Two Likelihood-Based Semiparametric Estimation Methods for Panel Count Data with Covariates

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1 Two Likelihood-Based Semiparametric Estimation Methods for Panel Count Data with Covariates Jon A. Wellner 1 and Ying Zhang 2 December 1, 2006 Abstract We consider estimation in a particular semiparametric regression model for the mean of a counting process with panel count data. The basic model assumption is that the conditional mean function of the counting process is of the form E{Nt) Z} = expβ T 0 Z)Λ 0 t) where Z is a vector of covariates and Λ 0 is the baseline mean function. The panel count observation scheme involves observation of the counting process N for an individual at a random number K of random time points; both the number and the locations of these time points may differ across individuals. We study semiparametric maximum eudo-likelihood and maximum likelihood estimators of the unknown parameters β 0, Λ 0 ) derived on the basis of a nonhomogeneous Poisson process assumption. The eudo-likelihood estimator is fairly easy to compute, while the maximum likelihood estimator poses more challenges from the computational perspective. We study asymptotic properties of both estimators assuming that the proportional mean model holds, but dropping the Poisson process assumption used to derive the estimators. In particular we establish asymptotic normality for the estimators of the regression parameter β 0 under appropriate hypotheses. The results show that our estimation procedures are robust in the sense that the estimators converge to the truth regardless of the underlying counting process. 1 Research supported in part by National Science Foundation grant DMS , NIAID grant 2R01 AI , and an NWO Grant to the Vrije Universiteit Amsterdam 2 Corresponding author AMS 1980 subject classifications. Primary: 60F05, 60F17; secondary 60J65, 60J70. Key words and phrases. asymptotic distributions, asymptotic efficiency, asymptotic normality, consistency, counting process, empirical processes, information matrix, maximum likelihood, Poisson process, eudo-likelihood estimators, monotone function. 1

2 1. Introduction Suppose that N = {Nt) : t 0} is a univariate counting process. In many applications, it is important to estimate the expected number of events E{Nt) Z} which will occur by the time t, conditionally on a covariate vector Z. In this paper we consider the proportional mean regression model given by Λt Z) E{Nt) Z} = e βt 0 Z Λ 0 t), 1.1) where Λ 0 is a monotone increasing baseline mean function. The parameters of primary interest are β 0 and Λ 0. Suppose that we observe the counting process N at a random number K of random times 0 T K,0 < T K,1 < < T K,K. We write T K T K,1,..., T K,K ), and we assume that K, T K Z) G Z) is conditionally independent of the counting process N given the covariate vector Z. We further assume that Z H on R d with some mild conditions on H for the identifiability of our semiparametric regression model given in Section 3. The observation for each individual consists of X = Z, K, T K, NT K,1 ),..., NT K,K )) Z, K, T K, N K ). 1.2) This type of data is referred to as panel count data. Throughout this manuscript, we will assume that we observe n i.i.d. copies, X 1,..., X n, of X. Panel count data arise in many fields including demographic studies, industrial reliability, and clinical trials; see for example Kalbfleisch and Lawless 1985), Gaver and O Muircheataigh 1987), Thall and Lachin 1988), Thall 1988), Sun and Kalbfleisch 1995), and Wellner and Zhang 2000) where the estimation of either the intensity of event recurrence or the mean function of a counting process with panel count data was studied. Many applications involve covariates whose effects on the underlying counting process are of interest. While there is considerable work on regression modeling for recurrent events based on continuous observations see, for example Lawless and Nadeau 1995), Cook, Lawless, and Nadeau 1996), and Lin, Wei, Yang, and Ying 2000)), regression analysis with panel count data for counting processes has just started recently. Sun and Wei 2000) and Hu, Sun and Wei 2003) proposed estimating equation methods, while Zhang 1998, 2002) proposed a eudo-likelihood method for studying the proportional mean model 1.1) with panel count data. 2

3 To derive useful estimators for this model we will often assume, in addition to 1.1), that the counting process N, conditionally on Z, is a non-homogeneous Poisson process. But our general perspective will be to study the estimators and other procedures when the Poisson assumption may be violated and we assume only that the proportional mean assumption 1.1) holds. Such a program was carried out by Wellner and Zhang 2000) for estimation of Λ 0 without any covariates for this panel count observation model. The outline of the rest of the paper is as follows: In Section 2, we describe two methods of estimation, namely maximum eudo-likelihood and maximum likelihood estimators of β 0,Λ 0 ). The basic picture is that the eudo-likelihood estimator is computationally relatively straightforward and easy to implement, while the maximum likelihood estimators are considerably more difficult, requiring an iterative algorithm in the computation of the profile likelihood. In Section 3, we state the main asymptotic results: strong consistency, rate of convergence and asymptotic normality of ˆβ n maximum likelihood estimators and ˆβ n, for the maximum eudo-likelihood and ˆΛ n ) and ˆβ n, ˆΛ n ) of β 0, Λ 0 ) assuming only the ˆβ n, proportional mean structure 1.1), but not assuming that N is a Poisson process. These results are proved in Section 5 by use of tools from empirical process theory. Although eudo-likelihood methods have been studied in the context of parametric models by Lindsay 1988) and Cox and Reid 2004), not much seems to be known about their behavior in non- and semi-parametric settings such as the one studied here, even assuming that the base model holds. In Section 4, we present the results of simulation studies to demonstrate the robustness of the methods and compare the relative efficiency of the two methods. An application of our methods to a bladder tumor study is presented in this section as well. A general theorem concerning asymptotic normality of semiparametric M-estimators and a technical lemma upon which the proofs of our main theorems rely, are stated and proved in Sections 6 and 7, respectively. 2. Two Likelihood-Based Semiparametric Estimation Methods Maximum Pseudo-likelihood Estimation: To derive our estimators we assume that conditionally on Z, N is a non-homogeneous Poisson process with mean function given by 1.1). The eudo-likelihood method for this model uses the marginal 3

4 distributions of N, conditional on Z, P Nt) = k Z) = Λt Z)k k! exp Λt Z)) and ignores dependence between Nt 1 ), Nt 2 ) to obtain the log-eudo-likelihood: l n β, Λ) = n K i i=1 { } N i) T i) i) K i,j) log ΛT K i,j ) + Ni) T i) K i,j )βt Z i e βt Z i ΛT i) K i,j ). Let R R d be a bounded and convex set, and let F be the class of functions F {Λ : [0, ) [0, ) Λ is monotone nondecreasing, Λ0) = 0}. 2.1) Then the maximum eudo-likelihood estimator ˆβ n, ˆΛ n ) of β 0, Λ 0 ) is given by ˆβ n, ˆΛ n ) argmax β,λ) R F ln β, Λ). This can be implemented in two ste via the usual eudo-) profile likelihood. For each fixed value of β we set and define l,profile n β) ln β, ˆβ n ). Note that l ˆΛ n, β) argmax Λ F l n β, Λ), 2.2) ˆΛ n, β)). Then ˆβ n = argmax β R ln,profile β), and ˆΛ n = ˆΛ n, n β, Λ) depends on Λ only at the observation time points. By convention, we define our estimator to be the one that has jum ˆΛ n only at the observation time points to insure uniqueness. The optimization problem in 2.2) is easily solved and the detail of the solution can be found in Zhang 2002). Maximum Likelihood Estimation: Under the assumption that conditionally on Z, N is a non-homogeneous Poisson process, the likelihood can be calculated using the conditional) independence of the increments of N, Ns, t] Nt) Ns), and the Poisson distribution of these increments: P Ns, t] = k Z) = to obtain the log-likelihood: where l n β, Λ) = n K i i=1 [ Λs, t] Z)]k k! exp Λs, t] Z)) { } N i) K i j log Λ K i j + N i) K i j βt Z i e βt Z i Λ Ki j N Kj NT K,j ) NT K,j 1 ), j = 1,..., K, Λ Kj ΛT K,j ) ΛT K,j 1 ), j = 1,..., K. 4

5 Then ˆβ n, ˆΛ n ) argmax β,λ) R F l n β, Λ). This maximization can also be carried out in two ste via profile likelihood. For each fixed value of β we set ˆΛ n, β) argmax Λ F l n β, Λ), and define ln profile β) l n β, ˆΛ n, β)). Then ˆβ n = argmax β R ln profile β) and ˆΛn = ˆΛ n, ˆβ n ). Similarly, the estimator ˆΛ n is defined to have jum only at the observation time points. To compute the estimate ˆβ n, ˆΛ n ), we adopt a doubly iterative algorithm to update the estimates alternately. The sketch of the algorithm consists of the following ste: S1. Choose the initial β 0) = ˆβ n, the maximum eudo-likelihood estimator. S2. For given β p) p = 0, 1, 2, ), the updated estimate of Λ 0, Λ p) is computed by the modified iterative convex minorant algorithm proposed by Jongbloed 1998) on the likelihood l n β p), Λ). Initialize this algorithm using Λ p 1) and stop the iteration when l n β p), Λ new ) l n β p), Λ current ) l n β p), Λ current ) η. In the very first step, we choose the starting value of Λ by interpolating ˆΛ n linearly between two adjacent jump points to make it monotone increasing and so the likelihood l n β, Λ) is well defined. S3. For given Λ p), the updated estimate of β, β p+1) is obtained by optimizing l n β, Λ p) ) using the Newton-Raphson method. Initialize the algorithm using β p) and stop the iteration when β new β current η. S4. Repeat Ste 2 and 3 until the following convergence criterion is satisfied: l n β p+1), Λ p+1) ) l n β p), Λ p) ) l n β p), Λ p) ) η. As in the case of eudo-likelihood studied in Zhang 2002), it is easy to verify that for any given monotone nondecreasing function Λ, the likelihood l n β, Λ) is a concave function of the regression parameter β with a negative definite Hessian matrix. Using this fact, we can easily show that the iteration process increases the value of the likelihood, i.e. l n β p+1), Λ p+1) ) l n β p), Λ p) ) 0, for p = 0, 1,. The iterative algorithms proposed via the profile eudo-likelihood or the profile likelihood approach converge very well and the convergence does not seem to be effected by the starting point in our simulation experiments described in Section 4. However, this algorithm is not efficient, especially for the maximum likelihood estimation method. It generally needs a considerable number of iterations to achieve the convergence criterion as stated in S4. Meanwhile, computing the profile 5

6 estimator ˆΛ n given in S2 involves the modified iterative convex minorant algorithm which also needs a large number of iterations to converge with the criterion stated in S2. Our simulation experiment with sample size of n=100 shows that computing the maximum likelihood estimator with η = needs about 1800 minutes to converge in a PC Intel Xeon CPU 2.80 GHz) with the algorithm coded in R. Compared to the profile likelihood algorithm, the profile eudo-likelihood algorithm is computationally less demanding, since the profile eudo estimator ˆΛ n has an explicit solution, as shown in Zhang 2002), and hence does not involve any iteration. As result, computing the maximum eudo-likelihood estimator is much faster than computing the maximum likelihood estimator. 3. Asymptotic theory: Results In this section, we study the properties of the estimators ˆβ n, ˆΛ n ) and ˆβ n, ˆΛ n ). We establish strong consistency and derive the rate of convergence of both estimators in some L 2 - metrics related to the observation scheme. asymptotic normality of both ˆβ n and ˆβ n under some mild conditions. We also establish the First we give some notations. Let B d and B denote the collection of Borel sets in R d and R, respectively, and let B [0,τ] = {B [0, τ] : B B} and B 2 [0, τ] = B [0,τ] B [0,τ]. On [0, τ], B [0,τ] ) we define measures µ 1, µ 2, ν 1, ν 2, and γ as follows: for B, B 1, B 2 B [0,τ] and C B d, set ν 1 B C) = P K = k Z = z) C k=1 ν 2 B 1 B 2 C) = P K = k Z = z) C k=1 γb) = R d k=1 k P T k,j B K = k, Z = z)dhz), k P T k,j 1 B 1, T k,j B 2 K = k, Z = z)dhz), P K = k Z = z)p T k,k B K = k, Z = z)dhz). We also define the L 2 -metrics d 1 θ 1, θ 2 ) and d 2 θ 1, θ 2 ) in the parameter space Θ = R F as d 1 θ 1, θ 2 ) = d 2 θ 1, θ 2 ) = { 1/2 β 1 β Λ 1 Λ 2 2 L 2 µ 1 )}, { 1/2 β 1 β Λ 1 Λ 2 2 L 2 µ 2 )}, 6

7 where µ 1 B) = ν 1 B R d ) and µ 2 B 1 B 2 ) = ν 2 B 1 B 2 R d ). To establish consistency, we assume that: C1. The true parameter θ 0 = β 0, Λ 0 ) R F where R is the interior of R. C2. For all j = 1,..., K, K = 1, 2,..., the observation times T K,j are random variables, taking values in the bounded interval [0, τ] for some τ 0, ). The measure µ l H on [0, τ] l R d, B l [0, τ] B d ) is absolutely continuous with respect to ν l for l = 1, 2, and EK) <. C3. The true baseline mean function Λ 0 satisfies Λ 0 τ) M for some M 0, ). C4. The function M 0 defined by M 0 X) K N Kj logn Kj ) satisfies P M 0 X) <. C5. The function M 0 defined by M 0 X) K N Kj log N Kj ) satisfies P M 0 X) <. C6. Z supph), the support of H, is a bounded set in R d. Thus there exists z 0 > 0 such that P Z z 0 ) = 1.) C7. For all a R d, a 0, and c R, P a T Z c) > 0. Condition C7 is needed together with µ l H ν l identifiability of the semiparametric model. Theorem 3.1. from C2 to establish Suppose that Conditions C1-C7 hold and the conditional mean structure of the counting process N is given by 1.1). Then for every b < τ for which µ 1 [b, τ]) > 0, ) d 1 ˆβ n, ˆΛ n 1 [0,b] ), β 0, Λ 0 1 [0,b] ) In particular, if µ 1 {τ}) > 0, then ) d 1 ˆβ n, ˆΛ n ), β 0, Λ 0 ) 0 a.s. as n. 0 a.s. as n. Moreover, for every b < τ for which γ[b, τ]) > 0, d 2 ˆβ n, ˆΛ ) n 1 [0,b] ), β 0, Λ 0 1 [0,b] ) 0 a.s. as n. In particular, if γ{τ}) > 0, then d 2 ˆβ n, ˆΛ ) n ), β 0, Λ 0 ) 0 a.s. as n. 7

8 Remark 3.1. Some condition along the lines of the absolute continuity part of C2 is needed. For example, suppose that Λ 0 t) = t 2, β 0 = 0, Λt) = t, and β = 1. Then if we observe at just one time point T so K = 1 with probability 1), and T = e Z with probability 1, then Λ 0 T )e β 0Z = ΛT )e βz almost surely and the model is not identifiable. C2 holds, in particular, if K, T K ) is independent of Z. The condition on the measure µ 2 H in C2 and C5 are not needed for proving consistency of ˆθ n = ˆβ n, ˆΛ n ), while the condition on the measure µ 1 H in C2 and C4 are not needed for proving consistency of ˆθ n = ˆβ n, ˆΛ n ). To derive the rate of convergence, we also assume that: C8. For some interval O[T ] = [σ, τ] with σ > 0 and Λ 0 σ) > 0, P K {T K,j [σ, τ]}) = 1. C9. P K k 0 ) = 1 for some k 0 <. C10. For some v 0 0, ) the function Z E e v 0Nτ) Z ) is uniformly bounded for Z Z. C11. The observation time points are s 0 separated : i.e. there exists a constant s 0 > 0 such that P T K,j T K,j 1 s 0 for all j = 1,..., K) = 1. Furthermore, µ 1 is absolutely continuous with respect to Lebesgue measure λ with a derivative µ 1 satisfying µ 1 t) c 0 > 0 for some positive constant c 0. C12. The true baseline mean function Λ 0 is differentiable and the derivative has a positive and finite lower and upper bounds in the observation interval, i.e. there exists a constant 0 < f 0 < such that 1/f 0 Λ 0 t) f 0 < for t O[T ]. C13. For some η 0, 1), a T V arz U)a ηa T EZZ T U)a a.s. for all a R d, where U, Z) has distribution ν 1 /ν 1 R + Z). C14. For some η 0, 1), a T V arz U, V )a ηa T EZZ T U, V )a a.s. for all a R d, where U, V, Z) has distribution ν 2 /ν 2 R +2 Z). Theorem 3.2 In addition to the conditions required for the consistency, suppose C8, C9, C10, and C13 hold with the constant v 0 in C10 satisfying v 0 4k 0 1+δ 0 )2 with δ 0 = c 0 Λ 3 0 σ)/24 8f 0) and µ 1 {τ}) > 0. Then ) n 1/3 d 1 ˆβ n, ˆΛ n ), β 0, Λ 0 ) = O p 1). 8

9 Moreover, if conditions C11, C12, and C14 hold along with the conditions listed above but with the constant v 0 in C10 satisfying v 0 4k δ 0 ) 2 with δ 0 = c0 s 3 0 /48 82 f0 4 ) and γ{τ}) > 0, it follows that n 1/3 d 2 ˆβ n, ˆΛ ) n ), β 0, Λ 0 ) = O p 1). Remark 3.2. Conditions C8, C9, C10, C11 and C12 are sufficient for validity of Theorem 3.2, but they are probably not necessary. Conditions C9 and C10 are mainly used in deriving the rate of convergence when the counting process N is allowed to be general but satisfying the mean model 1.1)). C8 says that all the observations should fall in a fixed interval in which the mean function is bounded away from zero and C9 indicates that the number of observations is bounded. These conditions are generally true in clinical applications. Condition C10 holds for all v 0 > 0, if the counting process is uniformly bounded which can be justified in many applications) or forms a Poisson process, conditionally on covariates. The first part of C11 requires that two adjacent observation times should be at least s 0 apart, an assumption which is very reasonable in practice. The second part of C11 implies that the total observation measure µ 1 has a strictly positive intensity or density). C12 requires that the true baseline mean function should be absolutely continuous with bounded intensity function. While C12 is a reasonable assumption in practice, it may be stronger than necessary. We assume C12 mainly for technical convenience in our proofs. Remark 3.3. The metrics d 1 and d 2 are closely related. Since k a j b j ) 2 k 2 k {a j a j 1 ) b j b j 1 )} 2 see Wellner and Zhang 1998) for a proof), the two metrics are equivalent under C9 and therefore the consistency and rate of convergence results for the Maximum Likelihood Estimator ˆβ n, ˆΛ n ) hold under the metric d 1 as well. Remark 3.4. Condition C13 can be justified in many applications. By the Markov inequality, it is easy to see that condition C7 implies that EZZ T ) is a positive-definite matrix. under the probability measure ν 1 /ν 1 R + Z). Let E 1 and V ar 1 denote expectations and variances If we assume that V ar 1 Z U) is a positive-definite matrix, and we set λ 1 = max{eigenvaluee 1 ZZ T U)} and λ d = min{eigenvaluev ar 1Z U))}, then 0 < λ d λ 1. Therefore, for any a R d, a T V ar 1 Z U)a a T λ d a = λ d λ 1 a T λ 1 a λ d λ 1 a T E 1 ZZ T U)a. 9

10 Thus, condition C13 holds by taking η λ d /λ 1. Note that both λ 1 and λ d depend on U in general and the argument here works assuming that this ratio has a positive lower bound uniformly in U. We can justify C14 similarly. Although the overall convergence rate for both the maximum eudo- and likelihood estimators is of the order n 1/3, the rate of convergence for the estimators of the regression parameter, as usual, may still be n 1/2. Similar to the results of Huang 1996) for the Cox model with current status data, we can establish asymptotic normality of both ˆβ n and ˆβ n. Theorem 3.3 Under the same conditions assumed in Theorem 3.2, the estimators ˆβ n and and ˆβ n are asymptotically normal: n ˆβn β 0 ) d Z N d 0, A 1 B A 1) T ), 3.1) n ˆβ n β 0 ) d Z N d 0, A ) 1 B A ) 1) T ), 3.2) where B = E C j,j Z) [ Z RT K,j, T K,j ) ] 2, j,j =1 A = E Λ 0Kj e βt 0 Z [Z RK, T K,j 1, T K,j )] 2, B = E C j,j Z) [Z R K, T K,j )] [ Z R K, T K,j ) ] T, j,j =1 A = E Λ 0Kj e βt 0 Z [Z R K, T K,j )] 2 in which RK, T K,j, T K,j ) E R K, T K,j ) E Ze βt 0 Z K, T K,j, T K,j Ze βt 0 Z K, T K,j ) /E ) /E ), e βt 0 Z K, T K,j, T K,j ) e βt 0 Z K, T K,j, C j,j Z) = Cov [ N Kj, N Kj Z, K, T K ], C j,j Z) = Cov [ NT Kj ), NT Kj ) Z, K, T K,j, T K,j ], Λ 0Kj = Λ 0 T K,j ), and Λ 0,K,j = Λ 0 T K,j ) Λ 0 T K,j 1 ). If the counting process is, conditionally given Z, a non-homogeneous Poisson process with conditional mean function given as specified, then C j,j Z) = Λ 0Kj e βt 0 Z 1{j = j }. It follows that B = A = Iβ 0 ), the information matrix computed in Wellner, Zhang, and Liu 2004), and hence A 1 B A 1) T = I 1 β 0 ). 10

11 This implies that the estimator ˆβ n under the conditional Poisson process is asymptotically efficient. However, since C j,j Z) = e βt 0 Z Λ 0Kj j ), B A. This shows that the semiparametric maximum eudo-likelihood estimator be asymptotically efficient under the Poisson assumption. ˆβ n will not There is, however, a natural Poisson regression model for which the maximum eudo-likelihood estimator is asymptotically efficient: if we simply assume that the conditional distribution of NT K,1 ),..., NT K,K )) given K, T K,1,..., T K,K, Z) is that of a vector of independent Poisson random variables with means given by ΛT K,j Z) = expβ T 0 Z)Λ 0T K,j ) for j = 1,..., K, then C j,j Z) = Cov [ NT K,j ), NT K,j ) Z, K, T K,j, T K,j ] = ΛTK,j Z)1{j = j }. Hence B = A = I P oissregr β 0 ) and ˆβ is asymptotically efficient for this alternative model. In practice, this occurs when NT K,1 ), NT K,2 ),, NT K,K )) consist of cluster Poisson count data in which the counts within a cluster are independent. 4. Numerical Results 4.1. Simulation Studies We generated data using the same schemes as those given in Zhang 2002). Monte- Carlo bias, standard deviation, and mean squared error of the maximum eudolikelihood and maximum likelihood estimates are then compared. Scenario 1. In this scenario, the data is {Z i, K i, T i) K i, N i) K i ) : i = 1, 2,..., n} with Z i = Z i,1, Z i,2, Z i,3 ) where, conditionally on Z i, K i, T Ki ), the counts N i) K i were generated from a Poisson process. For each subject, we generate data by the following scheme: Z i,1 Unif0, 1), Z i,2 N0, 1), Z i,3 Bernoulli0.5); K i is sampled randomly from the discrete set, {1, 2, 3, 4, 5, 6}; Given K i, T i) K i = T i) K i,1, T i) K i,2,..., T i) K i,k i ) are the order statistics of K i random observations generated from Unif1, 10) and rounded to the second decimal point to make the observation times possibly tied. The panel counts N i) K i = N i) T i) K i,1 ), Ni) T i) K i,2 ),..., Ni) T i) K i,k i )) are generated from the Poisson process with the conditional mean function given by Λt Z i ) = 2t expβ T 0 Z i), i.e. N i) T i) K i,j ) Ni) T i) i) K i,j 1) P oisson{2t K i,j T i) K i,j 1 ) expβt 0 Z i )}, 11

12 where β 0 = β 1, β 2, β 3 ) T = 1.0, 0.5, 1.5) T. For this scenario, we can directly calculate the asymptotic covariance matrices given in Theorem 3.3, Σ A ) 1 B A ) 1) T = 1582/17787)W 1 and Σ = A 1 B A 1) T = A 1 = 1260/19179)W 1 respectively, where W = E{e βt 0 Z [Z EZe βt 0 Z )/Ee βt 0 Z )] 2 }. Since it is difficult to evaluate the matrix W analytically, we calculated it numerically using Mathematica Wolfram 1966)) to obtain the following approximate results for the asymptotic covariance matrices: Σ ) and Σ ) We conducted simulation studies with sample sizes of n = 50 and n = 100, respectively. For each case, the Monte-Carlo sample bias, standard deviation and mean squared error for the semiparametric estimators of the regression parameters are reported in Table 1. We also include the asymptotic standard errors obtained from 4.1) and 4.2) in Table 1 to compare with the Monte-Carlo sample standard deviations. The results show that the sample bias for both estimators is small, the standard deviation and mean squared error are smaller for the maximum likelihood method compared to the eudo-likelihood method and the latter decrease as n 1/2 and n 1 respectively as sample size increases. Moreover, the standard errors of estimates based on asymptotic theory are all close to the corresponding standard deviations based on the Monte-Carlo simulations. All of these provide numerical support for our asymptotic results in Theorem 3.3. Based on the results of 1000 Monte-Carlo samples, we plot the pointwise means, 2.5- and 97.5-percentiles of both estimators of the baseline mean function Λt) = 2t in Figure 1. It clearly shows that both estimators seem to have negligible bias and the maximum likelihood estimator has smaller variability compared to the maximum eudo-likelihood estimator. When sample size increases, the variability of both estimators decreases accordingly. Scenario 2. In this scenario, the data is {Z i, K i, T i) K i, N i) K i ) : i = 1, 2,..., n} with Z i = Z i,1, Z i,2, Z i,3 ) and, conditionally on Z i, K i, T i) K i ), the counts N i) K i 12

13 Table 1: Results of the Monte-Carlo simulation studies for the regression parameters estimates based on 1000 repeated samples for data generated from the conditional Poisson process n = 50 n = 100 Pseudo- Pseudo- Likelihood Likelihood Likelihood Likelihood Estimate of β 1 BIAS SD ASE MSE Estimate of β 2 BIAS SD ASE MSE Estimate of β 3 BIAS SD ASE MSE

14 were generated from a mixed Poisson process. For each subject, Z i, K i, T i) K i ) are generated in exactly the same way as in Scenario 1. The panel counts are, however, generated from a homogeneous Poisson process with a random effect on the intensity: given subject i with covariates Z i and frailty variable α i independent of Z i ), the counts are generated from the Poisson process with intensity λ + α i ) expβ T 0 Z i), where λ = 2.0 and α i { 0.4, 0, 0.4} with probabilities 0.25, 0.5, and 0.25, respectively. In this scenario, the counting process given only the covariates is not a Poisson process. However, the conditional mean function of the counting process given the covariates still satisfies 1.1) with Λ 0 t) = 2t and thus our proposed methods are expected to be valid for this case as well. The asymptotic variances given in Theorem 3.3 for this scenario are and Σ = A ) 1 B A ) 1) T = W W 1 W W 1 ) T Σ = A 1 B A 1) T = W W 1 W W 1 ) T, respectively, where W = E{e 2βT 0 Z [Z EZe βt 0 Z )/Ee βt 0 Z )] 2 }. Using Mathematica Wolfram 1966)) to calculate the asymptotic covariance matrices numerically yields: Σ ) and Σ ) As in Scenario 1, we conducted simulation studies with sample sizes of n = 50 and n = 100, respectively. For each case, the Monte-Carlo sample bias, standard deviation and mean squared error for the semiparametric estimators of the regression parameters are computed with 1000 repeated samples. The results are shown in Tables 2. In Figure 1, we also plot the pointwise means, 2.5 percentiles, and percentiles of both estimators of the unconditional baseline mean function Λ 0 t) = 2t based on the results obtained from 1000 Monte-Carlo samples. We observe the same phenomenon as appeared in Scenario 1: for the regression parameters, both 14

15 standard deviation and mean squared error using the maximum likelihood method are smaller than those using the eudo-likelihood method while the bias is relatively small; for the baseline mean function, both estimators have a negligible bias but the maximum likelihood estimator has less variability than the maximum eudolikelihood estimator. We also note that the variability of semiparametric estimators are relatively larger than their counterpart in Scenario 1. This may be caused by violation of the assumption of a conditional Poisson process given only the covariates. We also include the asymptotic standard errors of the regression parameter estimates based on 4.3) and 4.4) in Table 2. Again the standard errors derived from the asymptotic theory are all close to the standard deviations based Monte-Carlo simulations. These simulation studies provide numerical support for the statement that the proposed semiparametric estimation methods are robust against the underlying conditional Poisson process assumption. These methods are valid as long as the proportional mean function model 1.1) holds. We have also conducted several analytical analyses to compare the semiparametric efficiency between the maximum eudo-likelihood and maximum likelihood estimation methods. There is considerable evidence that the maximum likelihood method based on the Poisson process assumption) is more efficient than the eudo-likelihood method both on and off the Poisson model, with large differences occurring when K is heavily tailed. The detailed analytical results are presented in Wellner, Zhang, and Liu 2004) A Real Data Example Using the semiparametric methods proposed in the preceding sections, we analyze the bladder tumor data, extracted from Andrews and Herzberg 1985, pp ). This data set comes from a bladder tumor study conducted by the Veterans Administration Cooperative Urological Research Byar et al., 1977). In the study, a randomized clinical trial of three treatments, placebo, pyridoxine pills and thiotepa instillation into the bladder was conducted for patients with superficial bladder tumor when entering the trial. At each follow-up visit, tumors were counted, measured and then removed if observed, and the treatment was continued. The treatment effects, especially the thiotepa instillation, on suppressing the recurrence of bladder tumor have been explored by many authors, for example, Wei, Lin, and Weissfeld 1989), Sun and Wei 2000), Wellner and Zhang 2000) and Zhang 2002). In this paper, we study the proportional mean model that has been proposed by 15

16 Table 2: Results of the Monte-Carlo simulation studies for the regression parameter estimates based on 1000 repeated samples for data generated from the mixed Poisson process n = 50 n = 100 Pseudo- Pseudo- Likelihood Likelihood Likelihood Likelihood Estimate of β 1 BIAS SD ASE MSE Estimate of β 2 BIAS SD ASE MSE Estimate of β 3 BIAS SD ASE MSE

17 Sun and Wei 2000) and Zhang 2002), E{Nt) Z} = Λ 0 t) expβ 1 Z 1 + β 2 Z 2 + β 3 Z 3 + β 4 Z 4 ), 4.5) where Z 1 and Z 2 represent the number and size of bladder tumors at the beginning of the trial, and Z 3 and Z 4 are the indicators for the pyridoxine pill and thiotepa instillation treatments, respectively. We choose β 0) = 0, 0, 0, 0) to start our iterative algorithm and η = for the convergence criteria to stop the algorithm. Since the asymptotic variances are difficult to estimate, we adopt the bootstrap procedure to estimate the asymptotic standard error of the semiparametric estimates of the regression parameters. We generated 200 bootstrap samples size and calculated the proposed estimators for each sampled data set. The sample standard deviation of the estimates based on these 200 bootstrap samples is used to estimate the asymptotic standard error. The inference based on the bootstrap estimator for asymptotic standard error is given in Table 3. The semiparametric maximum eudo-likelihood and maximum likelihood estimators of the baseline mean function are plotted in Figure 2. Both methods yield the same conclusion that the baseline number of tumors the number of tumors observed when entering the trial) significantly affect the recurrence of the tumor at level 0.05 p-value= and , respectively, for the maximum eudo-likelihood and maximum likelihood methods), and the thiotepa instillation treatment appears to reduce the recurrence of tumor significantly. p-value= and , respectively for the maximum eudolikelihood and maximum likelihood methods). In Figure 2, we can see that the maximum likelihood estimator of the baseline mean function is substantially smaller than the maximum eudo-likelihood estimator which preserves the phenomenon we have observed in nonparameteric estimation methods for this data set studied in Wellner and Zhang 2000). We also notice that the maximum likelihood method, in contrast to what we have observed through both the simulation and analytical studies, yields larger standard errors compared to the eudo-likelihood method. Violation of the proportional mean model 4.5) for this data set could be the explanation for this result, since Zhang 2006) plotted the nonparameteric eudo-likelihood estimators of the mean function for each of three treatments and found that the estimators cross over. While plotting the nonparametric estimators of the mean function for the grou defined by covariates is a reasonable first step in an exploration of the validity of the 17

18 Z 1 Pseudo-Likelihood Z 2 Pseudo-Likelihood Z 3 Pseudo-Likelihood Z 4 Pseudo-Likelihood Table 3: Semiparametric inference for the bladder tumor study based on 200 bootstrap samples from the original data set. Variable Method ˆβ ˆ se ˆβ) ˆ ˆβ/ se ˆβ) p-value Likelihood Likelihood Likelihood Likelihood

19 proposed model, it would be preferable to proceed via more quantitative measures, such as appropriate goodness-of-fit statistics. The construction of goodness-of-fit test statistics for regression modelling of panel count data remains an open problem for future research. All the numerical experiments in this paper were implemented in R. The computing programs are available from the second author. 5. Asymptotic theory: Proofs We use empirical process theory to study the asymptotic properties of the semiparametric maximum eudo-likelihood and maximum likelihood estimators. The proof of Theorem 3.1 is closely related to the proof of Theorems 4.1 of Wellner and Zhang 2000). The rate of convergence is derived based on the general theorem for the rate of convergence given in Theorem of van der Vaart and Wellner 1996). The asymptotic normality proofs for both ˆβ n and ˆβ n are based on the general theorem for M-estimation of regression parameters in the presence of a nonparametric nuisance parameter which is stated and proved) in Section 6. Proof of Theorem 3.1: Zhang 2002) has given a proof for the first part of the theorem concerning the semiparametric maximum eudo-likelihood estimator. Unfortunately, his proof of theorem 1 on pages 47 and 48 is not correct in particular the conditions imposed do not suffice for identifiability as claimed). Here we give proofs for both the maximum eudo-likelihood and maximum likelihood estimators. We first prove the claims concerning the eudo-likelihood estimators ˆβ n, ˆΛ n ). Let M n θ) = n 1 l n β, Λ) = P n m θ X) and M θ) = P m θ X), where K θ X) = { NKj log Λ Kj + N Kj β T Z Λ Kj expβ T Z) }. m First, we show that M has θ 0 = β 0, Λ 0 ) as its unique maximizing point. Computing the expectation conditionally on Z, K, T K ) yields [ M θ 0 ) M θ) = Λu) expβ T Λ0 u) expβ0 T z)h z) ] Λu) expβ T dν 1 u, z), z) where hx) = x logx) x + 1. The function hx) satisfies hx) 0 for x > 0 with equality holding only at x = 1. Hence M θ 0 ) M θ) and M θ 0 ) = M θ) if and only if Λ 0 u) expβ T 0 z) Λu) expβ T z) = 1 a.e. with respect to ν ) 19

20 This implies that β = β 0 and Λu) = Λ 0 u) a.e. with respect to µ 1 5.2) by C2 and C7. Here is a proof of this claim: Let f 1 u) = Λu) Λ 0 u), f 2 u) = Λ 0 u), h 1 z) = expβ T z), h 2 z) = expβ T z) expβ T 0 z). Then 5.1) implies that Λ 0 u) expβ T 0 z) = Λu) expβt z) a.e. ν 1, or, equivalently 0 = {Λu) Λ 0 u)}e βt z + Λ 0 u)e βt z e βt 0 z ) = f 1 u)h 1 z) + f 2 u)h 2 z) a.e. ν 1. Since µ 1 H is absolutely continuous with respect to ν 1 by assumption C2, equality holds in the last display a.e. with respect to µ 1 H. By multiplying across the identity in the last display by ab, integrating with respect to the measure µ 1 H, and then applying Fubini s theorem, it follows that 0 = f 1 adµ 1 h 1 bdh + f 2 adµ 1 h 2 bdh for all measurable functions a = au) and b = bz). The choice of a = f 1 1 A for A B 1 and b = h 1 1 B for B B d yields 0 = f1 2 1 A dµ 1 h 2 11 B dh + f 1 f 2 1 A dµ 1 h 1 h 2 1 B dh ; the choice of a = f 2 1 A for the same A B 1 ) and b = h 2 1 B for the same set B B d ) yields 0 = f 1 f 2 1 A dµ 1 h 1 h 2 1 B dh + f A dµ 1 h 2 21 B dh. Thus we have f A dµ 1 h 2 11 B dh = f 1 f 2 1 A dµ 1 h 1 h 2 1 B dh = f A dµ 1 h 2 21 B dh for all A B 1 and B B d. By Fubini s theorem, this yields f1 2 h 2 1dµ 1 H) = f2 2 h 2 2dµ 1 H). A B for all such sets A, B. But this implies that the measures γ 1 and γ 2 defined by γ j A B) = A B f j 2h2 j dµ 1 H), j = 1, 2, are equal for all the product sets A B, 20 A B

21 and hence, by a standard monotone class argument, we conclude that γ 1 = γ 2 as measures on [0, τ] R d, B 1 [0, τ] B d ). It follows that f1 2u)h2 1 z) = f 2 2u)h2 2 z) a.e. with respect to µ 1 H. Thus we conclude that or, in other words, f1 2u) f2 2u) = h2 2 z) h 2 1 z) a.e. on {u, z) : f1 2 u) > 0, h 2 1z) > 0}, ) Λu) 2 Λ 0 u) 1 = 1 expβ 0 β) T z)) 2 a.e. with respect to µ 1 H. This implies that 5.2) holds in view of C7. Integrating across this identity with respect to µ 1 yields ) Λu) 2 Λ 0 u) 1 dµ 1 u) = 1 expβ 0 β) T z)) 2 µ 1 [0, τ]) a.e. H, and hence the right side is a constant a.e. H. But this implies that β = β 0 in view of C7. Combining this with the last display shows that 5.2) holds. For any given ɛ > 0, let θ n = ˆβ n, 1 ɛ)ˆλ n +ɛλ 0 ) = ˆβ n, Since M n ˆθ n ) M n θ n ) = M n ˆθ n + ɛ0, Λ 0 where 0 lim { = P n P n ɛ 0 M n ˆθ n + ɛ0, Λ 0 ɛ N Kj ˆΛ T exp ˆβ n Z) 1 ˆΛ = ˆΛ n T K,j ). This yields { Λ 0Kj N Kj + ˆΛ ˆΛ n )) M n ˆθ n ) } } T ˆΛ exp ˆβ n Z) ˆΛ n )), it follows that Λ 0Kj ˆΛ n )+ɛ0, Λ 0 T ˆΛ ) exp ˆβ n Z), ) T P n N Kj + Λ 0Kj exp ˆβ n Z) CP n N K,j + Λ 0 T K,j )) a.s. CP N K,j + Λ 0 T K,j )), ˆΛ n ). by C1-C3 and the strong law of large numbers. Here C represents a constant. In the sequel C appearing in different lines may represent different constants.) The 21

22 limit on the right side is finite. On the other hand, { lim sup P n Λ 0Kj N Kj n ˆΛ + lim sup n P n C lim sup n ˆΛ n b)p n } T ˆΛ exp ˆβ n Z) 1 [b,τ] T K,j )ˆΛ T exp ˆβ n Z) 1 [b,τ] T K,j ) = C lim sup n ˆΛ n b) µ 1 [b, τ]). Hence ˆΛ n t) is uniformly bounded almost surely for t [0, b] if µ 1 [b, τ]) > 0 for some 0 < b < τ or for t [0, τ] if µ 1 {τ}) > 0. By the Helly-Selection ) Theorem and the compactness of R F, it follows that ˆθ n = ˆβ n, ˆΛ n has a subsequence ) ˆθ n = ˆβ n, ˆΛ n converging to θ + = β +, Λ + ), where Λ + is an increasing bounded function defined on [0, b] for a b < τ and it can be defined on [0, τ] if µ 1 {τ}) > 0. Following the same argument as in proving Theorem 4.1 of Wellner and Zhang 2000), we can show that M θ + ) M θ 0 ). Since M θ 0 ) M θ + ), by the argument above 5.1), we conclude that M θ + ) = M θ 0 ). Then 5.2) implies that β + = β 0 and Λ + = Λ 0 a.e. in µ 1. Finally, the dominated convergence theorem yields the strong consistency of ˆβ n, ˆΛ n ) in the metric d 1. Now we turn to the maximum likelihood estimator. Let M n θ) = n 1 l n β, Λ) = P n m θ X) and Mθ) = P m θ X), where m θ X) = { NKj log Λ Kj + N Kj β T Z Λ Kj expβ T Z) }. Much as in the eudo-likelihood case, M has θ 0 = β 0, Λ 0 ) as its unique maximum point, and β = β 0 and Λv) Λu) = Λ 0 v) Λ 0 u) a.e. with respect to µ ) The proof of consistency then proceeds along the same lines as for the eudolikelihood estimator; see Wellner and Zhang 2005) for the detailed argument. The uhot is that ˆβ n, ˆΛ n ) is almost surely consistent in the metric d 2. Proof of Theorem 3.2: We derive the rate of convergence by checking the conditions in Theorem of van der Vaart and Wellner 1996). Here we give 22

23 a detailed proof for the first part of the theorem and for the second, we point out the differences in the proof from the first. Let K θ X) = { NKj log ΛT K,j ) + N Kj β T Z ΛT K,j ) expβ T Z) } m with N Kj = NT K,j ) and M θ) = P m θ X). We have M θ 0 ) M θ) = E Z,K,T K ) ΛT K,j ) expβ T Z)h { Λ0 T K,j ) expβ0 T Z) } ΛT K,j ) expβ T. Z) Since hx) 1/4)x 1) 2 for 0 x 5, for any θ in a sufficiently small neighborhood of θ 0 M θ 0 ) M θ) 1 { 4 E Z,K,T K ) ΛT K,j ) expβ T Λ0 T K,j ) expβ0 T Z) Z) } 2 ΛT K,j ) expβ T Z) 1 C {Λu)e βt z Λ 0 u)e βt 0 z } 2 dν 1 u, z) 5.4) by C1, C2 and C6. Let gt) = Λ t U) expβ T t Z) with Λ t = tλ + 1 t)λ 0 and β t = tβ + 1 t)β 0 for 0 t 1 with U, Z) ν 1. Then ΛU) expβ T Z) Λ 0 U) expβ0 T Z) = g1) g0), and hence, by the mean value theorem, there exists a 0 ξ 1 such that g1) g0) = g ξ). Since g ξ) = expβ T ξ Z) [ Λ Λ 0 )U) + {Λ 0 + ξλ Λ 0 )} U)β β 0 ) T Z ] = expβ T ξ Z) [ Λ Λ 0 )U) { 1 + ξβ β 0 ) T Z } + β β 0 ) T ZΛ 0 U) ], from 5.4) we have { } P m θ 0 X) m θ X) C [Λ Λ0 )u) { 1 + ξβ β 0 ) T z } + β β 0 ) T zλ 0 u) ] 2 dν1 u, z) = ν 1 {g 1 h + g 2 } 2, where g 1 U, Z) β β 0 ) T ZΛ 0 U), g 2 U) = Λ Λ 0 )U), and hu, Z) = 1 + ξλ Λ 0 )U)/Λ 0 U) in the notation of Lemma 8.8, page 432, van der Vaart 2002). To 23

24 apply van der Vaart s lemma we need to bound [ν 1 g 1 g 2 )] 2 by a constant less than one times ν 1 g 2 1 )ν 1g 2 2 ). For the moment we write expectations under ν 1 as E 1. But by the Cauchy-Schwarz inequality and then computing conditionally on U we have [E 1 g 1 g 2 )] 2 = {E 1 [E 1 g 1 g 2 U)]} 2 E 1 {g 2 2}E 1 {[E 1 g 1 U)] 2 } E 1 {g 2 2}E 1 {Λ 2 0U)E 1 [{β β 0 ) T Z} 2 U]} = E 1 {g 2 2}E 1 {Λ 2 0U)E 1 [β β 0 ) T Z Z E 1 Z U)) 2 β β 0 ) U]} 1 η)e 1 {g 2 2}E 1 {Λ 2 0U)β β 0 ) T E 1 ZZ T U)β β 0 ) T } = 1 η)e 1 {g 2 2}E 1 {g 2 1}, where the last inequality follows from C13. By van der Vaart s lemma, ν 1 {g 1 h + g 2 } 2 C{ν 1 g 2 1) + ν 1 g 2 2)} = C{ β β Λ Λ 0 2 L 2 µ 1 ) } = Cd2 1θ, θ 0 ). To derive the rate of convergence, next we need to find a φ n σ) such that E sup d 1 θ,θ 0 )<σ G n m θ X) m θ 0 X)) Cφ n σ). { } We let M 1 δ θ 0) = m θ X) m θ 0 X) : d 1 θ, θ 0 ) < δ be the class of differences. We shall find an upper bound for the bracketing entropy numbers of this class. We also let F δ = {Λ F : Λ Λ 0 L2 µ 1 ) δ}. Since F δ is a class of monotone nondecreasing functions, by Theorem of van der Vaart and Wellner 1996), for any ɛ > 0, there exists a set of brackets: [Λ l 1, Λr 1 ], [Λl 2, Λr 2 ],..., [Λl q, Λ r q] with q exp M/ɛ), such that for any Λ F δ, Λ l i t) Λt) Λr i t) for all t O[T ] and some 1 i q, and {Λ r i u) Λl i u)}2 dµ 1 u) ɛ 2. Here we use the fact that µ 1 is a finite measure under our hypotheses, and hence can be normalized to be a probability measure.) For sufficiently small ɛ > 0 and δ > 0, we can construct the bracketing functions so that Λ r i t) Λl i t) γ 1 and Λ l i t) γ 2 with γ 1, γ 2 > 0 for all t O[T ] and 1 i q. Here is the proof for this claim: For any Λ F δ, the result of Lemma 7.1 implies that Λ 0 t) ɛ 1 Λt) Λ 0 t) + ɛ 1 for a sufficiently small ɛ 1 > 0 ɛ 1 can be chosen as δ/c) 2/3 in view of Lemma 7.1) and for all t O[T ]. For any 1 i q, there is a Λ F δ such that Λ r i Λ L 2 µ 1 ) ɛ and Λ Λ l i L 2 µ 1 ) ɛ, which implies that Λ r i Λ 0 L2 µ 1 ) ɛ ɛ = ɛ 2 + δ 2 ) and Λ l i Λ 0 L2 µ 1 ) ɛ. By Lemma 7.1, this yields that Λ r i t) Λ 0t) + ɛ 2 and Λ l i t) Λ 0t) ɛ 2 for a sufficient small 24

25 ɛ 2 > 0. ɛ 2 can be chosen as ɛ /C) 2/3 ) Therefore our claim is justified by letting γ 1 = 2ɛ 2 and γ 2 = Λ 0 σ) ɛ 2, in view of C8. Since β R, a compact set in R d, we can construct an ɛ-net for R, β 1, β 2,..., β p with p = [ M /ɛ d)] such that for any β R there is a s such that β T Z β T s Z ɛ and expβ T Z) expβ T s Z) Cɛ. Therefore we can construct a set of brackets for M 1 δ θ 0) as follows: where and [m l i,s m l m r X), mr i,s X)], for i = 1, 2,..., q; s = 1, 2,..., p, i,s X) = K [ N Kj log Λ l it K,j ) + N Kj β T s Z ɛ) Λ r i T K,j ) { expβ T s Z) + Cɛ }] m θ0 X) K i,s X) = [ NKj log Λ r i T K,j ) + N Kj βs T Z + ɛ) Λ l it K,j ) { expβ T s Z) Cɛ }] m θ0 X). In what follows, we show that f i,s X) 2 P,B = mr i,s X) ml i,s X) 2 P,B Cɛ2, where P,B is the Bernstein norm defined by f P,B = {2P e f 1 f ) } 1/2 see van der Vaart and Wellner, 1996, page 324). Since 2e x 1 x) x 2 e x for x 0, it follows that f 2 P,B P e f f 2). Therefore, f i,s X) 2 P,B P e f i,sx) f i,s X) 2). By writing out f i,s X) = m r i,s Since K f i,s X) N KK log Λ r i T K,j ) log Λ l it K,j ) + 2ɛ) + expβ T s Z) ) Λ r i T K,j ) Λ l it K,j ) + Cɛ X) ml i,s X), we find that ) Λ r i T K,j ) + Λ l it K,j ). log y = log x + x + ξy x)) 1 y x) for 0 < x y, some ξ [0, 1], 5.5) 25

26 we find that log Λ r i T K,j ) log Λ l it K,j ) + γ 1 2 ) Λ r i T K,j ) Λ l it K,j ) by construction of Λ l i. Hence, by C9 and our claim above, we conclude further that K log Λ r i T K,j ) log Λ l i T K,j) + 2ɛ ), K Λ r i T K,j ) Λ l i T K,j) ), and K Λ r i T K,j ) + Λ l i T K,j) ) are all uniformly bounded in O[T ]. More explicitly, taking ɛ Λ 0 σ), noting that this implies δ C2 1 Λ 0 σ)) 3/2 δ 0 with C = c 0 /24f 0 )) 1/2 by Lemma 7.1, and using the relations ɛ 2 = ɛ /C) 2/3, ɛ = ɛ 2 + δ 2 ) 1/2 δ, and ɛ Λ 0 σ), we find that log Λ r i T K,j ) log Λ l it K,j ) + 2ɛ) 2 ) 2ɛ ɛ 4k δ Λ 0 σ) ɛ 2 0 )2. Therefore, by arguing conditionally on Z, K, T K ) and using C10, f i,s X) 2 P,B P e fi,sx) f i,s X) 2) ) 2 CP log Λ r i T K,j ) log Λ l it K,j ) + 2ɛ evn KK N 2 KK + exp2β T s Z) ) 2 Λ r i T K,j ) Λ l it K,j ) + Cɛ 2. By C6, C10, and Taylor expansion for log Λ r i T K,j) at Λ l i T K,j) as shown above, we have ) 2 f i,s X) 2 P,B C E K,T K ) Λ r i T K,j ) Λ l it K,j ) + ɛ 2 Cɛ2. This shows that the total number of ɛ-brackets for M 1 δ θ 0) will be of the order M/ɛ) d e CM /ɛ) ) and hence log N [ ] ɛ, M 1 δ θ 0 ), P,B C 1/ɛ). We can similarly verify that P f θ X)) 2 Cδ 2 for any f θ X) = m θ X) m θ 0 X) M 1 δ θ 0). Hence by Lemma of van der Vaart and Wellner 1996), EP G n M 1 δ θ 0 ) C J ) [ )] [ ] δ, M 1 J [ ] δ, M 1 δ θ 0 ), P,B 1 + δ θ 0 ), P,B δ 2, n 26

27 where J [ ] δ, M 1 δ θ 0), P,B ) = δ = C ) 1 + log N [ ] ɛ, M 1 δ θ 0 ), P,B dɛ δ ɛ dɛ C ɛ 1/2 dɛ Cδ 1/2. 0 δ 0 0 Hence φ n δ) = δ 1/2 1 + δ 1/2 /δ 2 n)) = δ 1/2 + δ 1 / n. Then it is easy to see that φ n δ)/δ is a decreasing function of δ, and n 2/3 φ n n 1/3 ) = n 2/3 n 1/6 +n 1/3 n 1/2 ) = 2 n. So it follows by Theorem of van der Vaart and Wellner 1996) that ) n 1/3 d 1 ˆβ n, ˆΛ n ), β 0, Λ 0 ) = O P 1). For the maximum likelihood estimator ˆβ n, ˆΛ n ) the proof of the rate of convergence result as stated in Theorem 3.2 proceeds along the same lines as the rate result for the maximum eudo-likelihood estimator given above, but with g 1 U, V, Z) β β 0 ) T Z Λ 0 U, V ), g 2 U, V ) = Λ Λ 0 )U, V ), and hu, V, Z) = 1+ξ Λ Λ 0 )U, V )/ Λ 0 U, V ) in the application of van der Vaart s lemma 8.8. For details see Wellner and Zhang 2005). Proof of Theorem 3.3: We give a detailed proof for the first part of the theorem, and only outline the differences in the proof for the second. the theorem by checking the conditions A1-A6 of Theorem 6.1. We prove Note that A1 holds with γ = 1/3 because of the rate of convergence given in Theorem 3.2. The criterion function with only one observation is given by m β, Λ; X) = K {N Kj log Λ Kj + N Kj β T Z e βt Z Λ Kj }, and thus we have m K 1 β, Λ; X) = ZN Kj ΛT K,j ) expβ T Z)) K m 2 β, Λ; X)[h] = ) NKj expβ T Z) h Kj Λ Kj m K 11 β, Λ; X)[h] = Λ Kj ZZ T expβ T Z) K m 12 β, Λ; X)[h] = m 21 β, Λ; X)[h] = Z expβ T Z)h Kj 27

28 and K m 22 β, Λ; X)[h, h] = N Kj Λ 2 h Kj h Kj, Kj where Λ Kj = ΛT K,j ) and h Kj = T K,j 0 ht)dλt) for h L 2 Λ). A2 automatically holds by the model assumption 1.1). For A3, we need to find a h such that Ṡ 12 β 0, Λ 0 )[h] Ṡ 22 β 0, Λ 0 )[h, h] for all h L 2 Λ 0 ). Note that = P {m 12 β 0, Λ 0 ; X)[h] m 22 β 0, Λ 0 ; X)[h, h]} = 0, P {m 12 β 0, Λ 0 ; X)[h] m 22 β 0, Λ 0 ; X)[h, h]} [ = E Ze βt 0 Z [ = E K,T K,Z) Therefore, an obvious choice of h is Hence N Kj Λ 0Kj ) 2 h Kj ] h Kj Ze βt 0 Z eβt 0 Z h Kj Λ 0Kj ] h Kj. h Kj = Λ 0Kj EZe βt 0 Z K, T K,j )/Ee βt 0 Z K, T K,j ) Λ 0Kj R K, T K,j ). m β 0, Λ 0 ; X) = m 1 β 0, Λ 0 ; X) m 2 β 0, Λ 0 ; X)[h ] { ) ) } = Z N Kj e βt 0 Z NKj Λ 0Kj e βt 0 Z Λ 0Kj R K, T K,j ) Λ 0Kj = N Kj e βt 0 Z Λ 0Kj ) [Z R K, T K,j )], A = Ṡ 11 β 0, Λ 0 ) + Ṡ 21 β 0, Λ 0 )[h ] [ ] = E Λ 0Kj e βt 0 Z ZZ T e βt 0 Z Λ 0Kj R K, T K,j ) = E K,T K,Z) Λ 0Kj e βt 0 Z [Z R K, T K,j )] Z T = E K,T K,Z) Λ 0Kj e βt 0 Z [Z R K, T K,j )] 2, 28

29 and with B = Em β 0, Λ 0 ; X) 2 = E K,T K,Z) C j,j Z) [Z R K, T K,j )] [Z R K, T K,j )] T, C j,j Z) = E j,j =1 ) ) ] [N Kj e βt 0 Z Λ 0Kj N Kj e βt 0 Z Λ 0Kj Z, K, T K,j, T K,j. To verify A4, we note that the first part automatically holds, because S 1n ˆβ n, ˆΛ n ) = P n m 1 ˆβ n, ˆΛ n ; X) = 0 since ˆβ n satisfies the eudo-score equation. Next we shall show that ˆΛ S 2n ˆβ n, n )[h ] 5.6) = P n 1 { } T ˆΛ N Kj ˆΛ exp ˆβ n Z) Λ 0Kj R K, T K,j ) = o P n 1/2 ) with ˆΛ = ˆΛ n T K,j ). Since ˆβ n, ˆΛ n ) maximizes P n m θ X) over the feasible region, consider a path θ ɛ = ˆβ n, ˆΛ n + ɛh) for h F. Then d lim ɛ 0 dɛ P nm θ ɛ X) = P n 1 { } T ˆΛ N Kj ˆΛ exp ˆβ n Z) h Kj = 0. ˆΛ Now choose h Kj = EZ expβt 0 Z) K, T K,j)/Eexpβ0 T Z) K, T K,j). demonstrate 5.6), it suffices to show that I = P n 1 { } T ˆΛ N Kj ˆΛ exp ˆβ n Z) Λ 0Kj ˆΛ = o P n 1/2 ), )α Kj Then to where α Kj = EZ expβ T 0 Z) K, T K,j)/Eexpβ T 0 Z) K, T K,j). But I can be decomposed as I = I 1 I 2 + I 3, where N Kj I 1 = P n P ) ˆΛ Λ 0Kj ˆΛ 29 )α Kj,

30 and I 3 = P T I 2 = P n P ) exp ˆβ n Z)Λ 0Kj ˆΛ )α Kj 1 { } T ˆΛ N Kj ˆΛ exp ˆβ n Z) Λ 0Kj ˆΛ )α Kj. We show that I 1, I 2 and I 3 are all o P n 1/2 ). Let φ 1 X; Λ) = φ 2 X; β, Λ) = N Kj Λ Kj Λ 0Kj Λ Kj )α Kj, expβ T Z)Λ 0Kj Λ Kj )α Kj, and define two classes Φ 1 η) and Φ 2 η) as follows: and Φ 1 η) = { φ 1 : Λ F and Λ Λ 0 L2 µ 1 ) η } Φ 2 η) = {φ 2 : β, Λ) R F and d 1 β, Λ), β 0, Λ 0 )) η}. Using the same bracketing entropy arguments as used in deriving the rate of convergence, it follows that both Φ 1 η) and Φ 2 η) are P -Donsker classes under conditions C1, C6 and C8. Moreover, for the seminorm ρ P f) = { P f P f) 2} 1/2, under conditions C1, C6, C8 and C9, we have sup φ1 Φ 1 η) ρ P φ 1 ) 0 and sup φ2 Φ 2 η) ρ P φ 2 ) 0 if η 0. Due to the relationship between P -Donsker and asymptotic equicontinuity see Corollary of van der Vaart and Wellner 1996)), this yields I 1 = o P n 1/2 ) and I 2 = o P n 1/2 ). For I 3, we have I 3 = P = E = E { } N Kj ˆΛ exp ˆβ n Z) ˆΛ Λ 0Kj { Λ 0Kj expβ0 T T Z) ˆΛ exp ˆβ n Z) ˆΛ ˆΛ )α Kj } Λ 0Kj { Λ 0Kj ˆΛ )eβt 0 Z + ˆΛ eβt 0 Z e βt Z ) { } Cd 2 1 ˆβ n, ˆΛ n ), β 0, Λ 0 ), ˆΛ 30 } ˆΛ )α Kj Λ 0Kj ˆΛ )α Kj