survival analysis || asymptotic properties of several nonparametric multivariate distribution...

Asymptotic Properties of Several Nonparametric Multivariate Distribution FunctionEstimators under Random CensoringAuthor(s): Gregory CampbellSource: Lecture Notes-Monograph Series, Vol. 2, Survival Analysis (1982), pp. 243-256Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/4355464 .

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ASYMPTOTIC PROPERTIES OF SEVERAL NONPARAMETRIC MULTIVARIATE

DISTRIBUTION FUNCTION ESTIMATORS UNDER RANDOM CENSORING

Gregory Campbell

Laboratory of Statistical & Mathematical Methodology National Institutes of Health

1. Introduction

The problem of nonparametric estimation of a multivariate distribution

function in the presence of random censoring is considered. The multivariate

lifetimes could represent the times to death of animals in fixed-sized litters,

the failure times of components in a multicomponent system, the observations

of participants of a matched triples study, or the onset times to stages of a

disease in a patient. In the special bivariate case, there are the numerous

examples of paired data on eyes, lungs, kidneys, twins or married couples. It

is possible that the censoring is univariate or multivariate. Whereas the cen-

soring of times to death of animals in litters born at random times yet trun-

cated at a fixed time is an example of univariate censoring, the truncation at

a fixed time of measures on the participants in a matched triple study would

provide trivariate independent censoring. The study of lifelengths of twins

and married couples would provide an example of bivariate censoring with

possible dependence between the two censoring variables.

The estimation of one-dimensional distribution function estimators with

randomly censored data has been extensively developed. The product-limit esti-

mator was proposed by Kaplan and Meier (1958). Under suitable conditions,

asymptotic normality and weak convergence of this estimator was established

by Breslow and Crowley (1974) and strong uniform almost sure convergence was

proved by Foldes and Rejt?* (1981).

243



244

The bivariate problem has merited some attention recently* Campbell (1981)

estimated the bivariate distribution function under bivariate censoring for

discrete or grouped data via the EM algorithm. Leurgans, Tsai, and Crowley

(this volume) have proposed an estimator for univariate censoring that utilizes

Freund's bivariate exponential distribution. Campbell and Foldes (1982) have

proposed several estimators based on hazard gradient estimators and on products

of one-dimensional product-limit estimators. It is the weak convergence of the

latter estimators which is the purpose of this paper.

A path-dependent distribution function estimator based on the hazard gra-

dient is introduced in ?2 after some notational development. The result of

strong uniform almost sure convergence of the estimator, which was proved in

Campbell and Foldes (1982), is presented.

A topological discussion in ?3 precedes an important lemma on empirical

processes in two-dimensional time. The main theorems of ?4 prove the weak con-

vergence of the suitably normalized estimator. The discussion in the final

section considers estimators with different paths as well as estimators which

are products of product-limit estimators. The extension from two to k di-

mensions is noted.

2. Notation and the Estimator

For simplicity of exposition, bivariate observations are considered.

r ? ?? Let iX.-f._-t denote a sequence of independent random variables, X = (X ,X ),

from the continuous bivariate distribution function F. Each X represents the

lifetimes of a pair of (possibly dependent) items. Let ?C }. , denote a se-

quence of independent random variables, C. = (C#1,C ?), from the continuous

bivariate distribution function G. It is assumed that {X^ }. ., and {C.}. ., are ~i i=l ~i l-l

mutually independent.

In general X and C. are not both observable. Define



245

V^W5

?1 =

\? 1 cj?>

where I (x) is 1(0) if ? e (?)A. Note that e corresponds to whether ? is an

uncensored value (e =1 ) or censored value (e =0 ). It is assumed that ?. =

(? ,Z2 ) and e. = (e ,e ) are observable. Let ? denote the distribution

function of Z.. Define the bivariate survival function ~i

F(t) ? F(t_,t2)

? P(X1>t_, X2>t2) ,

and with abuse of notation, for ts (t-,t_) define

F(tx,t2) -

P(X1>t1, X2<t2);F(t1,t2) =

P(X1<t1, X2>t2) .

Similar functions can be defined for G and H. By independence of X and C

(1) H(trt2) -

F(t_,t2) G(t_,t2)

for all tlft2.

The hazard gradient approach of Marshall (1975) was employed by Campbell

and Foldes (1982) to estimate the distribution function as indicated below. The

cumulative hazard function is given by

(2) R(t) = -An F(t) .

Assuming R is absolutely continuous with partial derivatives that exist almost

everywhere, let r(t) denote the gradient of R(t). Then R(t) can be represented

as the path-independent integral of r(z) from (0,0) to t. In particular, for



246

the linear path (0,0) to (t ,0) to (t^t^ ,

A R(t) r (u,0)du +

r2(t1,v)dv

3R(s) 3R(s) Here r..(s) = , r (s) = ~

for s = (s ,s ); i.e., 1 ~ dS- ? ~ ?S? ~ 1 ?

R(t) (F(u,0)) ?

duF(u,0) +

h

(F(trv)) x

dvF(tlfv) ,

where d F(u,s) and d F(s,v) denote Lebesque-Stieljes integration over u and

v, respectively, with s fixed.

Define

(3)

Vt) -Piz^, z2>t2, ei-i)

K2(t) =

P(Z1>t1, Z2<t2, e2 = 1) =

G(u,t2)duF(u,t ) ;

G(tl,v)dvF(tl,v)

Then

(4) R(t) = ?

(H(u,0)) ?

d^^O) +

OKt^v))"1 dvK2(tljV)

Estimate H, K1 and K? by the empiricals

1 n

Hn(H>= ? ?^iilV z2i<t2)

1=1

i=l



247

where a (t) = I and a (t) = I _ li ~ <zii?V Z2i V ?li 1; 2l -

(Zli V Z2i^t2>e2i~1)?

Then R(t) is estimated from {?.}? , and {e.}? . by ~i i=l ~i i=l

(5) Rn(t) =

|o (H^u-,0))-1 duKln(u,0)

+ |o2(Hn(t1>v-))-1

??^^,?) ,

and F(t) by

(6) Fn(t) =

exp{-Rn(t)}

If F and G are continuous and if ? and ? are such that H(T ,T ) > 0,

Campbell and Foldes (1982) proved

sup

0<tl-Tl

0<t2lT2

|Fn(t) -F(t)| .o(/?jLn)a,t

3. Weak Convergence of Empiricals in Two-Dimensional Time

The study of the weak convergence of empirical processes in multi-

dimensional time culminated in articles by Neuhaus (1971) and Straf (1972), The

approach of Neuhaus (1971) is the reference for the topological discussion

below.

For simplicity one can reduce the domain of the bivariate distribution

function F from [0,??) x [0,??) to the unit square, [0,1] x [0,1] by the trans-

formation u =F(t-,?) and u=F(tJt ) = P(X <t |x ??, as suggested in

Durbin (1970). The approach of Neuhaus (1971) is to restrict the real-valued

functions from the unit square. For the point t=(t-,t2) inside the unit square,,



248

let Q-i?Q9?Qo>Q/ denote the four open quadrants in the square determined by t,

where Q- is the upper right quadrant. The space D9 is the set of all real

functions from the unit square such that if {t } denotes a sequence in Q. such

that lim t =t then lim f(t ) exists for i = 1,2,3,4 and for i = l its limiting

value is f(t). Let ? denote the class of all continuous functions from [0,1]

onto itself. Let ?=(? ,? )e?*? and |t| denote Euclidean distance in the

plane. Define the metric d (which can be thought of as an extension of the

one-dimensional Skorohod metric) for f,g in D~ as

d(f,g) = infie: there exists ? = ? with sup | X(t)-t |<_ e e>0

~ ~e t

^ ~ ~

and sup|f(t) -g(X(t))| ? e} t

Then (D ,d) is a separable complete metric space, unlike the space of dis-

continuous functions in the unit square with the metric d or the sup metric.

Therefore, the Prohorov development of weak convergence is applicable.

Let

U (t) = /?G(? (t?,t9) -

H(t-,t9)) ; ? ~ ? ? ? 1 ?

(7)

vjn(t) = /?G(? (t) - K.(t)) , j=l,2

LEMMA :

Let T.,T0 be such that H(T- ,T0) > 0. Then as n->~ (U (t), V, (t) , V0 (t)) L d l ? ? ~ In ~ zn ~

converges weakly to a trivariate, two-dimensional-time Gaussian process

(U(t), V (t), V (t)) with mean (0,0,0) and covariance structure given below,

where a Ab = min(a,b) , a V b = max(a,b) , and s = (s ,s ) :



249

Cov(U(s),U(t)) = H(s1Vt1, s2vt2)

- H(Sl,s2) ??t.^) ;

Cov(V1(s),V1(t)) =

K1(s1At1, s2Vt2) -

K1(s1,s2) K1(t1,t2) ;

Cov(V2(s),V2(t)) =

K^s^yt^, s2At2) -

K2(Sl,s2) ?2(??^2) ;

Cov(U(s),V1(t)) = {

Cov(U(s),V2(t)) = \

-H(Sl,s2) K^t^) if

s1>t1

(1(t1,s2Vt2)-K(s1>s2Vt2)-H(s1,s2) K1(t1,t2) if

s1<t1;

-H(S;L,s2) K2(tx,t2) if

s2>t2

Cov(V1(s),V2(t)) = \

K2(slVtl,t2) ~K2(slVtl,S2) ""H(sl,S2) VVV if

82<?2;

-K1(s1,s2) K2(tl9t2) if

s1<t1 or

s2>t2

J(s1,t2) +

J(t1,s2) -

J(S;L,s2) -

J(t1,t2)

K1(S1,S2) K2(ti't2) if

sl - V S2 - ?2 '

where Jit^t^

= PiZ^t^ Z2<t2, e?

= 1, e2 = 1) .

PROOF:

The finite dimensional distributions converge to multivariate normal distri-

butions by an application of the multivariate central limit theorem to the

three-dimensional variables

( *(7 > t ? > t ? H(t ,t ) ,

(?1? 1' 2i V ? ?

?(???? Z2i >

V e11-1)"?1(?) , I(Z11>t1. Z21<t2, e21=1) -K2(H>)?

A simple calculation on these indicator variables yields the covariance struc-

ture of the lemma. In order to prove tightness in three dimensions, the tight-

ness result of Neuhaus (1971, pp. 1292-5) for a one-dimensional empirical

function of a multidimensional time parameter is applied for U ,V and V



250

separately. Then tightness of the distributions of (U ,V ,V ) follows

immediately.

4. Main Theorems

The theorems in this section proceed in a fashion similar to the proof

of weak convergence in one dimension in Breslow and Crowley (1974). The random

variables (U ,V ,V9 ) and (U,V-,V?) can be replaced by random variables with

the same finite dimensional distribution but which also satisfy the condition

that d((U ,V- ,V0 ), (U,V-,Vn)) converges to zero almost surely, where d also ? in zn ? ?

represents the extension to D? xD? xD? of the metric d on D?.

THEOREM 1:

If F and G are continuous bivariate distribution functions and if ? ,T9<??

are such that H(T ,T ) > 0, then for t=(t ,t ) with 0<t <T , 0<t <T ,

? (Rn(t)

- R(t))

converges weakly to a two-dimensional-time Gaussian process W(t) given by:

W(t) = Ax(t)

+ B?(t)

+ A2(t)

+ B2(t) ,

where

(8)

f"1 - -2 A (t) = - (H(u,0))

? U(u,0)d ? (u,

B1(t) =

(H(t1,0))"1 V1(t1,0) - ?

? (H(u,0))"2 V1(u,0)duH(u,

0)

0)

A2(t) =- t2 - -2 Z

(H(t ,v)) Z

U(t ,v)d ? (t v) 0 ? ? ? ?

B2(t) =

(H(t;L,t2))""2 V2(t1,t2) t2 - -2

(H(trv)) V2(t1,v)dvH(t1,v)



251

PROOF:

From equations (4) and (5)

? (Rn(t)-R(t))

= /?G[ ?

(Hn(u ,0)) x

duKln(u,0) - (H(u.O))"?1? dîu.O)

'(Hn(t1,v-))-1 dvK2n(trv) (H(tlfv))"A dvK2(t;L,v)]

Now

/~?G[ ?

(Hn(u",0))_1 duKln(u,0) - (H(u,0))

1 dKjfa.O)]

(H(u, ,-2

0)) '

[H(u,0) - Hn(u",0)] dîu.O)

(H(u,0))-1 d(K (u,0) - ? (u,0)) u in 1

[H(u,0) - ? (u ,0)] =-?- - 1 ] d L(u,0)

J0 n Lh(u,0) Hn(u",0) H^fci.O) J

U L

t

f jz-~-~? I d <K <u,0) - ? (u,0))( JO Lh> ,0) H(u,0)J

U ln 1 I

Aln(H> +

?1?(?> +

E1?<B> +

Ei*n<!P '

where

Aln(tJ - ? 1(H(u,

J0 0))

' Un(u,0) dÔi.O)

,-1 Bln(~)

= <H<V0))

* vin<t1??)

-

E1n(t) = ?

In ^ /? vn

tl u;(tro)

0 Hn(u ,0) H?(u,0)

1 (H(u,0))

2 Vln(u,0) duH(u,0)

d^Cu.O)

ft, U (u,0) ft ? (u,u; ^(t) - - -? d (K.(u,0) - ? (u,0)) . ln

JO H,(u~,0) H(u,0) U ln ?



252

In a similar manner,

0 (WV ? dvK2n(tl'v)

- (H.(t1,v))'1 d^C^.v?J

where

A2n(B> +

B2n(tJ +

E2n(tJ +

E2n(E> ?

A2n(i> ?

'(H(tl,v))"2 Un(ti;v) dvK2(t1>V)

B2n(t) = (H(t

t2

1??G? WV -

?0 (S(tl'v))"2 V2n(tl'V) dHtVv)

E2n(?>

t2 U2(t.,?) ? 1 -2, ? ? (t-,v ) ? (t.,?) ni ?

dvK2(trv)

E2n<H) = -

-2 U (t.,v) n 1

O H(t,v ) H(t-,v) ni 1 dv(K2n(trv)

- K2(tl?v?

?

Now as n tends to infinity, E. (t) and E. (t) converge in probability to

zero in the supremum metric by an argument similar to that of Breslow and

Crowley (1974) and hence converge in probability to zero in the metric d, for

3=1,2. Further, A. (t) converges almost surely to A.(t) and B. (t) to B.(t)

in the sup metric and hence in d, for j =1,2. Therefore, /n~(R (t) -R(t))

converges weakly to W(t). That W(t) is a Gaussian process with mean 0 follows

immediately from the Lemma. The covariance structure of W(t) can be calculated

from (8) and from the covariance structure of (U,V ,V9).

THEOREM 2:

If F and G are continuous bivariate distribution functions and if ? ,??<??

are such that H(T ,T ) < ?>, then for 0 < t < ? , 0 < t < ?



253

/n" (Fn(t)

- F(t))

converges weakly to a two-dimensional-time Gaussain process W*(t) which has

mean 0 and covariance

Cov(W*(S;L,s2), W*(trt2)) =

F(S;L,s2) F?t^^,^) Cov(W(S;L,s2), Wit^t^)

PROOF:

By (6)

/? (F (t) - F(t)) = /"?" (e n ^ - e~K^t})

A Taylor's series expansion yields

*(*?? -R*(t) (9) /7(Fn(t)-F(t))=-e"K^; /T(Rn(t)-R(t))-e

n~ /? (R^t)

- R(t)T ,

where sup|R* - r| _< sup|R -r|. The second term on the right of (9) converges

to zero in probability in the sup norm and hence in d. Thus, /?"(F (t) -F(t)) ? ~ ~

converges weakly to -F(t) W(t) which is a Gaussian process with mean 0 and

and desired covariance.

5. Other Estimators

Campbell and Foldes (1982) introduced another path-dependent estimator.

For N(t)=n H (t), define the estimator S (t) of F(t): ~ ? ~ ? ~ ~

? /N(Z1?iO) Vli(V0) ?

/N(tl,Z21) Vzi'VV s (t) = p ???- n ?^???

n~ 1=1 V N(ZU,0)4-1 J i=l\

N(t1,Z2i)+l



254

provided N(t) >0. This is the product of two one-dimensional products-limit

estimators, one a marginal estimator for the first coordinate, and the second

a conditional one on the second coordinate given ? > t . It has been proved

that for ? ,T2<?>

such that ??^,?^) >0

sup |f (t) - S (t)| = 0?-) a.s. x. ^m ? ~ ? ~ ' \ n/ 0<t <T

o<t2<T2

Therefore S (t) inherits strong uniform almost sure consistency as well as weak

convergence from F (t).

The estimators F (t) and S (t) depend on the path from (0,0) to (t.,0) ? ~ ? ~ I

to (t-,t9). Since R(t) and F(t) are path independent, it is possible to have

also developed estimators for the linear path from (0,0) to (0,t?) to (t-,t ),

In general these estimators differ from F (t) and S (t). However, the strong

consistency and weak convergence results follow in the same way. The covariance

structure of the limiting Gaussian process does depend on the path.

The extension of these estimators and their asymptotic properites from two-

dimensional time to k(>2)-dimensional time is straightforward. In k-dimensions

there are k! piecewise linear paths similar to the two mentioned above. Strong

uniform almost consistency follows readily. It is convenient to use the

definition of D in Neuhaus (1971) to develop the weak convergence results.

The drawback of the estimators F and S is that the estimators are not ? ?

necessarily survival distribution functions. Although monotonicity is assured

along the path of definition, it is not guaranteed along other ever-increasing

paths nor is it the case that non-negative mass is assigned to rectangles. The

fact that the estimator is uniformly strongly consistent for the properly be-

haved function F minimizes this problem for large samples. In that S can be

thought of as a generalized maximum likelihood estimator along the designated

path, one could obtain such a generalized maximum likelihood estimator subject



255

to the constraint that the estimator is a distribution function. It is con-

jectured that these asymptotic results will not be changed for such an esti-

mator .

REFERENCES

Breslow, N. and Crowley, J. (1974). A large sample study of the life table

and product limit estimates under random censorship. Annals of Statistics

2, 437-453.

Campbell, G. (1981). Nonparametric bivariate estimation with randomly cen-

sored data. Biometrika 68, 417-422.

Campbell, G. and Foldes, A. (1982). Large-sample properites of nonparametric

bivariate estimators with censored data. Proceedings of the International

Colloquium on Nonparametric Statistical Inference, 1980, North Holland.

To appear.

Durbin, J. (1970). Asymptotic distributions of some statistics based on the

bivariate sample distribution function. In Nonparametric Techniques in

Statistical Inference, Ed. M.L. Puri. Cambridge University Press,

pp. 435-449.

Foldes, A. and Rejt?, L. (1981). Strong uniform consistency for nonparametric

survival curve estimators from randomly censored data. Annals of Statistics

9, 122-129.

Kaplan, E.L. and Meier, P. (1958). Nonparametric estimation from incomplete

observations. Journal of the American Statistical Association 53, 457-481.

Marshall, A.W. (1975). Some comments on the hazard gradient. Stochastic

Processes and Their Applications 3, 293-300.

Neuhaus, G. (1971). On weak convergence of stochastic processes with multi-

dimensional time parameter. Annals of Mathematical Statistics 42, 1285-

1295.



256

Straf, M.L. (1972). Weak convergence of stochastic processes with several

parameters. Proceedings of the Sixth Berkeley Symposium on Mathematical

Statistics and Probability,V.II, University of California Press, Berkeley,

California, pp 187-221.



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