating e based on the rank vector.
Under the
proportional hazards model, the distribution of the rank vector depends on the
parameter e but not on the df F.
Thus the reduction to rank estimators
eliminates dependence on the (unspecified) df F and produces a parametric
setting
~ith
only the parameter
e.
.~
expression for the Fisher information
can be hTitten explicitly using the probability distribution of the rank vector
R:
(5.1)
(cf. Savage (1956), p. 599).
Ho~ever,
vector complicates its evaluation.
the dependence among members of the rank
Instead of calculating the Fisher
information directly, we address the question of optimality by studying the
as)~totic
alternative
behavior of the likelihood ratio of the rank vector hnen the
h)~othesis
is a local alternative.
representation theorem shows that our
th~-step
.~p1~·ing ~jek's
estimator achieves full
asymptotic efficienC)p among alZ regular rank estimators of
For h
rank vector
£
(1970)
e.
R, set eN :: e + h(mn/~) -~ and define the likelihood ratio of the
!. !.
by
30
for m ~
JIb
and n
for which m ~
Jib
~
!1t,
and n
lJb
where
~!1t
and
!1t
denote the smallest pair of integers
iJrq:>lies that 6N > O. Thus,
~(h,!.l
measures the
likelihood of the observed rank vector lDlder the proportional hazards model
with parameter 6 relative to a local alternative which converges to 6 at the
rate (DDl/N) -~ •
TIiEORfM 5.1 (Asymptotic LognormaUty of ~(h,~)). Foro'aH h
~,
and n
~
nh' the ZikeZihood ratio of the
~nk
vectoro
~
E:
lR .. m ~
may be ezproessed as
(5.2)
I.Jheroe
2 f1
rN = C8 20 *N
'
r = (6 20;)-1,
O;~
= f~{AN
~ f~{A
+
and
6(1_A N)I C8 - 1)/e}-1 dI
eC1_A)I ce - 1)/8}-1 dI
+
= 0;2
,
and I.Jheroe, undero the prooporotionaZ r.azards modeZ l.Jith parametero
~(~) ~d N(O,r)
TN(h,R)
-
as m,n
~
00 ..
provided AN
~
~
P
e,
and
0 for aU h
E:
lR
A E: (0,1).
In other words, for all h
E:
R lDlder the proportional hazards model with
parameter 6, the likelihood ratio of the rank vector converges in distribution
2
2
to a random variable Z for which log Z - N( -~ h r , h r). Since
31
v
EZ • E exp{log Z} • 1, then LeCam's first lemma (H4jek and Sidak (1967), p. 203)
implies the fol1o\\ing result.
COROLLARY 5.1.
For aU h ( :R" the prcbabi lity measures Pe and PeN are
contiguous.
The proof of Theorem 5.1 is dela)'ed until the end of this section.
,.
Hajek (1970) proves a general representation theorem for the limiting law
of a regular sequence of estimators of a k-dimensiona1 vector of parameters.
For the special case of regular sequences of rank estimators of the parameter
in the proportional hazards model, Hajek's
5.1.
h)~otheses
~
are verified in Theorem
The fol1rn\ing result is a direct application of Hajek's theorem.
,.
:...'::~=: Zir.":~tiY!.2
a~
an
d.-" L; i.e., for aZZ x
continuitE! ,?c:'l:te x
£
R
L(x) •
lJhere
~(.
-1
Ir )
K(u) is a df on
denotee the
m tr.at
In other words, if
€
0: ti-:e
oo
f_~
,:o~aZ
IR,
df L.
,
Q(x-u r
-1
::'7: en
)dK(u)
d.t 1.::'tr. mean zero ar.C= val'i:mce r -1 " and
depends or: the choice of the sequence
~
~.
is a regular sequence of rank estimators of
under the proportional hazards model wi th parameter eN'
e, then
32
0;)
for all hER, where N(O,r- l ) • N(O ,a 2
K.
and A are independent and A has df
Furthenoore, the random variable A depends on the choice of the sequence of
estimators.
In particular, the asymptotic variance of a regular sequence of
rank estimators is botmded below by r- l :: a2
Our two-step estimator ~
0;.
A
J*
has asymptotic variance which achieves this botmd, and by the following result,
the sequence of two-step estimators is regular.
Hence,
A
aA
is asymptotically
J*
optimal among all regular sequences of rank estimators of
PROPOSITION 5.1.
A
1"ank estiT"lators a is
J
PROOF.
For eaah saore fUnction J in
1"eguz.ar~
provided AN'" A
E:
a.
)(3/2)~
the sequenae of
(0,1).
CAlr proofs of Theorems 2.2 and 5.1 rely on the specially
constructed empirical processes (discussed in Section 2).
From these two
proofs, we conclude that, for the specially constructed versions with J
)(3/2) and h ( lR, tmder a proportional hazards rode! with parameter
provided AN'" ).
in (2.16).
E:
(0,1), where
r :: (a 20;) -1
::
(a 20 2 (J.)) -1
E:
a,
and ZJ is defined
From our representation of ZJ and ZJ.' it follows that Z has a
bivariate normal distribution with
Since Var ZJ •
e20 2(J)
and Var ZJ. •
a20 2 (J.)
::
r- l ,
then from (1), (3), and
(5) of the Appendix, the correlation of ZJ and ZJ. is equal to the square
33
e
root of the asymptotic relative efficiency of
gJ
eJif;
to
Le.,
and hence the covariance is
Thus the covariance matrix of Z is
v
B)' LeCam's third lemma (cf. Hajek and Sidak l196:), p. 208), under a
proportional ha:ards model \dth parameter
for all J
E
)(3/2) and all h
RDIARK 5.1.
E
e~.
lR, provided
A~ -+ \
E
(0,1).
In our t\,'o-sample proportional hazards model \dth uncensored
data, Cox's (1972) partial likelihood corresponding to his general regression
model for censored data reduces to the likelihood of the rank vector
Cox's estimator
"eC
of
a (::
exp(S) in Cox's notation) is simply the
~,
and
~1axiJmJm
This sequence of estimators "
aC
is regular and asymptotically achieves the miniJmJm asymptotic variance r- l ::
a2 provided AN -+ A E (0,1). To verify this statement, denote the score
Likelihood Estimator O'ILE) based on the ranks.
a;,
function of the ranks by ~(e ,~)
:: alae
log P6 {~}.
With TNl (~),
rN'
and ~(~)
34
defined and investigated in Proposition 5.2 below, Theorem 5.1 (see (5.2) and
(5.5)), and Proposition 5.3 respectively, a Taylor expansion of the score
ftDlction yields
[mnN )~
" - S)
CSC
mn) -1 as
a ~(s,~)1
- -1 • [mn)
-~
= 1- [N
¥
~(e,~)
-
A € (0,1), where e is between "eC and e and where ZJ tlr
~(0,e2o;) is defined in (2.16). The regularity of @C follows from LeCam's
provided AX
-I-
third lemna, provided
PROOF OF
THEO~I
A~
-I-
5.1.
several useful identities.
A ! (0,1).
Before expanding the likelihood ratio, we introduce
In the spirit of (3.2), with
we have that
-1
(5.3)
(l-F)dHN = J*NCF,G)dF .
Hence, it follows that
(5.4)
and that
f
J*N(F,G) (l-G)dHN • !(l-G) (l-F)-l dF • !Cl-F)e-l dF • a-I
35
(5.5)
= a-I J J*~(F,G)(I-G)(I-F)J*N(F,G)~
= a- 2 J J*~(F,G)(I-F)dG
by the change of variables 1 - G = I.
by (5.3) and (2.1)
Finally, using (1.1) and (1. 2), \\'e find
that for Z = 1 and 2,
(~ )
(5.6)
-;'/2
N
kIl(~/(Mk+6\)))'
-1
= A"
•\
:\0\\'
.
£-1 ( ) (2·'.)/2,
(1-\,.)~
J J~,.(Fm,Gn ) (1-Gn ) 9. dH..,,\ •
j\
,\
consider the likelihood ratio.
.\
Using first the probability
distribution of the rank vector ~ in (5.1) and then the expansion log(l+x) • x x 2/2 + x 3/3(1+v)3 where v = ·.(x) is between zero and x, \\'e have that, for all
h
£
lR, m ~
lllt' and n
~
nh ,
36
.1'
- exp[n
logCl+hCmn/Nf~ a-I)
N
log(l+h(mn/Nf~
l
-
k-l
(N /c/. +aN )))]
k
k
lc
- exp[nh(mn/N) -~ a-I - nh 2 (mn/Nf 1 a- 2 / 2
(5.7)
+ nh 3 (mn/N)-3/2 a- 3 /3(I+V~I)3
- h(mn/Nf
~
N
L
k=1
2
+ h Cmn/N)-1
I
Nk/c,\+aNk )
CNk/CMk+aNk))2/2
k=1
where vi\1 is bet\,'een zero and h(mn/~)
h(mn/N) -~ N /CMk +~Nk)
k
-~
a
-1 and v (k) is between zero and
N2
for k - 1, ... , N.
Define
~(~) , n[W-r" e- 1 - [~r~
TN(h,~)
Tm (~)
=h 2 TN1 (~)/2
_ (mn) -1 ~
=
1["
L
k-l
_ (mn) -3/2
TN2(h,~ = 1["
- h
3
kt Nk/(Mk +eNk ) •
TN2(h,~)/3
(Nk/(~+aNk))
2
+ h
-1
3
TN3(h,~)/3
- AN (I-AN)
f
N
3
J l CNk/(~ +eNk Hl+v N2 (k))) ,
TN3(h,~) = nCmn/N)-3/2
3
e- /Cl+v
N1
)3 •
2
,
J*N(F,GHI-G)
and
2
dIN'
37
By adding and subtracting AN-1 (I-AN)
J
2
J*N(F,G)
(I-G) 2
of (5.7) and using (5.5), the likelihood ratio
~(h,~)
dl\
in the exponent
is now expressed in the
fom of (5.2).
Clearly rN ~ r provided
random variable
t.x(~)
~ (~)
.
~'N ~
A as m,n
~
ex>.
Next, consider the leading
Combining (5.4) and (5.6) "ith £. = 1, \,'e can re\\Tite
as
By Proposition 5.3 be1o"" tmder the proportional hazards model \dth parameter
e,
~(~) ~d
i'\(O,n provided J,:\
~ \ E:
(0,1).
as~~totica11y negligible
Finally, consider the
component
all k = 1,. ~ . , X, since v
is beb'een zero and h (mn/:\) _1.2
X1
bet\\'een zero and h(mn/X)
-~
e-1,
Nk/(,\+€N ), and Nk/(Mk+tNk):s 6
k
and n sufficiently large, \.1;\1 " '\2(k) <:
-~.
T~(h,~).
-1
v
N2
For
(k) is
,then form
Hence,
-. ° as
m,n
~
oc
° as
m,n
~
ex>
and
~
provided AN
~ ~ E:
(0,1).
,
By (5.6) with R.. 2, T (!9 can be expressed as
N1
38
. By Proposition 5.2 below, \Dlder the proportional hazards JOOc1el with parameter
a, Tm Q9
.p 0 provided AN· A
CO,l).
E
This completes the proof of Theorem
0
5.1.
Our proofs of Propositions 5.2 and 5.3 rely on Lemmas 5.1 and 5.2,
respectively.
((l-s)
and q
(l-t) }-1 for
+
E
Recall that the bo\Dlding f\Dlction R is defined by RCs, t) (s,t)
E
[0,1]2 and that the ,...eight f\Dlctions h
QU) are defined by h
IDNA 5.1.
(1-I)1- 6 and q
==
-8
Suppose G· F wi th 0 <
e<
00
= (l-I)~-c
Then~
fOl'
~
J R2 CF,G) (1-G)2
(5.9)
J Ri CF,GHl-G)i-1
h(G)dG <
00
fOl' i
= 1,2 ~
(5.10)
J Ri(F,G) (1_G)i-1
h(F)dG <
00
fOl' i
= 1,2
uni.~omZy
H(i-)
for some 6 > o.
and F con tinuous •
(5.8)
dG <
E
00
and
in F.
IDM-\ 5.2.
Suppose G
= F-8
e<
wi th 0 <
00
and F con tinuou8.
,!",I;en~ ,,~ol'
aZZ 0 < 6 < {2(l+(lt\e)}-1~
(5.11)
J R(F,G)q(G)dC!
(5.12)
J R2 (F,G)q(G) (l-G)dG
(5.13)
J R2 (F,G)q(G)h(G)dG
<
00
(5.14)
J R2 (F,G)qCG)hCF)dG
<
00
<
00
~
<
00
~
~
and
uniforml.y in F.
PROPOSITION 5.2.
Suppose
C· Fe
r.Jith 0 <
a<
00
and F continuous.
If
39
as m,n"'
uniformZy in F.
co
PROOF.
The lDlifonnity in F follows from Remark 2.1.
We find it useful to note that, since
°
< A* ~ A:\ ~ AIII < 1,
(5.15)
a~d,
for all (s,t)
?
!
[0,1]-,
(5.16)
Use the definitions of H!\ and
-TN -= f
TN
-= f
~
to decompose T\1 (~) as
2
2
Jic\.,(F
.\ m,Gn ) (I-G)
n dFm -
f
2
Jic,,(F,G)
(I-G) 2 dF and
,\
2
222
J*,.(F
(I-G) dG.
,\ m,Gn )(l-G)
n dG n - f JicN(F,G)
'
_ 2
Define the score flDlction I N by IN(s,t) = J*N(s,tHl-t) for (s,t) (
[0,1] 2• Clearly jN satisfies the differentiability condition (Jl); and, by
(5.15) and (5.16), ~ satisfies the bOlDldedness condition (J2) with grO\\'th rate
40
e'
1 and constant
~.
Noting that (with no change in the proof) the score
function J in Proposition 2.2 may be replaced by a sequence of score functions
{I } which satisfy (J1) and (J2) with some constant M > 0 and some growth rate
N
a < 2 unifonnly in N, we conclude that TN'"
• a. s. 0 as m,n'"
0lI •
After adding and subtracting terms, we have that
Khere , for i = 1, 2 ,
b".
.d
-= f
3-i
iG - GI (I-G) 2 dG ,
J*N
(F ,Gi
•
m n )J*~:(F,G)
j~
n
n
n
c".
.,1
-= f
2
J*,,(F,G)
(l-Gn ) 2-i (I-G) i-II Gn - GldG n '
.,
and
Hence, to prove the proposition, it suffices to show that each of the above
terms is as)11lptotically negligible in probability.
Since the subset
~\
Lemma A of Section 4 (on which the linear bounds hold) has P{nI} > 1 £:
of
£:
\';ith
> 0 arbitrary, it suffices to consider the as}11lptotic behavior of each term
on the subset 1'2 ,
1
By (5.15), (5.8), and the
SLL~, cL ...
0 as m,n'"
a.s.
Using (5.15) and Lemma A, we have that, for i • 1,2,
~
f
•.2 i-2 Ph(G)(G~,G)
s ~n
"'a.s. 0 as m,n'"
0lI
,
00 •
2
R (F,G)(l-G)h(G)dG n
41
where the convergence follows from Lerrma C of Section 4, (5.9), and the SLLN.
Hence
~i
"p 0 as m,n"
for i
co
II:
1,2.
In each of the remaining terms ~i and b~i for i • 1,2, since the
restricts the range of integration from R+ to [0, Y ], 'ft'e also
n
restrict our attention to the interval [O,Y ]. No,~, since the function R is
n
increasing in each coordinate, on the subset r2 we have
l
integrand dG
n
J 1r,,(F ,G ) s
~11r
R(Fm,Gn ) by (5.15)
$
~~
R(l, l-n(l-G))
., m n
= ~t1r n- 1 (1_G)-1 .
Hence, for i
1,2,
z
y
1n
.'1
b..... .. 1n
1"1
1,1
JI'on
3-i(F , G) J 1ri ,. (
I,G - Gl (l-G ) 2 dG
J*,.
F,G)
n
n
.,
m n"
n
f
s ~~3 ni-5 ph(G) (Gn,G)
..a. s. a
as m n ..
'
<Xl
iI
Ri
(F,G)(l-G)
- h(G)dGn
,
using Lemma·C of Section 4, (5.9), and the
SLL~
Hence, bNi "p 0 as m,n"
A similar argument Kith ph(G) (Gn,G)
co
for i
II:
1,2.
to justify the convergence.
replaced by CheF) (Fm,F) and (5.9) replaced by (5.10) shows that
m,n ...
co
for i
z
E:
"p a as
o
1,2 .
PROPOSITIOl\ 5.3.
suppose AN" A
~i
Suppose G ..
(0,1) as m,n"
Fe
<Xl.
"d N(O,r) as m,n"
co
lJith
Then
a<e<
<Xl
and F C!ontinuous~ and
42
PROOF. As in the proof of Proposition 5.2, the asserted \Dlifonnity
follows from Remark 2.1. Also, since AN
sufficiently large, 0 < A*
S
~
A E: (0,1) implies that, for m and n
AN < A* < 1, then both (5.15) and (5.16) hold.
As in the proof of Theorem 2.2, replace the empirical processes with the
specially constructed empirical processes wnich exhibit the strong convergence
properties (2.12). Use the definitions of HN and
terms to decompose - ~ (~) as the sum
(K~'J -K )
:\, *N J *N
(5.1 i)
where
K
~,J*N
and
~
and add and subtract
+
(1_).~,)3/2 A.'.
;\'":\
+
e)~~(1-,~)2
+
e-1 A:-.;~
D~
,
= J J*N(Fm,Gn)J*~(F,G)Um(F)(l-Gn)dGn
'
EN = J J*N(Fm,Gn)J*N(F,G)Vn(G) (l-Gn)dGn
'
K
J*N
~
eN'•
are defined in (2.7) and ,,'here
= J J*N(F
• ,G)Vn (G)dGn
,
and
By an integration by parts and the definition of J*N'
ON·
-eANlCl-AN)
J VnCG)d{J*N(F,G)(l-G)}
• J VnCG)d{J*N(F,G) (I-F)} •
BN -
43
We will show that, provided AN'" A ( (0,1),
(S.18)
...
P
((1-)')~ A +).~ B -).~ C
J 'Ie
J 'Ie
J 'Ie
-
(l-)')~ DJ ,A', B' , C' , DJ ) ,
'Ie
'Ie
J· 1 (S,t)
for CS,t) ( [0,1]2, and where
A'
= J J;CF,G)U(F)CI-G)dG
.,
,
B' = J J;CF,G)VCG)C1-G)dG,
C'
= J J.(F,G)V(G)dG
Using (3.2) and C2.1) with H).
= ).F
+
and
.
(l-)')G shows that
J.(F,G)CI-G)dH). • (I-F)-l (l-G)dF • a-I dG ,
and hence that
44
(5.19)
-
where C • is defined in (2.15).
J
"" -a- 1 {J
:II
e
Combining (5.17), (5.18), and (5.19) gives
J.(F,GH1-G)dF} ZJ.
-a-Z{J J.(F,G) (l-F)dG} ZJ
where ZJ. is defined in (2.16) and
0; :
•
2
a (J.)
is defined in (3.1).
Hence Zr:,
is normally distributed with mean zero, and by (2.17),
To prove the proposition, it remains only to verify (5.18).
Provided AN
~
A ( (0,1), the convergence in the first and last coordinates
of (5.1) follows from the proofs of Propositions 2.5 and 2.4 respectively, with
J replaced by the sequence {J.N} since J.
and bounding constant
m,n ...
N
( )(3/2) with growth rate a • 1
M., both independent of
N, and since J.
N
~
J. as
III •
Because the argt.Dllent for
AN
is similar to that for
EN,
we omit it.
In view of the convergence J. N ... J., provided AN ~ A, it suffices to show
that (EN-B', CN-C') "'p (0,0) where the J. in B' and C' is :replaced with J.N•
\'Ie perfonn the replacement with no change in notation.
subtracting terms, we find that
By adding and
45
IBN - 13' I S
ANBNl + e(1-~)BN2 + BN3 + BN4 + BN5
ICN - C' I
~l
S
+
and
CN2 '
where
R"l
-;\
= J J;,.(F,G)J*'T(F
.,
... m,Gn ) IFm -
-!\..
R'.~
= J J;,:(F,G)J*,.(Fm,Gn)IGn
R"3
-:\
= J J;,.(F,G)
Ivn (G)
,\
1\
.\
FI
- G!
Ivn (G) I(l-Gn )dGn
Ivn (G) !(l-Gn)dGn ,
- reG) I (l-Gn )dGn ,
~4 = J J:~(F,G) !V(G) IIGn - GldGn '
.'
B~._
.\;'
= If
CI'
= J J*,,(F,G)
irn (G)
.\
-:\ l
J;,.(F
,\ ,G)V(G)(l-G)d (Gn - G) i ,
The arguments for
B~S
and
- V(G)!dGn ,
C~2
and
are similar to the one given for 1\5 in
the proof of Proposition 2.4 (see Section 4).
By replacing (4.7) in the
previous argument \·:ith (5.12) and (5.11) respectively, \\'e conclude that
and CX2
..
0 as m,n"
B~5"
0
ex: •
The arguments given in the proof of Proposition 2.4 (see Section 4) for
1\4' 1\3' and Cxl can easily be modified to handle the tenns BN4'
eNl
respectively.
BN3'
and
First, in each of the original arguments replace the
condition (J2) by the inequality (5.15).
Next, replace (4.9), (4.7), and (4.5)
in the three original arguments by (5.13), (5.12), and (5.11) respectively.
Finally, replace the three original conclusions of asymptotic negligibility of
Bx4'
~3'
and CNl by the corresponding conclusions of asymptotic negligibility
of BN4' BN3' and CKI respectively.
46
Finally, consider
BN1
and
BNZ.
Since JltN(Fm,GnHl-Gn):S;
~
by (5.16),
then by (4.1) and (5.15) we have
BNt = I
J;N(F,G)JltN(Fm,Gn)\Fm - FIIVn(G)I(l-Gn)dGn
2
:s; r.~ l-f Ph(F) (Fm,F) I R (F,G)h(F)Q(G)dG
w
n
....a
. s.
0 as m,n ....
ClO
,
where the convergence follows from Lenma C of Section 4, (5.14), and the SLLN.
Bya similar argument with Ph(F)(Fm,F) replaced by ph(G)(Gn,G) and (5.14)
replaced by (5.13), we have B'T2....
O.
N a.s.
Proposition 5.3.
This completes the proof of
o
APPENDIX
Suppose
J(3/2).
G= Fa
with 0 <
a<
CD
and F continuous.
Suppose J and
J are
in
In the following calculations, we repeatedly use (2.1), (2.13), and
the identity F = I - (I-F).
(l)Cov[f J(F,G)U(F)dG
=
II
,I j(F,G)U(F)dG]
J(F(u) ,G(u)).J(F(v) ,G(v)) [F(u) " F(v) - F(u)F(v)]dG(u)dG(v)
= f{I~
J(F(u) ,G(u))F(u)dG(u)}J(F(v) ,G(V)) (I-F(v))dG(v)
+
I{I; J(F(u)
,G(U)) (l-F(u))dG(u)}j(F(v) ,G(v))F(v)dG(v)
II
• I<I; J(F(u)
, G(u))dG(u)}J(F(v) , G(v)HI-F(v))dG(v)
- I <I~ J(F(u)
+
I{f,v J(F(u)
- I{r; J(F(u)
, G(u) HI-F(u) )dG (u)}J(F (v) , G(v)) (l-F (v))dG (v)
,G(U)) (l-F(u))dG(u)}j(F(v) ,G(v))dG(v)
,G(u)) (I-F(u))dG(u)}.J(F(v) ,G(v)) (I-F(v))dG(v)
47
..
f{I~ J(F(u) , G(u))dG(u)}J(F(v) , G(v)) (l-F(v))dG(v)
+
In; J(F(u) , G(u))dG(u)}J(F(v) , G(v)) (l-F(v))dG(v)
- (f
(2)
VarU
J(F,G) (l-F)dG] •
J(F,G) (l-F)dG] •
J(F,G)U(F)dG] ... 2 J{J~ J(F(u) , G(u))dG(u)}J(F(v) , G(v))(l-F(v))dG(v)
- [J
(3)
(f
Cov[f U(F)d[J(F,G)(l-G)] ,
... JJ J(F(v),
J(F,G)(1-F)dG]2 by (1)
Kith
J ..
J •
J J(F,G)U(F)dG]
G(v)) (F(u)
II
F(v) - F(u)F(v)]d[J(F(u), G(u))(l-G(u))]dG(v)
... J{/~' F(u)d[J(F(u) , G(u)) (l-G(u))}}J(F(v) ,G(v)) (l-F(v))dG(v)
+ J{f;(l-F(u))d[J(F(U) ,G(u))(l-G(u))]}J(F(v) , G(v))F(v)dG(v)
. . J J(F,G)J(F,G)F(l-F) (l-G)dG
- J{j~
J(F(u) , G(u)) (l-G(u))dF(u)}J(F(v) , G(v)) (l-F(v))dG(v)
- ! J(F,G)J(F,G)F(l-F) (l-G)dG
+
J{(: J(F(u)
, G(u))(l-G(u))dF(urJ(F(,·) , G(v))dG(v)
- !U; J(F(u) , G(u)) (l-G(u))dF(u)}.J(F(v) , G(v)) (l-F("))dG(v)
... JU
v ....
a
J(F(u) , G(u))dG(u)}J(F(v) , G(v))(l-G(v))dF(v)
- e[J
J(F,G) (l-G)dF] •
where the second to last equality uses (6).
[J
J(F,G) (l-G)dF]
48
(4)
Cov[f
U(F)d[J(F,G) (I-G)] ,
•
f J(F,G)U(F)dG]
J<f~ J(F(u) , G(u))dG(u)}J(F(v) , G(v)) (l-G(v))dF(v)
- a[f J(F,G)(l-G)dF]2 by (3) with J. J .
(5)
Var[J U(F)d[J(F,G) (l-G)]]
• f f[F(u)
It.
F(v) - F(u)F(v)]d[J(F(u) , G(u)) (l-G(u))]
• d [ J (F (v) , G(v) ) (1- G(v) ) )
.. Var[J(F(X) ,G(X))(l-G(X))] '''here X - F
. ! f J 2 (F,G)(1-G)(1-F)dG - [f J(F,G)(1-G)dF)2 .
(6)
lim !J(F(x) , G(x)) I (l-F(x)) (l-G(x))
X-+ClO
s ~1 • lim Ra(F(x) , G(x) )(l-F(x) )(l-G(x))
X-+ClO
.. M • 1im(s+se)-a sl+e
s~O
sM. lim s(2-a) (llt.e)
s"'O
.. 0 .
Acknowledgement.
This paper is based on part of the author's 1980 Ph.D.
dissertation, prepared at the University of Rochester under the direction of
Jon A. Wellner.
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Mass.
ProbabiZity.
Addison-Wesley Publishing Co., Reading,
49
~.
COX, D.R. (1972).
Regression models and life tables (with discussion).
J. Roy. Statist.
COX, D.R. (1975).
~~,
500.
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