Asymptotic Properties of Kaplan-Meier Estimator. for Censored Dependent Data. Zongwu Cai. Department of Mathematics

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1 To appear in Statist. Probab. Letters, 997 Asymptotic Properties of Kaplan-Meier Estimator for Censored Dependent Data by Zongwu Cai Department of Mathematics Southwest Missouri State University Springeld, MO 6584 August, 997 ABSTRACT In some long term studies, a series of dependent and possibly censored failure times may be observed. Suppose that the failure times have a common marginal distribution function, and inferences about it are of our interest. The main result of this paper is that of providing, under certain regularity conditions, the Kaplan-Meier estimator can be expressed as the mean of random variables, with a remainder of some order. In addition, the asymptotic normality of the Kaplan-Meier estimator is derived. Short Title: Kaplan-Meier estimator under dependence. AMS 99 Subject Classication: Primary 62G5; secondary 62M9, 6G, 62G3. Keywords: Asymptotic normality;?mixing; Censored dependent data; Kaplan-Meier estimator; Strong representation.

2 . Introduction and Preliminary Results Let T ; ; T n be a sequence of the true survival times for the n individuals in the life table. The random variables (r.v.s) are not assumed to be mutually independent (see assumption (A) for the kind of dependence stipulated); it is assumed, however, they have a common unknown continuous marginal distribution function (d.f.) F (x) = P (T i x) such that F () =. Let the r.v.s T i be censored on the right by the censoring r.v.s Y i, so that one observes only (Z i ; i ), Z i = T i ^ Y i and i = I(T i Y i ); i = ; : : : ; n; here and in the sequel, ^ denotes minimum and I(A) is the indicator r.v. of the event A. In this random censorship model, the censoring times Y i, i = ; ; n, are assumed to be independently identically distributed (i.i.d.) with d.f. G(y) = P (Y i y) such that G() = ; they are also assumed to be independent of the r.v.s T i 's. The problem at hand is that of drawing nonparametric inference about F based on the censored observations (Z i ; i ), i = ; ; n. For this purpose, dene two stochastic processes on [; ) as follows: N n (t) = I(Z i t; i = ) = I(T i t ^ Y i ); the number of uncensored observations less than or equal to t, and Y n (t) = I(Z i t); the number of censored or uncensored observations greater than or equal to t. Kaplan-Meier (K-M) estimator e Fn is taken to be? e Fn (t) = Y st? dn n(s) Y n (s) ; Then, the dn n (s) = N n (s)? N n (s?). As is known (see, for example, Gill (98)), for a d.f. F on [; ), the cumulative hazard function is dened by (t) = df (s)? F (s?) ; and (t) =? log(? F (t)) for the case that F is continuous. The empirical cumulative hazard function e n (t) is taken to be e n (t) = dn n (s) Y n (s) ;

3 which is referred to in the literature as the Nelson estimator of (t). For the case that the failure time observations are mutually independent, the K-M estimator has been studied extensively by many authors during the last three decades, such as, Breslow and Crowley (974), Peterson (977), Gill (98, 98, 983), Lo and Singh (986), Wang (987), Stute and Wang (993), and others. However, there are preciously few results available, for the case that these observations exhibit some kind of dependence. For example, Voelkel and Crowley (984) used an approach, based on semi-markov process, to establish a reasonable model in Cancer Research Clinical Trials that assumes each patient may either remain in an initial state, or progress, or respond and then possible relapse. Ying and Wei (994) explored consistency and asymptotic normality of e Fn (t) under?mixing context. An application of the right censoring model in Diabetes Control and Complications Trial was given by them for a special dependent case, in which survival times are highly stratied. Our basic aim in this article is to express the K-M estimator as the mean of bounded random variables with a remainder of order O? n?=2 (log n)? for some >, for the case in which the underlying failure times are assumed to be?mixing whose denition is given below. In addition, asymptotic normality of the K-M estimator is represented in Section 3. Let F k i (X) denote the?eld of events generated by fx j ; i j kg. For easy reference, recall the following denition. Denition Let fx i ; i = ; ; 2; : : :; g denote a sequence of r.v.s. Given a positive integer n, set o (n) = sup njp (A \ B)? P (A)P (B)j : A 2 F k (X) and B 2 F k+n (X) : The sequence is said to be?mixing (strongly mixing) if the mixing coecient (n)! as n!. Among various mixing conditions used in the literature,?mixing is reasonably weak, and has many practical applications.?mixing. Many stochastic processes and time series are known to be Withers (98) obtained various conditions for a linear process to be?mixing. Under certain weak assumptions autoregressive and more generally bilinear time series models are strongly mixing with exponential mixing coecients. Auestad and Tjstheim (99) provided illuminating discussions of the role of?mixing for model identication in non-linear time series analysis. 2

4 For the sake of simplicity, the assumptions used in this paper are listed below. It should be pointed out that, throughout the paper, the letter C is used indiscriminately as a generic constant, and all limits are taken as n! unless otherwise specied. Assumptions: (A) ft j ; j g is a stationary sequence of?mixing r.v.s with continuous distribution function F. (A2) The censoring time variables fy j ; j g are i.i.d. with continuous distribution function G, and are independent of ft j ; j g. (A3) (n) = O(n? ) for some > 3. In this section, a number of lemmas are presented to be employed later in subsequent parts of the paper. Lemma Let fx n ; n g be a sequence of?mixing r.v.s with mixing coecient (n), independent of an i.i.d. sequence of r.v.s fy n ; n g, then f(x n ; Y n ); n g is a sequence of?mixing r.v.s with mixing coecient 4(n). In particular, so is fx n ^ Y n ; n g. Proof k + n). For any sets A 2 F k (X; Y ) = (X j ; Y j ; j k) and B 2 F n+k (X; Y ) = (X j; Y j ; j jp (A \ B)? P (A)P (B)j = je(i(a)i(b))? EI(A)EI(B)j = je[e(i(a)i(b)jf k (X; Y ))]? EI(A)EI(B)j = jefi(a)[e(i(b)jf k (X; Y ))? EI(B)]gj EjE(I(B)jF k (X; Y ))? EI(B)j: Since E(I(B)jF k (X; Y )) = E(I(B)jX ; : : : ; X k ; Y ; : : : ; Y k ); and Y i, i k, are independent of X i, i k, and Y i, i k + n, then we have E(I(B)jF k (X; Y )) = E((X j ; j k + n)jx ; : : : ; X k ); (X j ; j k + n) = E(I(B)jX j ; j k + n): 3

5 Clearly, jj and E(I(B)jF k (X; Y )) is measurable with respect to (w.r.t.) F k (X). Let = sgn[e(i(b)jf k (X; Y ))? EI(B)] = sgn[e((x j ; j k + n)jf k (X))? EI(B)]; then, is measurable w.r.t. F k (X) and jj. Therefore, by Theorem 7.2. in Ibragimov and Linnik (97, p.36) and the fact that (X j ; j k + n) is measurable w.r.t. F k+n (X), one has that jp (A \ B)? P (A)P (B)j Cov(; (X j ; j k + n)) 4(n): This completes the proof of the lemma. First, we proceed with the introduction of some additional notation and the statement of some preliminary results. To this end, let H denote the d.f. of the Z i 's, so that H =? H = F G = (? F )(? G); and dene (the possibly innite) times F, G and H by F = inffy : F (y) = g; and likewise for G and H. Then, H = F ^ G (see, for example, Stute and Wang (993)). By setting we have Let F (t) = Z = (F; G) = P (T Y ) = F (t) = P (Z t; = ); F (t ^ z)dg(z) = [? G(z)]dF (z): Z Z F (z)dg(z) = [? G(z)]dF (z) and assume that >. Clearly, F (t)= is the conditional distribution function of Z given =. Dene F = infft : F (t) = g: Then it is easily seen that = F F ^ G, so that H = F. Thus, it follows that, for continuous F, Finally, let (t) = df (s) H(s) : N n (t) = N n (t)=n and Y n (t) = Y n (t)=n: 4

6 Consider the uniform convergence rate of the empirical cumulative hazard function e n. Namely, we have the following result. Theorem Under assumptions (A)-(A3), sup t Y n (t)? H(t) = O(a n ); () and sup t N n (t)? F (t) = O(a n ); (2) =2 log log n a n = : (3) n Consequently, we have for any < < H : sup e n (t)? (t) = O(a n ): (4) t In order to prove Theorem, we need the following lemma, which is Theorem 3.2 in Cai and Roussas (992), stated here without proof. Lemma 2 Let fx n g, n, be a stationary?mixing sequences of r.v.s with d.f. F and mixing coecient (n) = O (n? ) for some > 3, and let F n be the empirical d.f. based on the segments X ; ; X n. Then sup jf n (x)? F (x)j = O(a n ): x2< We proceed with the proof of Theorem by utilizing Lemma 2. Proof of Theorem It follows from Lemma that fz i ; i g and f(z i ; i ); i g are two stationary sequences of?mixing r.v.s. Then () and (2) follow by Lemma 2 and the fact that both? Y n and N n are empirical functions. An application of Lemma 2 in Gill (98) yields sup e n (t)? (t) 2 (N n ; F ) + (Y n ; H) N n () + (N n ; F ) t Y n () Y n () Y n ()? (Y n ; H) ; (5) is the supreme metric on [; ]. Therefore, (4) holds true from (5), () and (2). This completes the proof of the theorem. 5

7 Lemma 3 (Theorem 3 in Dhompongsa (984)) Under assumptions (A) and (A3), there exists a Kiefer process fk(s; t); s 2 <; t g with covariance function E [K(s; t)k(s ; t )] =?(s; s ) min(t; t ) and?(s; s ) is dened by?(s; s ) = Cov(g (s); g (s )) + X k=2 [Cov(g (s); g k (s )) + Cov(g (s ); g k (s))]; g k (s) = I(Z k assumption (A3), s)? H(s), such that, for some > depending only on, given in sup Y n (t)? H(t)? K(t; n)=n = O(b n ); t2< b n = n?=2 (log n)? : (6) 2. Strong Representation Results Let g(x) = Z x and for positive reals z and x, and = or, let (H(s))?2 df (s); (z; ; x) = g(z ^ x)? I(z x; = )=H(z): Observe that E((Z i ; i ; x)) = and Cov((Z i ; i ; s); (Z i ; i ; t)) = g(s ^ t): Now, let us write Z dn n (s) t e n (t)? (t) = Y n (s)? df (s) H(s) Z = Y n (s)? t df (s) + d N n (s)? F (s) H(s) H(s) Y n (s)? d N n (s)? F (s) H(s) = I + I 2 + I 3 say: (7) 6

8 It follows from () that the rst term I in (7) turns into I = a n is dened by (3). Consequently, I + I 2 = = =? n??2?? H(s) H(s)? Y n (s) df (s) + O a 2 n ;? H(s)?2? H(s)? Y n (s) df (s) + dn n (s)? H(s) Y n (s) H(s) 2 df (s) + O? a 2 n d? N n (s)? F (s) + O a 2 n H(s) (Z i ; i ; t) + O(a 2 n ): (8) To estimate I 3?, divide the interval [; ] for < H, into subintervals [x i ; x i+ ], i = ; ; k n, k n = O a? n, and = x < x 2 < < x kn + = are such that H(x i+ )? H(x i ) = O(a n ). It is easily seen from () and (2) that Z ji 3 t j = Y n (s)? d(n n (s)? F (s)) H(s) 2 max ikn + k n sup x C max ikn + C max ikn sup Y? y2[xi;xi+] Y? n sup y2[xi;xi+] = I 3 + I 32 + O? a 2 n n (y)? Y? n (x i )? H? (y) + H? (x i ) (x)? H? (x) max ikn Y n (y)? Y n (x i )? H(y) + H(x i ) N n (x i+ )? N n (x i )? F (x i+ ) + F (x i ) N n (x i+ )? N n (x i )? F (x i+ ) + F (x i )? + O a 2 n say: (9) Thus, it follows from Lemma 3 that I 3 = C max ikn sup y2[xi;xi+] K(y; n)? K(x i ; n) =n + O(b n ): By the law of the iterated logarithm for the Kiefer processes (see, for example, Theorem.4.2, page 79 in Csorgo and Revesz (98)), we have I 3 = O log log kn nk n =2! + O(b n ) = O(b n ): () Likewise, I 32 = O(b n ): () 7

9 Therefore, () in conjunction with (7)-() establishes the following result. Theorem 2 Under assumptions (A)-(A3), e n (t)? (t) =? n (Z i ; i ; t) + r n (t); for any < H, and b n is dened by (6). sup jr n (t)j = O(b n ) t Since H(Z n:n )?! by Theorem, then for any < < H we have that < < Z n:n for suciently large n. Therefore, Lemma in Breslow and Crowley (974) gives which implies that: <? log(? e Fn (t))? e n (t) < n? Y n(t) ny n (t) ;? log(? Fn e ); n e n? Y n() C()=n (2) ny n () for suciently large n, < C() < is independent of n. Using Taylor expansion, we have ef n (t)? F (t) =? F (t)? (? Fn e (t)) = e?(t)? e log(?ef n(t)) h i = e?e n (t) en (t)? (t) + e?e i n h? (t) log(? Fn e (t))? n e (t) ; (3) and, from (2) e n ; en ; ; (4) e n ; e n? log(? e Fn ); e n C()=n (5): Therefore, it follows from (4), (2)-(5) that: ef n (t)?f (t) = e?(t) h en (t)? (t) i +O(=n)+O 2 en ; i = F (t)h? en (t)? (t) +O an 2 : Then, the following theorem has been established. 8

10 Theorem 3 Under assumptions (A)-(A3), ef n (t)? F (t) =? F (t) n (Z i ; i ; t) + R n (t); for any < H, b n is dened in (6). sup jr n (t)j = O(b n ) t 3. Asymptotic Normality We now present our asymptotic normality of the K-M estimator based on our strong representation result. It is easy to see from Lemma that f(z i ; i ; t)g i is a sequence of stationary?mixing bounded random variables. In order to obtain the asymptotic normality for K-M estimator, we just apply Theorem in Ibragimov and Linnik (97) and establish the following results: Theorem 4 Under assumptions (A)-(A3), p n h en (t)? (t)i D?!N? ; 2 (t) for t 2 [; ] for any < H, 2 (t) = V ar((z ; ; t)) + 2 X j=2 Cov((Z ; ; t); (Z j ; j t)): Theorem 5 Under assumptions (A)-(A3), p h n efn (t)? F (t)i D?!N ; 2 (t)f 2 (t) for t 2 [; ] for any < H. References Auestad, B. and Tjstheim, D.(99). Identication of nonlinear time series: First order characterization and order determination. Biometrika 77, 669{687. 9

11 Breslow, N and Crowley, J.(974). A large sample study of the life table and product limit estimators under random censorship. Ann. Statist. 2, 437{453. Cai, Z. and Roussas, G.G.(992). Uniform strong estimation under?mixing, with rates. Statist. Probab. Lett. 5, 47{55. Csorgo, M. and Revesz, P.(98). Strong Approximations in Probability and Statistics, Academic Press, New York. Dhompongsa, S.(984). A note on the almost sure approximation of the empirical process of weakly dependent random variables, Yokohama Math. J. 32, 3-2. Gill, R.D.(98). Censoring and Stochastic Integrals. Mathematical Centre Tracts No. 24, Mathematisch Centrum, Ansterdam. Gill, R.D.(98). Testing with replacement and the product limit estimator. Ann. Statist. 9, 853{86. Gill, R.D.(983). Large sample behavior of the product limit estimator on the whole line. Ann. Statist., 49{56. Ibragimov, I.A. and Linnik, Yu.V.(97). Independent and Stationary Sequences of Random Variables. Walters-Noordho, Groningen, the Netherlands. Kaplan, E.L. and Meier, P.(958). Nonparametric estimation from incomplete observations J. Amer. Statist. Assoc. 53, 457{48. Lo, S.H. and Singh, K.(986). The product-limit estimator and the Bootstrap: Some asymptotic representations. Probab. Theor. Rel. Fields 7, 455{465. Peterson, A.V.(977). Expressing the Kaplan-Meier estimator as a function of empirical subsurvival functions. J. Amer. Statist. Assoc. 72, 854{858. Stute, W. and Wang, J.L.(993). A strong law under random censorship. Ann. Statist. 2, 59{67. Voelkel, J. and Crowley, J.(984). Nonparametric inference for a class of semi-markov process with censored observation. Ann. Statist. 2, 42{6. Wang, J.G.(987). A note on the uniform consistency of the Kaplan- Meier estimator. Ann. Statist. 5, 33{36. Withers, C.S.(98). Conditions for linear processes to be strong mixing. Z. Wahrsch. verw. Gebiete 57, 477{48. Ying, Z. and Wei, L.J.(994). The Kaplan-Meier estimate for dependent failure time observations. J. Multivariate Anal. 5, 7{29.

Editorial Manager(tm) for Lifetime Data Analysis Manuscript Draft. Manuscript Number: Title: On An Exponential Bound for the Kaplan - Meier Estimator

Editorial Manager(tm) for Lifetime Data Analysis Manuscript Draft. Manuscript Number: Title: On An Exponential Bound for the Kaplan - Meier Estimator Editorial Managertm) for Lifetime Data Analysis Manuscript Draft Manuscript Number: Title: On An Exponential Bound for the Kaplan - Meier Estimator Article Type: SI-honor of N. Breslow invitation only)