QUANTILE ESTIMATION FOR LEFT TRUNCATED AND RIGHT CENSORED DATA

Transcription

1 Statistica Sinica 10(2000), QUANTILE ESTIMATION FOR LEFT TRUNCATED AND RIGHT CENSORED DATA Xian Zhou, Liuquan Sun and Haobo Ren Hong Kong Poltechnic Universit, Academia Sinica and Peking Universit Abstract: In this paper we stud the estimation of a quantile function based on left truncated and right censored data b the kernel smoothing method. Asmptotic normalit and a Berr-Esseen tpe bound for the kernel quantile estimator are derived. Monte Carlo studies are conducted to compare the proposed estimator with the PL-quantile estimator. Ke words and phrases: Asmptotic normalit, Berr-Esseen tpe bound, quantile function, kernel estimation, truncated and censored data. 1. Introduction In medical follow-up or in engineering life testing studies one ma not be able to observe the variable of interest, referred to hereafter as the lifetime. Among the different forms in which incomplete data appear, right censoring and left truncation are common. Left truncation ma occur if the time origin of the lifetime precedes the time origin of the stud. Onl those individuals that fail after the start of the stud are being followed, otherwise the are left truncated. Those that are followed are further subject to right censoring during the followup period. See for example, an AIDS stud b Struthers and Farewell (1989) where lifetime is the incubation period, and the truncation variable is the time from infection until entr to the stud. Formall, let (X, T, Y ) denote random variables, where X is the lifetime with continuous distribution function (d.f.) F, T is the random left truncation time with arbitrar d.f. G, andy is the random right censoring time with arbitrar d.f. H. It is assumed that X, T, Y aremutuall independent and, without loss of generalit, that the are nonnegative. In the random left truncation and right censoring (LTRC) model one observes (Z, T, δ) if Z T,whereZ = X Y =min(x, Y )andδ = I(X Y ) is the indicator of censoring status. When Z<Tnothing is observed. Let α P (T Z) > 0, and let W be the d.f. of Z, i.e., 1 W =(1 F )(1 H). Let (Z i,t i,δ i ),i =1,...,n be an independent and identicall distributed (i.i.d.) sample of (Z, T, δ) which is observed (i.e., Z i T i ). Let C(x) =P (T x Z T Z) =α G(x)(1 W (x )), and consider its empirical estimate C n (x) =n n I(T i x Z i ).

2 1218 XIAN ZHOU, LIUQUAN SUN AND HAOBO REN For LTRC data, it is important to be able to obtain nonparametric estimates of various characteristics of the distribution function F. Tsai, Jewell and Wang (1987) gave the nonparametric maximum likelihood estimator of F itself, called the product-limit (PL) estimator, as { 1 ( ˆF n (x) = Z i x 1 [ncn (Z i )] ) δ i,x<z (n), (1.1) 1, x Z (n), where Z (n) =max(z 1,...,Z n ). The properties of ˆF n have been studied b Wang (1987), Gu and Lai (1990), Lai and Ying (1991), Gijbels and Wang (1993), and Zhou (1996), among others. Nonparametric estimates of the densit and hazard rate for F have been studied b Gijbels and Wang (1993), Sun and Zhou (1998), Uzunoḡullari and Wang (1992), Gu (1995) and Sun (1997). One characteristic of the distribution function F that is of interest is the quantile function. It plas an important role in various statistical applications, especiall in data modeling, reliabilit and medical studies. Let Q(t) F (t) = inf{x : F (x) t}, 0<t<1, be the quantile function of F. We focus here on estimating the quantile function based on LTRC data. A natural estimator of Q is the PL-quantile function defined as ˆF n (x) = inf{u : ˆFn (u) x}, 0 <x<1. In case Q is a continuous function, it ma be more suitable to use a smooth estimator rather than the step function ˆF n, since smoothing reduces random variation in the data and allows a better displa of interesting features of the lifetime distribution function. Numerous smooth quantile function estimators have been proposed for complete samples. In the censored model, Padgett (1986), Lio, Padgett and Yu (1986), Xiang (1995a, b) studied kernel-tpe quantile function estimators. None of their results included the case where both left truncation and right censoring are involved however. On the other hand, Gürler, Stute and Wang (1993) provide a strong representation of the empirical quantile function for left truncated data, together with asmptotic normalit and the Law of Iterated Logarithm (LIL), which can be easil extended to the LTRC model. In this paper, for the LTRC model, we propose and stud the kernel quantile function estimator of the form 1 ˆQ n (t) =h n 0 ( x t ) ˆF n (x)k dx, (1.2) where K is a kernel function and { } n 1 is a sequence of positive bandwidths with 0asn. The first aim of this article is to derive the asmptotic normalit of the kernel quantile function estimator. The qualit of the approximation to the distribution

3 QUANTILE ESTIMATION FOR TRUNCATED AND CENSORED DATA 1219 of ˆQ n is an important issue of both theoretical and practical interest (for example, construction of confidence intervals based on the asmptotic normalit). Thus we also establish a Berr-Esseen tpe bound of ˆQ n. A small Monte Carlo simulation stud shows that, for certain values of the bandwidth, ˆQn performs better ˆF n than in the sense of having smaller estimated mean squared errors (MSE). For an d.f. L, denote the left and right endpoints of its port b a L = inf{x : L(x) > 0} and b L ={x : L(x) < 1}, respectivel. In the current model, as discussed b Gijbels and Wang (1993) and Zhou (1996), we assume that a G a W,b G b W and a W df (x) G 2 (x) <. (1.3) Condition (1.3) requires that the left truncation is not too heav. kernel function K(x), assume (K1) K is of bounded variation with port [, 1]; (K2) K is Lipschitz continuous of order one; (K3) For some integer r 2, 1 j! 1 if j =0, x j K(x)dx = 0 if j =1,...,r 1, c r 0 ifj = r. As for the These assumptions are the usual ones encountered in the kernel method of curve estimation, as in Padgett (1986) and Uzunoḡullari and Wang (1992). This article is organized as follows. Main results are stated in Section 2. Some simulation results are presented in Section 3. Proofs of the results are deferred to the Appendix. 2. Main Results We first give the asmptotic normalit of ˆQ n. Theorem 1. Under (K1) and (K3), assume that F is r times continuousl differentiable in a neighborhood of Q(t) with F (Q(t)) = f(q(t)) > 0, a w < Q(t) <b w,wherer is the same as that in (K3). If n 1/2 h r n 0 as n, n 1/2 ( ˆQ n (t) Q(t)) N(0,σ 2 (t)) in distribution as n, (2.1) where the variance is σ 2 (t) = (1 t)2 Q(t) df (x) f 2 (Q(t)) a W C(x)(1 F (x)). (2.2)

4 1220 XIAN ZHOU, LIUQUAN SUN AND HAOBO REN To appl this result to make some standard inferences in hpothesis testing and construction of confidence intervals, an estimator of the variance in (2.2) is needed and a natural nonparametric estimator is given below. Let W 1 (x) = P (Z x, δ =1 T Z) =α x G(u)(1 H(u ))df (u). Then it can be checked that σ 2 (1 t)2 Q(t) dw 1 (x) (t) = f 2 (Q(t)) a W C 2 (x). (2.3) Thus σ 2 (t) can be estimated consistentl b replacing the densit f in (2.3) b a densit estimate from Gijbels and Wang (1993), and all other unknown quantities in (2.3) b their empirical estimates. We now give a Berr-Esseen tpe bound for ˆQ n to assess the qualit of the normal approximation in Theorem 1. Theorem 2. Under (K1), (K2) and (K3), assume that F is r-time continuousl differentiable in a neighborhood of Q(t) with F (Q(t)) = f(q(t)) > 0, a W <Q(t) <b W. If a G <a W and nh 2 n / log n as n, there exists a positive constant M = M(t) such that for all n 1, P {n 1/2 ( ˆQ ( ) n (t) Q(t)) σ(t)} Φ() M n /2 h n log n+n 1/2 h r n+hn 1/2, (2.4) where σ(t) is defined b (2.2). 3. Simulation Studies A small Monte Carlo stud was performed in order to provide some smallsample comparisons of the kernel estimator ˆQ n with the PL-quantile ˆF n in terms of mean squared error. The stud also provides some insight into the choices of that might be used to estimate Q with smaller mean squared error than ˆF n. For the LTRC model, the following cases are simulated. (i) F (x) =1 exp( x), H(x) =1 exp( x) andg(x) =1 exp( 2x), with 50% censoring and 50% truncation respectivel. (ii) F (x) = 1 exp( x), H(x) = 1 exp( 3x/7) and G(x) = 1 exp( 30x/7), with 30% censoring and 25% truncation respectivel. The triangular densit function K(x) = (1 x )I( x 1) was used as the kernel function for the estimator ˆQ n. The ratio of the mean squared error of ˆF n (t) to that of the smoothed estimator ˆQ n (t) were computed for various values of t (0, 1) and sample size n = 100. For each case, 1000 replications were done on S-PLUS on a PC-pentium 200 computer. The simulations were run for various values of and the results show that ˆQ n gives reasonable performance for 0.40 onl. On the other hand, the results for below 0.10 do not var

5 QUANTILE ESTIMATION FOR TRUNCATED AND CENSORED DATA 1221 much. Some of the simulation results for between 0.05 and 0.40 are presented in Tables 1-2 below, and these show that, for each value t listed, there is a range of window widths such that the estimator ˆQ n has smaller estimated mean squared error than the PL-quantile estimator. In particular, this is true for the median estimators ˆQ n (0.5) and ˆF n (0.5). From a range of values, we found that =0.15 comes close to giving the smallest discrepanc between the kernel estimator and the quantile. Figures 1-2 show the plots of the quantile, kernel estimator ˆQ n and the PL-quantile estimator ˆF n with =0.15, from which we see that ˆQ n looks better than except for large values of t. As one would expect for censoring, truncation and bound effect, the performance of either estimator ( ˆQ n or ˆF n ) at large values of t is not as good as that for values near 0.5. ˆF n Table 1. Ratios of mean squared errors for Case (i): 50% censoring and 50% truncation (n = 100). t Note: Ratio=MSE( ˆF n )/MSE( ˆQ n). Table 2. Ratios of mean squared errors for Case (ii): 30% censoring and 25% truncation (n = 100). t Note: Ratio=MSE( ˆF n )/MSE( ˆQ n).

6 1222 XIAN ZHOU, LIUQUAN SUN AND HAOBO REN Q(t) t Figure 1. Plots of PL-quantile and the kernel estimator for case (i) with n =50and =0.15. Q(t) t Figure 2. Plots of PL-quantile and the kernel estimator for case (ii) with n =50and =0.15.

7 QUANTILE ESTIMATION FOR TRUNCATED AND CENSORED DATA 1223 Appendix. Proofs of Theorems First we present preliminar results needed in the proofs of the theorems. Let x d[w 1n (u) W 1 (u)] x C n (u) C(u) L n (x) = a W C(u) a W C 2 dw 1 (u). (u) From Theorem 1 of Gijbels and Wang (1993) and Theorem 2 of Zhou (1996), we have that for a W x b<b W, ˆF n (x) F (x) =(1 F (x))l n (x)+r n (x), (A.1) where R n (x) = O(n log 1+β n) a.s. (A.2) a W x b with β =0whena G <a W, and β>1/2 whena G = a W. Introduce the PL-process ˆα n (x) =n 1/2 ( ˆF n (x) F (x)). The following lemma, of interest in its own right, provides a general version of the oscillation behavior of ˆα n. Lemma A.1. Assume that F is Lipschitz continuous of order one on [c, d], a W <c d<b W. Let {a n,n 1} be a sequence of positive constants tending to zero wita n / log n as n. Then ( ( ˆα n (x) ˆα n () = O an 1/2 (log n)1/2) +O n /2 (log n) 1+β) a.s., c x, d, x a n where β is defined in (A.2). (A.3) Proof. It follows from (A.1) that x n 1 d[w 1n (u) W 1 (u)] 2 (αn (x) α n ()) = (1 F (x)) +(F () F (x))l n () C(u) x C n (u) C(u) (1 F (x)) C 2 dw 1 (u)+(r n (x) R n ()) (u) 4 := Γ ni (x, ). (A.4) First, b the continuit of F together with a partitioning argument similar to that in Burke, Csörgő andhorváth (1988), it can be shown that ( Γ n1 (x, ) = O n /2 an 1/2 (log n)1/2) a.s. (A.5) c x, d, x a n Next, in view of (1.3), the process L n () is the sum of two empirical processes over VC classes of functions with square integrable envelope on [c, d], so it satisfies

8 1224 XIAN ZHOU, LIUQUAN SUN AND HAOBO REN the LIL (see, e.g. Arcones and Giné (1995)), i.e., its over [c, d] isa.s. of the order (n log log n) 1/2. Moreover, note that C n C is the difference of two empirical processes and inf c d C() > 0. Therefore, using the LIL for empirical processes, we obtain that for all n sufficientl large, Γ ni (x, ) = O c x, d, x a n ( n /2 a 1/2 n (log n)1/2) a.s., i =2, 3, (A.6) and b (A.2), ( ) Γ n4 (x, ) = O n log 1+β n c x, d, x a n a.s. (A.7) Lemma A.1 now follows from (A.4) (A.7). ProofofTheorem1.Theproof of Theorem 1 relies on the weak asmptotic representation for the PL-quantile function as given in (A.8) below. Let a W <Q(t 1 ) Q(t 2 ) <b W. If F has a continuous and positive densit f on [Q(t 1 ) η, Q(t 2 )+η] forsomeη>0, it follows from similar arguments as in Gürler, Stute and Wang (1993) that, as n, t 1 t t 2 ˆF n ˆF n (t) Q(t) t ˆF n (Q(t)) f(q(t)) = o p(n /2 ). Let Q n (t) =h ( 1 n 0 Q(x)K x t )dx. Using (A.1) and (A.8), we get ˆQ n (t) Q n (t) =n n t+hn t where [ I(Zi x, δ i =1) ϕ i (x) =(1 F (x)) C(Z i ) 1 [ ϕ i (Q(x))] K f(q(x)) x I(T i u Z i ) a W C 2 (u) (A.8) ( x t ) dx + o p (n /2 ), ] dw 1 (u). (A.9) Thus an application of Liapunov s form of the Central Limit Theorem gives Note that n 1/2 ( ˆQ n (t) Q n (t)) N(0,σ 2 (t)) in distribution as n. (A.10) ˆQ n (t) Q(t) = It follows from a Talor expansion that Q n (t) Q(t) = hr n r! [ ˆQn (t) Q n (t)] + [ Qn (t) Q(t) ]. (A.11) 1 x r K(x)Q (r) (t + θx)dx

9 QUANTILE ESTIMATION FOR TRUNCATED AND CENSORED DATA 1225 for some 0 θ 1, where Q (r) is the rth derivative of Q. Thus n 1/2 Q n (t) Q(t) = O(n 1/2 h r n) 0 as n. (A.12) Theorem 1 then follows immediatel from (A.10) (A.12). The following lemma is useful in proving Theorem 2. Lemma A.2. Let U, V be two random variables. Then for an a>0, P (U + V ) Φ() P (U ) Φ() + a + P ( V >a), (A.13) 2π P (U V ) Φ() P (U ) Φ() + P ( V 1 >a)+a. (A.14) Proof. The first inequalit follows from Lemma 2 of Chang and Rao (1989), while the second is a consequence of Michel and Pfanzagl (1971). In the sequel M = M(x) denotes a positive constant which will var in different contexts. Proof of Theorem 2. Using a change of variable theorem (cf. Shorack and Wellner (1986, p.25)), we have ˆQ n (t) Q ( ( ˆF n(x) t)/ ) n (t) = dx = S n1 + S n2, (A.15) where S n1 = S n2 = From (A.1), write 0 0 [ ( ˆFn(x) t)/hn 0 (F (x) t)/ K(u)du ( ) ( ) F (x) t ˆFn (x) F (x) K dx, (F (x) t)/ S n1 = n ( K(u) K n ( F (x) t )) ] du dx. V (n) i + r n (A.16) with V (n) ( F (x) t ) i = ϕ i (x)h n K dx, 0 ( F (x) t ) r n = h n K R n (x)dx, 0 where ϕ i and R n are defined b (A.9) and (A.1), respectivel. Standard calculations ield EV (n) i =0, σ 2 n Var(V (n) i )= (1 t)2 f 2 (Q(t)) Q(t) a W dw 1 (x) C 2 (x) + O()=σ 2 (t)+o( ) (A.17)

10 1226 XIAN ZHOU, LIUQUAN SUN AND HAOBO REN and E V (n) i 3 (1 t) d 3 Q(t) dw 1 (x) 1 f 3 (Q(t)) a W C 3 (x) <, where d 1 > 0 is a constant. Thus, b the Berr-Esseen Theorem (see Chow and Teicher (1978, p.299)), for all n 1, n P {n 1 2 It follows from (A.14) that P {n /2 n P {n /2 n V (n) i V (n) i V (n) i <σ n } Φ() Mn/2. <σ(t)} Φ() (A.18) <σ n } Φ() + P { σn σ(t) 1 >a} + a (A.19) for an a>0. It is eas to see that for an random variable V and 0 <ε<1/2, P { V 1 > 2ε, V 0} 2P { V 1 >ε}. Hence, from the fact that for,z > 0therelation 1 >zholds if 1 > z, and from (A.17), we have P { σn σ(t) 1 >a} P { σn 2 σ 2 (t) 1 >a 2 } 2P{ σnσ 2 2 (t) 1 >a 2 /2} 2P {d 2 >a 2 /2} (A.20) for some constant d 2 > 0. In view of (A.18) (A.20) and putting a =(2d 2 ) 1/2, we get P {n 1 2 n V (n) i σ(t)} Φ() M(n /2 + hn 1/2 ). (A.21) When a G <a W, from (1.15) of Gijbels and Wang (1993) we obtain that for each x>0and0 b<b W, P { n R n () >x+4θ 2 } M(e λx +(x/50) 2n +e λx3 ), 0 b (A.22) where θ>0 and λ>0 are some constants. Note that for large n, there exists a constant b such that 0 Q(t + u) b<b W for all u 1. Hence for all n sufficientl large, it follows from (A.22) that P {n 1 2 r n n /2 h n log n} 1 1 = P { f(q(t + u)) K(u)R n(q(t + u))du (n ) log n} P { n R n () d 3 h n log n} Mn, (A.23) 0 b

11 QUANTILE ESTIMATION FOR TRUNCATED AND CENSORED DATA 1227 where d 3 > 0isaconstant. Forsomeτ>0with1+τ (1 t)/ (this holds if n is large ), write S n2 = n1 + n2 + n3, (A.24) where n1 = h 1+τ n τ K(x)dx, n2 h τ n t/ K(x)dx with = (1 t)/hn 1+τ K(x)dx and n3 = ( ˆFn(Q(t+x)) t)/ 1 K(x) = (K(u) K(x))du. f(q(t + x)) x Since K has bounded port [, 1], ( ) (1 t)/hn 1 ˆF n2 1+τ f(q(t + x)) I n (Q(t + x)) t < 1 ( ˆFn(Q(t+x)) t)/ (K(u) K(x))du dx x ( ) (1 t)/hn 1 ˆFn 1+τ f(q(t+ x)) I (Q(t+(1+τ) )) (t+(1+τ) ) < τ ( ˆFn(Q(t+x)) t)/ (K(u) K(x))du dx. (A.25) x Using Theorem 1 of Zhu (1996), we have that for a G <a W b<b W and ε>0, P { a W x b ˆF n (x) F (x) >ε} d 4 exp( nd 5 ε 2 ), (A.26) where d 4 and d 5 are absolute constants. Therefore, (A.25) and (A.26) impl that for all n sufficientl large, P {n 1/2 n2 n /2 h n log n} P { ˆF n (Q(t+(1+τ) )) (t+(1+τ) ) τ } d 4 exp( nd 5 τ 2 h 2 n) Mn. (A.27) (Recall that nh 2 n/ log n, so that nd 5 τ 2 h 2 n > log n for large n.) Similarl, P {n 1/2 n3 n /2 h n log n} Mn. (A.28) Since K is Lipschitz continuous of order one, for some constant d 6 > 0, n1 d 6 h n x 1+τ ˆFn (Q(t + x)) (t + x) 2. Hence, it follows from (A.26) that P {n 1/2 n1 n /2 h n log n} { } P ˆF n (Q(t+ x)) (t+ x) 2 d /2 6 n /2 (log n) 1/2 Mn.(A.29) x 1+τ

12 1228 XIAN ZHOU, LIUQUAN SUN AND HAOBO REN Take U = n /2 n V (n) i,v = n 1/2 (r n + S n2 ), and a n = n /2 h n log n in (A.13), so that n 1 2 ( ˆQ n (t) Q n (t)) = U + V b (A.15) (A.16). It follows from (A.21), (A.23), (A.24) and (A.27) (A.29) that P {n 1/2 ( ˆQ n (t) Q(t)) σ(t)} Φ() M(n /2 h n log n + h1/2 n ). Note that Φ(x) Φ() x, <x,<. Now, using (A.11), (A.12) and Theorem 3, we get = P {n 1 2 ( ˆQ n (t) Q(t)) σ(t)} Φ() P {n 1 2 ( ˆQ n (t) Q n (t)) σ(t)} Φ( )+Φ( ) Φ() P {n 1 2 ( ˆQ n (t) Q n (t)) σ(t)} Φ() + n 1 2 Q n (t) Q(t) /σ(t) M(n /2 h n log n + n 1/2 h r n + hn 1/2 ), where = n 1/2 ( Q n (t) Q(t))/σ(t). This completes the proof of Theorem 2. Acknowledgement This work was partiall ported b an RGC Earmarked Grant No. B-Q175 and a HKPOLYU Internal Research Grant No. G-S598. We are grateful to the Editor, Associate Editor, and an anonmous referee for their valuable comments and suggestions which led to the improvement of this paper. References Arcones, M. A. and Giné, E. (1995). On the law of the iterated logarithm for canonical U- statistics and processes. Stochastic Process. Appl. 58, Burke, M. D., Csörgő, S. and Horváth, L. (1988). A correction to and improvement of Strong Approximations of some biometric estimates under random censorship. Probab. Theor Related Fields 71, Chang, M. N. and Rao, P. V. (1989). Berr-Esseen bound for the Kaplan-Meier estimator. Commun. Statist. Theor Methods 18, Chow, Y. S. and Teicher, H. (1978). Probab. Theor. Springer-Verlag, New York. Gijbels, I. and Wang, J. L. (1993). Strong representations of the survival function estimator for truncated and censored data with applications. J. Multivariate Anal. 47, Gu, M. G. (1995). Convergence of increments for cumulative hazard function in mixed censorship-truncation model with application to hazard estimators. Statist. Probab. Lett. 23, Gu, M. G. and Lai, T. L. (1990). Functional laws of the iterated logarithm for the product-limit estimator of a distribution function under random censorship or truncation. Ann. Probab. 18, Gürler, U., Stute, W. and Wang, J. L. (1993). Weak and strong quantile representations for randoml truncated data with applications. Statist. Probab. Lett. 17, Lai, T. L. and Ying, Z. (1991). Estimating a distribution with truncated and censored data. Ann. Statist. 19, Lio, Y. L., Padgett, W. J. and Yu, K. F. (1986). On the asmptotic properties of a kernel tpe quantile estimator from censored samples. J. Statist. Plann. Inference 14,

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