Lifetime Data Anal DOI 10.1007/s10985-015-9325-0

Estimating the survival function based on the semi-Markov model for dependent censoring Ziqiang Zhao1 · Ming Zheng1 · Zhezhen Jin2

Received: 10 March 2014 / Accepted: 7 March 2015 © Springer Science+Business Media New York 2015

Abstract In this paper, we study a nonparametric maximum likelihood estimator (NPMLE) of the survival function based on a semi-Markov model under dependent censoring. We show that the NPMLE is asymptotically normal and achieves asymptotic nonparametric efficiency. We also provide a uniformly consistent estimator of the corresponding asymptotic covariance function based on an information operator. The finite-sample performance of the proposed NPMLE is examined with simulation studies, which show that the NPMLE has smaller mean squared error than the existing estimators and its corresponding pointwise confidence intervals have reasonable coverages. A real example is also presented. Keywords Semi-Markov model · Dependent censoring · NPMLE · Survival function

Electronic supplementary material The online version of this article (doi:10.1007/s10985-015-9325-0) contains supplementary material, which is available to authorized users.

B

Zhezhen Jin [email protected] Ziqiang Zhao [email protected] Ming Zheng [email protected]

1

Department of Statistics, School of Management, Fudan University, 670 Guoshun Road, Shanghai, China

2

Department of Biostatistics, Mailman School of Public Health, Columbia University, 722 West 168th Street, New York, NY, USA

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1 Introduction In survival analysis, the survival function of the failure time is commonly estimated by the Kaplan-Meier estimator and the Nelson-Aalen estimator. For these two estimators, a key assumption is that the censoring time and the survival time are independent. It is challenging to estimate the survival function under dependent censorship (Tsiatis 1975). In oncology studies, independent dropout yields independent censoring. In addition, subjects are often censored due to the ending of the study or to the onset of progressive disease (PD). The study-ending censoring is usually independent of the survival time while the PD-related censoring might be dependent on the survival time since the PD could be a precursor of both death and lost of follow-up. In other words, in the presence of PD, the patients have a much higher risk of death and are more likely to leave the study. Ignoring such dependence would yield biased and inconsistent estimation of the survival function. On the other hand, it is possible to improve the estimation if the dependency information is properly used. Datta et al. (2000) considered nonparametric estimation using a three-stage irreversible illness–death model. In the case of dependent censoring, Lee and Tsai (2005) proposed a semi-Markov model and developed an empirical-type estimator of the survival function. The asymptotic variance of their proposed estimator, however, is too complicated to compute. In this paper, we present a nonparametric maximum likelihood estimator (NPMLE) of the survival function based on the semi-Markov model. We show that our proposed NPMLE converges weakly to a Gaussian process and achieves asymptotic efficiency. In addition, we develop a consistent estimator of its asymptotic covariance function based on an information operator, which can be easily calculated. The remainder of the paper is organized as follows. In Sect. 2, we will introduce the semi-Markov model. In Sect. 3, we will derive the NPMLE of the survival function. In Sect. 4, we will establish the asymptotic properties of the NPMLE, and construct a consistent estimator of its asymptotic covariance function based on an information operator. In Sect. 5, we will present simulation studies and re-analysis of the example in Lee and Tsai (2005). We will conclude with a short discussion in Sect. 6 and provide sketches of the proofs of theorems in the Appendix.

2 The semi-Markov model Let T be the survival time and U be the PD censoring time. The semi-Markov model proposed by Lee and Tsai (2005) assumes that: (0,2)

λT|U (t|u) = λ0

(1,2)

(t) I {t  u} + λ0

(t − u) I {t > u} , for t, u  0,

(1)

where λT|U is the conditional hazard function of T given U, I {·} is the indicator function (0,2) (1,2) taking the value 1 if the condition is satisfied and the value 0 otherwise, λ0 and λ0 are unknown hazard functions for death without PD and death with PD respectively.

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Semi-Markov model for dependent censoring Fig. 1 Transition

Note that the cumulative form of Model (1) is: (0,2)

ΛT|U (t|u) = Λ0

(1,2)

(min {t, u}) + Λ0

and that for any t, u  0, ST|U (t|u) = (0,1)

(0,2)

(t − u) I {t > u} , for t, u  0,

⎧ ⎨ S(0,2) (t)

t  u,

⎩ S(0,2) (u) S(1,2) (t − u) 0 0

t > u,

0

(1,2)

(0,1)

(0,2)

(2)

(1,2)

and S0 , S0 , S0 are the corresponding cause where Λ0 , Λ0 , Λ0 specific cumulative hazard functions and the corresponding cause specific survival (0,1) (0,2) (1,2) functions for λ0 , λ0 and λ0 , respectively. The superscript values 0, 1, 2 indicate three different states, with 0 being the state of alive without PD, 1 being the state of alive with PD and 2 being the state of death. More precisely, Model (1) corresponds to a non-homogeneous semi-Markov process J with the state space {0, 1, 2}: ⎧ ⎨ 0 T > t, U > t (Alive without PD at time t) J (t) = 1 T > t, U  t (Alive with PD at time t), for t  0 . ⎩ 2Tt (Died at time t) (0,1)

denotes the hazard function of U, i.e. the hazard With this specification, if λ0 (0,1) (0,2) (1,2) function for PD, it can be shown that λ0 , λ0 and λ0 are respectively the cause specific hazard functions for the transition from state 0 to state 1, state 0 to state 2, and state 1 to state 2. Figure 1 provides a plot of transition for illustration. In addition, we use C to denote independent censoring, which is independent of (T, U). Let X = min {T, C} and Δ = I {T  C}. When X  U, the subject is dead or censored before PD, i.e., (X, Δ) is observed while U is not. When X > U, PD occurs before both death and the independent censoring C and so the subject would leave the study at time U with a probability θ0 , i.e., U is observed and (X, Δ) is observed with probability θ0 , where θ0 is an unknown parameter.

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Let V = min {X, U} , R = I {X  U} and ξ be the indicator of observing (X, Δ). The observed data for a subject is (V, R, ξ, ξ X, ξ Δ). Throughout the paper, it is assumed that ξ is conditionally independent of (T, U, C) given R with P (ξ = 1|R = 1) = 1 and P (ξ = 1|R = 0) = θ0 . For a sample of size n, we observe (Vi , Ri , ξi , ξi Xi , ξi Δi ) , i = 1, . . . , n.

3 Nonparametric maximum likelihood estimation In general, without additional information, likelihood based estimation is not available for dependent censoring problems. Using the semi-Markov model specification, it is possible to develop likelihood based estimation. Next, we present the likelihood for the unknown parameters: 

 Λ(0,1) , Λ(0,2) , Λ(1,2) , Λ(c) , θ , (c)

where Λ(c) is the parameter for Λ0 which is the true cumulative hazard function of C. Note that Λ(c) and θ are nuisance parameters. Suppose that Λ(0,1) , Λ(0,2) , Λ(1,2) and Λ(c) are differentiable with corresponding derivatives: λ(0,1) , λ(0,2) , λ(1,2) and λ(c) . Let S(0,1) , S(0,2) , S(1,2) and S(c) be the corresponding survival functions of Λ(0,1) , Λ(0,2) , Λ(1,2) and Λ(c) . With these notations,  (0,1) , Λ(0,2) , Λ(1,2) , Λ(c) , θ can be derived. the likelihood of Λ For subjects with ξi = 1, (Vi , Xi , Δi , Ri , ξi ) is observed and its corresponding likelihood can be obtained based on the joint distribution of (V, X, Δ, R, ξ ). Specifically, when ξi = 1, Ri = 1 and Δi = 1, the likelihood is: λ(0,2) (Vi ) S(0,2) (Vi ) S(c) (Vi ) S(0,1) (Vi ) ; When ξi = 1, Ri = 1 and Δi = 0, the likelihood is: λ(c) (Vi ) S(c) (Vi ) S(0,2) (Vi ) S(0,1) (Vi ) ; When ξi = 1, Ri = 0 and Δi = 1, the likelihood is: λ(1,2) (Xi − Vi ) S(1,2) (Xi − Vi ) S(c) (Xi ) λ(0,1) (Vi ) S(0,1) (Vi ) S(0,2) (Vi ) θ ; When ξi = 1, Ri = 0 and Δi = 0, the likelihood is: λ(c) (Xi ) S(c) (Xi ) S(1,2) (Xi − Vi ) λ(0,1) (Vi ) S(0,1) (Vi ) S(0,2) (Vi ) θ. For subjects with ξi = 0, (Vi , Ri , ξi ) is observed and its corresponding likelihood is: λ(0,1) (Vi ) S(0,1) (Vi ) S(0,2) (Vi ) S(c) (Vi ) (1 − θ ) .

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Hence, the likelihood based on the observed data (Vi , Ri , ξi , ξi Xi , ξi Δi ) , i = 1, . . . , n, is proportional to ⎡ n

(1−Ri ) ⎣ λ(0,1) (Vi ) i=1



⎤  1 − Λ(0,1) (dy) ⎦

y∈[0,Vi )

⎡

n

Δi Ri ⎣ λ(0,2) (Vi ) ×



⎤  1 − Λ(0,2) (dy) ⎦

y∈[0,Vi )

i=1

⎡

n

Δi ⎣ λ(1,2) (Xi − Vi ) ×



⎤ξi (1−Ri )   1 − Λ(1,2) (dy) ⎦ .

y∈[0,Xi −Vi )

i=1

Unfortunately, this function is unbounded from above and the usual maximum like(0,1) (0,2) (1,2) lihood estimator (MLE) does not exist when Λ0 , Λ0 and Λ0 are restricted (0,1) (0,2) and to continuous functions. With discretized extensions by allowing Λ0 , Λ0 (1,2) to be discontinuous, the NPMLE is well defined in the sense of Kiefer and Λ0 (0,1) (0,2) and Wolfowitz (1956) and Scholz (1980). Specifically, we assume that Λ0 , Λ0 (1,2) are cadlag, piecewise constant and right continuous with left limits. Λ0 To obtain the NPMLE, we rewrite the likelihood function with the discretized Λ(0,1) , Λ(0,2) and Λ(1,2) :     (0,2)   (1,2)   (0,1) Λ(0,1) Ln Λ(0,2) Ln Λ(1,2) , Ln Λ(0,1) , Λ(0,2) , Λ(1,2) = Ln where for any cumulative hazard function Λ, (0,1) Ln (Λ)

⎡ n   ⎣ Λ {Vi } (1−Ri ) = i=1

(0,2) Ln (Λ)

⎡



⎤ (1 − Λ (dy))⎦ ,

y∈[0,Vi )

n   ⎣ Λ {Vi } Δi Ri (1 − Λ {Vi })1−Δi Ri = i=1

i=1

(1 − Λ (dy))⎦ ,

y∈[0,Vi )

(1,2) Ln (Λ)

⎡ n   ⎣ Λ {Xi −Vi } Δi (1−Λ {Xi − Vi })1−Δi =

⎤





⎤ξi (1−Ri ) (1 − Λ (dy))⎦

,

y∈[0,Xi−Vi )

with Λ {t} = Λ (t) − Λ (t−) for all t  0. (0,1) (0,2) (c) does not share jump points with Λ0 and Λ0 , It is also assumed that Λ0 which means that the time of PD censoring occurrence is different from the time of death or the time of independent censoring. Then, for any cumulative hazard function Λ,

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(0,1)

log Ln



+∞

(Λ)  0

 0

= (0,2)

log Ln



0



(1,2)

log Ln

+∞



0

(dt) +



(0)

Yn (t+) log [1 − Λ {t}]

(5)



(0,2)

 (0) (0,2) ˆ {t} {t} + log 1 − Λ log Λˆ (0,2) N Y (dt) (t+) n n n n

(6)

  Λˆ (0,2) , n (1,2)

log [Λ {t}] Nn

(dt) +



(1)

Yn (t+) log [1 − Λ {t}]

(7)

t 0

+∞

 0

=

(0,2)

t 0

+∞



(4)

  Λˆ (0,1) , n

log [Λ {t}] Nn

(0,2) log Ln

(Λ) 

(3)

t 0

0

=

(0)

Yn (t+) log [1 − Λ {t}]

t 0

+∞







(0,1)

 (0) {t} Nn (dt) + {t} log Λˆ (0,1) Yn (t+) log 1 − Λˆ (0,1) n n

(0,1) log Ln

(Λ) 

(dt) +

t 0

+∞



(0,1)

log [Λ {t}] Nn



(1,2)

 (1) {t} Nn (dt) + {t} log Λˆ (1,2) Yn (t+) log 1 − Λˆ (1,2) n n

(1,2) log Ln

t 0

  Λˆ (1,2) , n

(8)

where for any t  0, (0)

n 

(0,1)

i=1 n 

Yn (t) = Nn

(1,2)

Nn

(t) = (t) =

i=1 n 

(1)

I {Vi  t} , Yn (t) =

(0,2)

(1 − Ri ) I {Vi  t} , Nn

Λˆ (0,2) (t) = n Λˆ n(1,2) (t) =



(t) =

n 

Δi Ri I {Vi  t} ,

i=1

ξi Δi (1 − Ri ) I {Xi − Vi  t} ,

 

ξi (1 − Ri ) I {Xi − Vi  t} ,

i=1

i=1

Λˆ n(0,1) (t) =

n 

[0,t]

[0,t]

[0,t]

  

(0)

−1  (0)  (0,1) I Yn (y) > 0 Nn (dy) ,

(0)

−1  (0)  (0,2) I Yn (y) > 0 Nn (dy) ,

(1)

−1  (1)  (1,2) I Yn (y) > 0 Nn (dy) .

Yn (y) Yn (y) Yn (y)

The equalities for (3), (5), and (7) hold if and only if Λ is a pure jump function. The inequalities in (4), (6), and (8) follow from the fact that for any λ ∈ (0, 1) and any α, β > 0,

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α log λ + β log (1 − λ)  α log (α/(α + β)) + β log (β/(α + β)) .   (0,1) (0,2) (1,2) is the NPMLE of According to Criteria (B) of Scholz (1980), Λˆ n , Λˆ n , Λˆ n     (0,1) (0,2) (1,2) (0,1) (0,2) (1,2) Λ , since Ln Λˆ n , Λˆ n , Λˆ n > 0. ,Λ ,Λ 0

0

0

Let S0 be the marginal survival function of T. Note that for any t  0, 

(0,1)

S0 (t) = P (T > t) = −

ST|U (t|u) S0 (du) R+  (0,1) (0,2) (0,2) (1,2) (0,1) = S0 (t) S0 (t) − S0 (u) S0 (t − u) S0 (du) . [0,t]

Correspondingly, for any t  0, define Sˆ n (t) = Sˆ n(0,1) (t) Sˆ (0,2) (t) − n

 [0,t]

Sˆ (0,2) (u) Sˆ n(1,2) (t − u) Sˆ n(0,1) (du) , n

where for any t  0, Sˆ n(0,1) (t) =

1 − Λˆ n(0,1) (dy) , Sˆ (0,2) 1 − Λˆ (0,2) (t) = (dy) , n n y∈[0,t]

y∈[0,t]

1 − Λˆ n(1,2) (dy) . Sˆ n(1,2) (t) = y∈[0,t]

  (0,1) (0,2) (1,2) is the NPMLE of By the invariance property, it follows that Sˆ n , Sˆ n , Sˆ n   (0,1) (0,2) (1,2) S0 , S0 , S0 and Sˆ n is the NPMLE of S0 .

4 Asymptotic results In this section, we present the asymptotic properties of the NPMLE. 4.1 Regularity conditions We list regularity conditions in this subsection. (0) (0,1) (0,2) (c) For any t  0, define L0 (t) = S0 (t) S0 (t) S0 (t) and (1)

(1,2)

L0 (t) = −θ0 S0

 (t)

R+

(0,2)

S0

(c)

(0,1)

(u) S0 (u + t) S0

(du) .

The regularity conditions are as follows: A.1 For a sample of size n, (Ti , Ui , Ci , Xi , Δi , Vi , Ri , ξi ) , i = 1, . . . , n are n independent copies of (T, U, C, X, Δ, V, R, ξ ).

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A.2 Model (2) holds with Λ0 (0) = 0. A.3 The censoring time C is independent of (T, U). A.4 The indicator ξ is conditionally independent of (T, U, C) given R, with P (ξ = 1|R = 1) = 1 and P (ξ = 1|R = 0) = θ0 ∈ (0, 1) . A.5 There exists τ > 0 such that L0(0) (τ −) > 0 and L0(1) (τ −) > 0. (0,1) (0,2) (0,1) (c) A.6 For any t ∈ [0, τ ] , Λ0 {t} Λ0 {t} = Λ0 {t} Λ0 {t} = 0. Assumption A.1 is commonly satisfied. Assumption A.2 is a technical condition for model specification. Assumption A.3 indicates that C is independent censoring. Assumption A.4 assumes that there is a subgroup of the potentially dependent censored (0) subjects whose (X, Δ) are observed. Assumption A.5 is equivalent to L0 (0) > 0 and (1) L0 (0) > 0. The required τ is generally not unique and could be chosen as large as possible. Assumption A.6 is a mild technical condition which is weaker than continuity. 4.2 Asymptotic normality In this subsection, we establish the asymptotic normality and the asymptotic efficiency of the NPMLE estimators. The asymptotic normality is established by convergence to a tight Gaussian process in the space of uniformly bounded funcˆ element in ∞ ([0, τ ]), and tions on [0, τ ]. Specifically,  Sn is viewed as a random   (0,1) (0,2) (1,2) (0,1) (0,2) (1,2) and Sˆ n , Sˆ n , Sˆ n are viewed as random elements in Λˆ n , Λˆ n , Λˆ n ∞ ∞ ∞ ∞

∞ 3 ([0, τ ]) = ([0, τ ]) × ([0, τ ]) × ([0, τ ]), where ([0, τ ]) denotes the space of all uniformly bounded real valued functions defined on [0, τ ] equipped with the uniform norm. The asymptotic efficiency is shown by   convolution theorem. (0,1) (0,2) (1,2) . The first theorem gives the asymptotic normality for Λˆ n , Λˆ n , Λˆ n

Theorem 1 Under assumptions A.1–A.6,   (0,1) (0,2) ˆ (1,2) (1,2) n 1/2 Λˆ n(0,1) − Λ0 , Λˆ (0,2) − Λ , Λ − Λ n n 0 0 weakly converges to a tight zero-mean Gaussian process in ∞ 3 ([0, τ ]) with covariance function U0 , as n → ∞, where for any t, s ∈ [0, τ ],   (0,1) (0,2) (1,2) U0 (t, s) = diag U0 (t, s) , U0 (t, s) , U0 (t, s) , in which, for any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)} and any t, s ∈ [0, τ ],    −1  (i, j) (i, j) (i, j) (i) L0 (y−) 1 − Λ0 {y} Λ0 (dy) . U0 (t, s) = [0,min{t,s}]

  (0,1) (0,2) (1,2) . The second theorem gives the asymptotic normality for Sˆ n , Sˆ n , Sˆ n

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Theorem 2 Under assumptions A.1–A.6,   (0,1) (0,2) (1,2) − S0 , Sˆ n(1,2) − S0 n 1/2 Sˆ n(0,1) − S0 , Sˆ (0,2) n weakly converges to a tight zero-mean Gaussian process in ∞ 3 ([0, τ ]) with covariance function V0 , as n → ∞, where for any t, s ∈ [0, τ ],   (0,1) (0,2) (1,2) V0 (t, s) = diag V0 (t, s) , V0 (t, s) , V0 (t, s) , in which, for any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)} and any t, s ∈ [0, τ ], (i, j)

(i, j)

V0

(t, s) = S0  (i, j) W0 (t, s) =

(i, j)

(i, j)

(t) S0 (s) W0 (t, s), −1  −1  (i) (i, j) (i, j) L0 (y−) 1 − Λ0 {y} Λ0 (dy) .

[0,min{t,s}]

The next theorem establishes the asymptotic normality for Sˆn . Theorem 3 Under assumptions A.1–A.6, n 1/2 (Sˆ n − S0 ) weakly converges to a tight zero-mean Gaussian process in ∞ ([0, τ ]) with covariance function Ω0 , as n → ∞, where for any t, s ∈ [0, τ ], (0,1)

Ω0 (t, s) = Ω0  (0,1) Ω0 (t, s) =

(0,2)

(1,2)

(t, s) + Ω0 (t, s) + Ω0 (t, s) ,  (0,1) (0,2) (0,2) V0 (x−, y−) S0 (dy) S0 (d x)

(0,t] (0,s] (0,2)

− S0





(0) 

(0,2)

(0,1)

(0,t] (0,s]



V0

(1,2)

(x−, s − y) S0

(0,1)

(0,2)

(dy) S0

(0,2)

(d x)

(1,2)

− S0 V0 (0) (t − x, y−) S0 (dy) S0 (d x) (0,t] (0,s]  

2 (0,2) (0,1) (1,2) (1,2) + S0 V0 (0) (t − x, s − y) S0 (dy) S0 (d x), (0,2)

Ω0

(0,1)

(t, s) = S0

(0,1)

− S0

(0,1)

− S0

(1,2)

Ω0

 (t, s) =

(t)



(0,t] (0,s] (0,2)

(s) V0

[0,s]

 (s) 

 +

(0,1)

(t) S0

[0,t]



[0,t] [0,s]

(0,2)

(t, y) S0

(0,2)

(x, s) S0

V0

V0

(0,2)

[0,t] [0,s]

V0

(1,2)

V0

(t, s) (1,2)

(s − y) S0

(1,2)

(t − x) S0

(1,2)

(x, y) S0

(0,1)

(dy)

(0,1)

(d x)

(1,2)

(t − x) S0 (0,2)

(t − x, s − y) S0

(0,2)

(x) S0

(0,1)

(s − y) S0

(0,1)

(y) S0

(0,1)

(dy) S0 (0,1)

(dy) S0

(d x) ,

(d x) .

(0,1) (0,1) (0,2) (0,2) (1,2) (1,2) It is worth noting that, although Λˆ n − Λ0 , Λˆ n − Λ0 and Λˆ n − Λ0 themselves are martingales, they do not form a joint martingale, since no common σ field is available. Hence, the martingale limit theory cannot be directly used here. The proof of the above theorems are based on empirical process theory and are provided in the Appendix.

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4.3 Asymptotic efficiency In this subsection, we turn to the asymptotic nonparametric efficiency, which is formally defined by the convolution theorem (Theorem VIII.3.1 of Andersen et al. 1993). We explicitly specify the Hilbert space required by the convolution theorem. Define H = H(0,1) × H(0,2) × H(1,2) , where for (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)},   H(i, j) = h : [0,τ ]

  −1 (i, j) (i, j) (i) Λ0 (dy) < +∞ . [h (y)]2 L0 (y−) 1 − Λ0 {y}

    For any g = g (0,1) , g (0,2) , g (1,2) , h = h (0,1) , h (0,2) , h (1,2) ∈ H, define g, hH =

 (i, j) [0,τ ]

−1  (i, j) (i, j) (i) g (i, j) (y) h (i, j) (y) L0 (y−) 1 − Λ0 {y} Λ0 (dy) ,

where the summation is taken over {(0, 1) , (0, 2) , (1, 2)}. For any h ∈ H, define 1/2 hH = h, hH . It can be shown that H is a Hilbert space with the inner product ·, ·H and the norm ·H .   For any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)} and any h = h (0,1) , h (0,2) , h (1,2) ∈ H, define    (i, j) (i, j) 1 + n −1/2 h (i, j) (y) Λ0 (dy) , Λn,h (t) = [0,t]

    (0,1) (0,2) (1,2) (0,1) (0,2) (1,2) is: the likelihood ratio of Λn,h , Λn,h , Λn,h to Λ0 , Λ0 , Λ0 Rn (h) = Rn(0,1) (h) R(0,2) (h) Rn(1,2) (h) , n   where for any h = h (0,1) , h (0,2) , h (1,2) ∈ H, log Rn(0,1) (h) ⎡

=

n 

(1 − Ri ) log 1 + n −1/2 h (0,1) (Vi )

i=1

⎤   (0,1) (0,1) ⎣log 1 − Λn,h (dy) − log 1 − Λ0 (dy) ⎦ , + i=1 y∈[0,Vi ) y∈[0,Vi ) n

 Δi Ri log 1 + n −1/2 h (0,2) (Vi ) log R(0,2) (h) = n n 





i=1

+

n  i=1

    (0,2) (0,2) (1 − Δi Ri ) log 1 − Λn,h {Vi } − log 1 − Λ0 {Vi }

⎤     (0,2) (0,2) ⎣log 1 − Λn,h (dy) − log 1 − Λ0 (dy) ⎦ , + i=1 y∈[0,Vi ) y∈[0,Vi ) n 

123

⎡

Semi-Markov model for dependent censoring

log Rn(1,2) (h) =

n 



ξi (1 − Ri ) Δi log 1 + n −1/2 h (1,2) (Xi − Vi )

i=1

+ −

n  i=1 n 

  (1,2) ξi (1 − Ri ) (1 − Δi ) log 1 − Λn,h {Xi − Vi }   (1,2) ξi (1 − Ri ) (1 − Δi ) log 1 − Λ0 {Xi − Vi }

i=1

+

n 

−

  (1,2) 1 − Λn,h (dy)

y∈[0,Xi −Vi )

i=1 n 



ξi (1 − Ri ) log



ξi (1 − Ri ) log

  (1,2) 1 − Λ0 (dy) .

y∈[0,Xi −Vi )

i=1

Theorem 4 Under assumptions A.1–A.6, for any m > 0, h 1 , . . . , h m ∈ H, (log Rn (h 1 ) , . . . , log Rn (h m )) weakly converges to a Gaussian random vector in Rm with mean −

 1 h 1 2H , . . . , h m 2H 2

and covariance matrix ⎛

h 1 , h 1 H ⎜ .. ⎝ .

h m , h 1 H

... .. . ···

⎞ h 1 , h m H ⎟ .. ⎠. .

h m , h m H

Hence, the information operator I0 : H × H → R is: for any g, h ∈ H, I0 (g, h) = g, hH .   For any h = h (0,1) , h (0,2) , h (1,2) ∈ H, define 



κˆ n (h) = [0,τ ]

 (0,2) (0,2) (1,2) (1,2) ˆ ˆ Λ Λ . h (0,1) (y) Λˆ (0,1) (dy)+h (y) (dy)+h (y) (dy) n n n

Note that for any g, h ∈ H, the asymptotic covariance of κˆ n (g) and κˆ n (h) is  (i, j) [0,τ ]

 −1 (i, j) (i, j) (i) g (i, j) (y) h (i, j) (y) L0 (y−) 1 − Λ0 {y} Λ0 (dy)

(9)

123

Z. Zhao et al.

and can be interpreted as the “inverse” of the information operator I0 , where the over {(0, 1) , (0, 2) , (1, 2)}. The efficiency result for   summation is taken (0,1) ˆ (0,2) ˆ (1,2) ˆ and Sˆ n is stated in the following theorem. Λn , Λn , Λn   (0,1) (0,2) (1,2) and Sˆ n are efficient. Theorem 5 Under assumptions A.1–A.6, Λˆ n , Λˆ n , Λˆ n The proof of the above theorems are given in the Appendix. 4.4 Asymptotic covariance function estimation To carry out statistical inference, it is necessary to estimate the asymptotic covariance function. In this subsection, we provide uniform consistent estimators for the asymptotic covariance functions.     For any g = g (0,1) , g (0,2) , g (1,2) , h = h (0,1) , h (0,2) , h (1,2) ∈ H, define Iˆ n (g, h) =

 (i, j) [0,τ ]

−1 (i, j)  (i, j) g (i, j) (y) h (i, j) (y) 1 − Λˆ n {y} Nn (dy)

where the summation  is taken over {(0,  1) , (0,2) , (1, 2)}.  For any g = g (0,1) , g (0,2) , g (1,2) , h = h (0,1) , h (0,2) , h (1,2) ∈ H, it can be shown that Iˆ n (g, h) is the negative second order directional derivative of log Ln at    (0,1) (0,2) (1,2) along the direction Gn , Hn , where Λˆ n , Λˆ n , Λˆ n  (1,2) ˆ , Gn = g g d Λn g     Hn = , h (1,2) d Λˆ n(1,2) . h (0,1) d Λˆ n(0,1) , h (0,2) d Λˆ (0,2) n 

(0,1)

d Λˆ n(0,1) ,



(0,2)

d Λˆ n(0,2) ,



(1,2)

Since the function Ln plays the role of the likelihood, Iˆ n should be a reasonable estimator for I0 . The “inverse” operation in (9) leads to an estimator for the asymptotic covariance function of κˆ n :  (i, j) [0,τ ]

−1 (0)  (i, j) (i, j) g (i, j) (y) h (i, j) (y) 1 − Λˆ n {y} Yn (y) Λˆ n (dy)

    for all g = g (0,1) , g (0,2) , g (1,2) , h = h (0,1) , h (0,2) , h (1,2) ∈ H, where the summation is taken over {(0, 1) , (0, 2) , (1, 2)}. Hence, the estimators for U0 and V0 are given as follows: for any t, s ∈ [0, τ ],   Uˆn (t, s) = diag Uˆn(0,1) (t, s) , Uˆn(0,2) (t, s) , Uˆn(1,2) (t, s) ,   Vˆ n (t, s) = diag Vˆ n(0,1) (t, s) , Vˆ n(0,2) (t, s) , Vˆ n(1,2) (t, s) ,

123

Semi-Markov model for dependent censoring

where for any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)} and any t, s ∈ [0, τ ], (i, j) (i, j) (i, j) ˆ n(i, j) (t, s) , Vˆ n (t, s) = Sˆ n (t) Sˆ n (s) W  −1   (i)  (i, j) (i, j) (i, j) Λˆ n (dy) , 1 − Λˆ n {y} Yn (y) Uˆn (t, s) = n [0,min{t,s}]  −1  −1  (i) (i, j) (i, j) ˆ n(i, j) (t, s) = n Λˆ n (dy) . 1 − Λˆ n {y} W Yn (y) [0,min{t,s}]

The estimator for Ω0 is given as Ωˆ n : for any t, s ∈ [0, τ ], Ωˆ n (t, s) = Ωˆ n(0,1) (t, s) + Ωˆ n(0,2) (t, s) + Ωˆ n(1,2) (t, s) , where for any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)} and any t, s ∈ [0, τ ], Ωˆ n(0,1) (t, s) =





Vˆ n(0,1) (x−, y−) Sˆ (0,2) (dy) Sˆ (0,2) (d x) n n   − Sˆ (0,2) Vˆ n(0,1) (x−, s − y) Sˆ (1,2) (0) (dy) Sˆ (0,2) (d x) n n n (0,t] (0,s]   ˆ (1,2) (d x) − Sˆ (0,2) Vˆ n(0,1) (t − x, y−) Sˆ (0,2) (0) (dy) S n n n (0,t] (0,s] 

2  + Sˆ (0,2) Vˆ n(0,1) (t − x, s − y) Sˆ (1,2) (0) (dy) Sˆ (1,2) (d x) , n n n (0,t] (0,s]

(0,t] (0,s]

Ωˆ n(0,2) (t, s) = Sˆ (0,1) (t) Sˆ (0,1) (s) Vˆ n(0,2) (t, s) n n  Vˆ n(0,2) (t, y) Sˆ (1,2) − Sˆ (0,1) (t) (s − y) Sˆ (0,1) (dy) n n n [0,s]  ˆ (0,1) (d x) − Sˆ (0,1) Vˆ n(0,2) (x, s) Sˆ (1,2) (s) (t − x) S n n n [0,t]   + Vˆ n(0,2) (x, y) Sˆ (1,2) (t − x) Sˆ (1,2) (s − y) Sˆ (0,1) (dy) Sˆ (0,1) (d x) , n n n n [0,t] [0,s]

Ωˆ n(1,2) (t, s) =





[0,t] [0,s]

ˆ (0,2) (x) Sˆ (0,2) (y) Sˆ (0,1) (dy) Vˆ n(1,2) (t − x, s − y) S n n n

× Sˆ n(0,1) (d x) . The following theorem states the uniform consistency for Uˆn , Vˆ n and Ωˆ n . Theorem 6 Under assumptions A.1–A.6, as n → ∞, sup

Uˆn (t, s) − U0 (t, s) , sup

t,s∈[0,τ ]

sup

Vˆ n (t, s) − V0 (t, s) ,

t,s∈[0,τ ]

Ωˆ n (t, s) − Ω0 (t, s)

t,s∈[0,τ ]

converge to zero in probability. The proof of the theorem is given in the Appendix.

123

Z. Zhao et al.

Based on the asymptotic theory and the estimation of the asymptotic covariance function, various types of inference can be conducted. For example, for any fixed t0 ∈ [0, τ ], a 95 % pointwise confidence interval can be constructed as: 1/2 Sˆ n (t0 ) ± 1.96Ωˆ n (t0 , t0 ) .

5 Numerical studies 5.1 Simulation studies Four sets of simulation studies were carried out to examine the finite-sample performance of the proposed NPMLE. In the first and the third sets of the simulation studies, the independent censoring time C was specified as +∞. In the second and the fourth sets, the independent censoring time C was generated from a uniform distribution such that P (T  C) = 0.15. In the first and the second sets of the simulation studies, the PD censoring time U was generated from Uniform (0, η), while in the third and the fourth sets, the PD censoring time U was generated from the exponential distribution with hazard rate η. In all simulation studies, the survival time T was generated as  (1) if T(1)  U T , T= (2) U+T if T(1) > U where T(1) was generated from the exponential distribution with hazard rate 1 and T(2) was generated from the exponential distribution with hazard rate λ. The parameters were set to be λ ∈ {1, 2} , P (U < T) ∈ {0.3, 0.7} and θ0 ∈ {0.3, 0.6, 1}. For each scenario, we generated 1000 samples of size n = 100. The simulation results were summarized in Tables 1, 2, 3, 4, 5, 6, 7 and 8, which report the empirical bias (Bias), the empirical standard deviation (Std), the square root of the empirical mean squared error (sqrt[MSE]) and the average estimated standard deviation (AES) of the proposed estimator, as well as the empirical coverage rate (CR) of the corresponding 95 % pointwise confidence interval, at the 0.3th, 0.5th and 0.7th quantiles of T. The values of Bias, Std, sqrt(MSE) and AES were multiplied by 10,000. For comparison, the results from the Kaplan–Meier approach were also included in the tables. In addition, the empirical bias, the empirical standard deviation and the square root of the empirical mean squared error of Lee and Tsai’s estimator were included in Tables 1, 2, 5 and 6. Because there is no valid estimator of the variance of Lee and Tsai’s estimator, the AES and the CR are not reported for the Lee and Tsai’s estimator. The Lee and Tsai’s estimator was not included in Tables 3, 4, 7 and 8, because it does not incorporate the additional independent censoring. For the cases with C = +∞, in which the Lee and Tsai’s approach is applicable, the proposed NPMLE and the Lee and Tsai’s estimator were nearly unbiased and consistent. The NPMLE had smaller sqrt(MSE) than the Lee and Tsai’s estimator for all three quantiles, the improvement ranged from 0.6 to 7.9 %. The coverage of the 95 % pointwise confidence interval for the proposed NPMLE were close to the nominal level.

123

1.0

1.0

2

2

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

τ

473.924

−11.000

449.504

440.306

−9.700

9.900

523.249

457.866

−7.299

478.814

480.614

1.588

507.639

−3.479

−11.359

−16.654

467.485

507.468

457.686

−8.600

−9.100

9.379

456.914

472.164

505.172

477.190

493.339

514.706

449.049

19.700

29.273

11.651

11.145

15.460

20.922

16.322

449.613

474.051

440.413

479.104

523.251

457.924

480.749

507.651

467.579

457.777

507.541

457.339

473.071

505.306

477.320

493.581

515.131

449.346

9.900

−11.000

−9.700

133.208

103.364

36.817

282.399

183.352

89.851

−9.100

−8.600

19.700

30.350

9.105

9.971

4.205

19.302

17.504

449.504

473.924

440.306

504.190

533.389

463.476

502.682

510.943

471.065

457.686

507.468

456.914

486.644

517.079

487.063

483.332

519.672

452.963

Std

449.613

474.051

440.413

521.490

543.312

464.936

576.575

542.845

479.558

457.777

507.541

457.339

487.589

517.160

487.165

483.350

520.030

453.301

sqrt(MSE)

Bias

sqrt(MSE)

Bias

Std

Kaplan–Meier estimator

Lee and Tsai’s estimator

456.035

497.740

456.149

481.332

508.829

459.002

505.049

518.393

460.533

455.098

497.407

454.647

479.243

510.235

460.309

497.773

519.932

464.366

AES

0.960

0.949

0.956

0.935

0.931

0.940

0.927

0.932

0.935

0.948

0.947

0.949

0.935

0.945

0.932

0.954

0.952

0.946

CR

9.806

−8.285

−10.120

−15.314

8.289

−5.320

−12.246

−2.961

12.260

−10.352

−9.648

17.716

25.349

10.361

9.911

20.245

18.855

15.095

Bias

NPMLE

430.754

460.149

426.256

464.329

507.991

448.094

477.239

502.958

464.082

437.898

481.701

444.797

461.293

497.356

473.493

487.241

507.437

444.900

Std

430.865

460.224

426.376

464.582

508.059

448.126

477.396

502.966

464.244

438.020

481.798

445.150

461.989

497.464

473.596

487.662

507.788

445.156

sqrt(MSE)

476.470

485.975

443.491

488.198

493.961

446.429

518.828

512.860

453.541

446.506

480.689

444.825

467.520

489.322

447.447

511.333

508.858

453.465

AES

0.970

0.957

0.952

0.961

0.943

0.943

0.965

0.946

0.944

0.951

0.953

0.940

0.946

0.942

0.928

0.966

0.950

0.948

CR

The AES and the CR are not reported for the Lee and Tsai’s estimator, since the corresponding variance estimator has not been developed. The values of Bias, Std, sqrt(MSE) and AES have been multiplied by 10,000 Bias the empirical bias, Std the empirical standard deviation, sqrt(MSE) the square root of the empirical mean squared error, AES the average estimated standard deviation, CR the empirical coverage rate of the 95 % confidence interval

1.0

2

0.6

2

0.6

0.3

2

0.6

0.3

2

2

0.3

2

2

1.0

1.0

1

1

0.6

1.0

1

1

0.6

0.6

1

1

0.3

0.3

1

0.3

1

1

θ

λ

Table 1 Simulation results when C = +∞ and U is generated from the uniform distribution with P (T > U) = 0.3

Semi-Markov model for dependent censoring

123

123

1.0

1.0

2

2

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

τ

531.379

−5.362

511.517

457.627

483.329

445.847

−6.385

−8.200

−23.800

−14.800

527.470

460.885

645.529

11.827

7.979

1.222

602.643

495.129

−6.200

12.071

489.127

448.980

−2.100

467.809

−5.090

1.900

525.029

5.565

442.553

−2.828

4.242

658.791

716.866

−31.311

446.093

483.915

457.700

511.557

527.603

460.954

645.530

602.763

495.158

449.023

489.131

467.813

531.404

525.059

442.574

716.871

659.534

499.565

−14.800

−23.800

−8.200

193.050

247.354

138.658

466.958

513.093

245.676

−6.200

−2.100

1.900

−6.285

3.813

1.272

0.973

−33.695

−0.496

499.437

445.847

483.329

457.627

566.862

563.289

474.635

739.829

595.947

499.341

448.980

489.127

467.809

560.956

558.605

469.836

760.752

675.060

520.377

446.093

483.915

457.700

598.833

615.206

494.474

874.869

786.395

556.506

449.023

489.131

467.813

560.991

558.618

469.838

760.753

675.901

520.377

sqrt(MSE)

−11.337

Std

Bias

Std

Bias

sqrt(MSE)

Kaplan–Meier estimator

Lee and Tsai’s estimator

454.984

497.645

455.865

559.222

546.354

470.687

721.418

593.462

483.087

455.336

497.592

455.301

556.616

562.140

480.906

728.592

644.222

503.210

AES

0.960

0.956

0.953

0.938

0.914

0.931

0.881

0.851

0.895

0.950

0.945

0.941

0.946

0.950

0.946

0.918

0.925

0.930

CR

−18.720

−24.953

−15.557

−1.412

11.581

16.016

6.178

8.185

−5.022

−6.642

−5.547

392.816

436.142

409.854

485.327

494.984

438.841

634.232

589.272

479.782

414.769

447.852

510.060 430.119

−5.814

496.985

418.693

710.683

647.774

492.276

Std

−9.390

4.559

2.315

−3.802

−30.850

−14.576

Bias

NPMLE

393.261

436.856

410.149

485.329

495.120

439.133

634.262

589.329

479.808

414.822

447.886

430.221

510.093

497.006

418.699

710.694

648.508

492.492

sqrt(MSE)

406.233

449.685

413.332

479.425

491.471

428.777

623.757

581.017

467.545

427.758

454.722

420.918

521.131

511.491

437.047

697.288

631.552

474.698

AES

0.951

0.958

0.956

0.944

0.950

0.943

0.932

0.935

0.945

0.956

0.950

0.932

0.951

0.952

0.959

0.920

0.939

0.932

CR

The AES and the CR are not reported for the Lee and Tsai’s estimator, since the corresponding variance estimator has not been developed. The values of Bias, Std, sqrt(MSE) and AES have been multiplied by 10,000 Bias the empirical bias, Std the empirical standard deviation, sqrt(MSE) the square root of the empirical mean squared error, AES the average estimated standard deviation, CR the empirical coverage rate of the 95 % confidence interval

0.6

1.0

2

2

0.6

0.6

2

2

0.3

0.3

2

0.3

1.0

1

2

1.0

1

2

0.6

1.0

1

1

0.6

0.3

1

0.6

0.3

1

1

0.3

1

1

θ

λ

Table 2 Simulation results when C = +∞ and U is generated from the uniform distribution with P (T > U) = 0.7

Z. Zhao et al.

0.6

1.0

1.0

1.0

2

2

2

2

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

τ AES

CR

8.600

8.000

444.391

498.583

481.221

473.568

−4.200

533.052

466.143

520.929

519.638

466.208

168.701

124.852

54.006

285.552

186.877

73.204

517.326

457.368

−1.200

−9.700

477.638

456.393

−4.198

5.800

457.719

521.312

−55.071

−27.605

518.809

480.116

−8.948

−24.535

475.857

3.157

444.474

498.647

473.587

509.935

547.478

469.262

594.060

552.220

471.920

457.470

517.327

456.430

477.657

522.042

461.020

480.742

518.886

475.868

456.052

497.498

455.499

483.245

509.020

458.288

504.759

518.041

461.281

455.068

497.304

455.269

477.467

509.740

463.386

497.576

520.695

465.098

0.961

0.948

0.945

0.939

0.923

0.947

0.911

0.932

0.932

0.949

0.938

0.957

0.952

0.944

0.952

0.949

0.949

0.943

5.087

4.254

−4.680

16.348

18.940

11.512

9.750

423.137

478.715

459.618

447.174

508.319

451.738

494.252

508.992

461.198

−3.787 5.280

439.057

504.409

450.296

444.951

496.274

444.683

488.030

515.592

464.227

Std

−4.476

−0.957

2.850

−6.434

−23.875

−53.657

−26.005

−12.346

1.603

Bias

sqrt(MSE)

Bias

Std

NPMLE

Kaplan–Meier estimator

423.167

478.734

459.642

447.473

508.671

451.884

494.348

509.020

461.213

439.080

504.410

450.305

444.998

496.848

447.908

488.722

515.740

464.229

sqrt(MSE)

476.635

485.907

442.950

489.082

494.004

445.745

519.513

513.616

454.278

446.865

480.853

445.537

466.468

489.593

450.794

512.812

510.107

454.039

AES

0.974

0.955

0.934

0.963

0.936

0.937

0.964

0.955

0.936

0.947

0.945

0.951

0.960

0.939

0.943

0.961

0.952

0.944

CR

The Lee and Tsai’s estimator is not included, since the additional independent censoring is not incorporated in this approach. The values of Bias, Std, sqrt(MSE) and AES have been multiplied by 10,000 Bias the empirical bias, Std the empirical standard deviation, sqrt(MSE) the square root of the empirical mean squared error, AES the average estimated standard deviation, CR the empirical coverage rate of the 95 % confidence interval

0.6

0.6

2

2

0.3

0.3

2

0.3

2

2

1.0

1.0

1

1

0.6

1.0

1

1

0.6

0.6

1

1

0.3

0.3

1

0.3

1

1

θ

λ

Table 3 Simulation results when P (T > C) = 0.15 and U is generated from the uniform distribution with P (T > U) = 0.3

Semi-Markov model for dependent censoring

123

123

1.0

2

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

τ

500.114

593.135

453.780

514.198

465.653

0.100

−20.600

561.836

555.667

453.296

739.105

−18.300

190.462

217.812

97.707

420.009

468.726

510.455

468.413

−15.700

−12.200

268.558

556.991

475.711

21.931

−5.000

492.202

557.711

−1.934

−1.948

674.780

466.108

514.523

453.780

593.242

596.831

463.707

850.108

775.837

540.211

468.572

510.696

475.738

557.423

557.714

492.205

749.633

642.816

454.485

497.333

455.556

559.105

546.970

473.168

722.839

594.952

482.080

454.838

497.374

455.495

556.910

561.540

480.297

731.977

0.943

0.936

0.957

0.942

0.916

0.945

0.900

0.851

0.889

0.946

0.938

0.943

0.947

0.950

0.941

0.936

0.938

0.952

483.365 482.644

−7.811

−19.324

−7.882

415.083

450.739

407.523

413.581

−27.117 −17.538 1.700

587.409 641.460

−1.386 −13.806

466.171

433.071

461.219

432.805

510.031

501.484

440.665

714.376

649.068

491.383

Std

9.309

−21.023

−12.767

−5.471

21.264

11.016

−1.255

21.350

4.681

−7.032

674.774

748.811

35.088

504.172

−2.953

509.053

508.927

−11.321

CR

Bias

AES

Std

Bias

sqrt(MSE)

NPMLE

Kaplan–Meier estimator

415.533

450.807

407.527

482.708

483.683

414.469

641.608

587.410

466.264

433.581

461.395

432.840

510.474

501.605

440.666

714.695

649.085

491.433

sqrt(MSE)

405.605

449.654

412.883

479.955

492.237

430.981

621.044

581.482

466.745

427.045

454.286

420.981

521.362

511.521

437.101

698.724

629.126

473.164

AES

0.937

0.938

0.948

0.937

0.950

0.960

0.924

0.945

0.940

0.938

0.945

0.937

0.935

0.945

0.948

0.933

0.930

0.938

CR

The Lee and Tsai’s estimator is not included, since the additional independent censoring is not incorporated in this approach. The values of Bias, Std, sqrt(MSE) and AES have been multiplied by 10,000 Bias the empirical bias, Std the empirical standard deviation, sqrt(MSE) the square root of the empirical mean squared error, AES the average estimated standard deviation, CR the empirical coverage rate of the 95 % confidence interval

1.0

0.6

2

1.0

0.6

2

0.6

2

2

2

0.3

0.3

2

0.3

2

2

1.0

1.0

1

1

0.6

1.0

1

1

0.6

0.6

1

1

0.3

0.3

1

0.3

1

1

θ

λ

Table 4 Simulation results when P (T > C) = 0.15 and U is generated from the uniform distribution with P (T > U) = 0.7

Z. Zhao et al.

1.0

1.0

2

2

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

τ

5.600

485.494

452.751

517.350

−22.500

1.400

472.414

457.667

503.876

−8.363

−0.668

454.920

−12.466

−22.600

514.297

496.163

2.862

440.773

453.594

−1.447

18.762

6.200

456.154

475.155

19.800

−0.230

464.160

495.721

14.251

21.891

522.233

513.833

4.645

456.820

−2.288

10.296

452.753

517.839

458.224

472.414

503.945

455.091

496.165

514.305

441.172

453.636

485.526

456.583

475.155

495.925

464.676

513.838

522.254

456.936

1.400

−22.500

−22.600

167.725

113.093

43.213

318.350

222.344

116.418

6.200

5.600

19.800

0.286

14.596

19.899

−2.456

10.906

11.247

452.751

517.350

457.667

498.620

513.398

460.479

506.289

511.420

447.944

453.594

485.494

456.154

482.683

503.023

470.593

503.665

527.338

469.847

Std

452.753

517.839

458.224

526.074

525.707

462.502

598.059

557.662

462.825

453.636

485.526

456.583

482.683

503.235

471.014

503.671

527.451

469.981

sqrt(MSE)

Bias

sqrt(MSE)

Bias

Std

Kaplan–Meier estimator

Lee and Tsai’s estimator

455.626

497.298

456.493

486.202

512.080

460.518

512.735

523.404

462.360

455.838

497.628

454.665

481.632

513.288

461.643

505.226

526.662

468.018

AES

0.951

0.941

0.948

0.943

0.943

0.934

0.915

0.934

0.938

0.956

0.947

0.958

0.944

0.951

0.940

0.943

0.955

0.941

CR

489.006 435.387

−2.116

440.063

458.369

488.754

438.648

489.412

508.523

433.321

429.566

469.216

446.730

464.456

489.975

457.762

510.711

517.105

450.538

Std

−16.422

−18.267

−1.017

−10.435

−15.996

−3.142

2.159

19.215

7.415

4.393

19.156

1.168

12.513

20.407

−3.613

3.852

10.442

Bias

NPMLE

435.392

489.282

440.442

458.371

488.865

438.939

489.422

508.527

433.747

429.630

469.236

447.141

464.457

490.135

458.216

510.724

517.120

450.659

sqrt(MSE)

476.523

484.956

441.278

491.470

496.818

446.454

526.006

522.428

456.426

448.339

479.710

442.640

473.894

492.027

446.718

529.396

521.405

456.574

AES

0.955

0.944

0.939

0.971

0.955

0.944

0.961

0.950

0.953

0.963

0.944

0.952

0.954

0.947

0.942

0.952

0.953

0.944

CR

The AES and the CR are not reported for the Lee and Tsai’s estimator, since the corresponding variance estimator has not been developed. The values of Bias, Std, sqrt(MSE) and AES have been multiplied by 10,000 Bias the empirical bias, Std the empirical standard deviation, sqrt(MSE) the square root of the empirical mean squared error, AES the average estimated standard deviation, CR the empirical coverage rate of the 95 % confidence interval

0.6

1.0

0.6

2

2

0.6

2

2

0.3

0.3

2

0.3

2

2

1.0

1.0

1

1

0.6

1.0

1

1

0.6

0.6

1

1

0.3

0.3

1

0.3

1

1

θ

λ

Table 5 Simulation results when C = +∞ and U is generated from the exponential distribution with P (T > U) = 0.3

Semi-Markov model for dependent censoring

123

123

1.0

0.3

0.3

0.3

1

2

2

2

1.0

1.0

2

2

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

τ

24.438

685.975

443.302

523.877

618.911

622.933

447.090

−13.400

−24.212

−24.450

−14.467

−5.620

6.100

31.400

21.600

457.320

501.483

469.163

498.036

484.336

−11.100

−3.514

450.740

−17.500

511.504

526.254

−17.088

0.120

458.252

536.533

0.793

710.008

−17.919

20.986

457.361

502.465

469.660

498.048

511.504

447.125

623.101

619.393

524.437

443.505

484.463

451.080

526.532

536.832

458.253

710.318

686.410

538.657

6.100

31.400

21.600

266.248

259.435

153.853

598.470

503.502

289.561

−13.400

−11.100

−17.500

−13.209

457.320

501.483

469.163

542.172

540.600

465.542

678.103

612.361

503.138

443.302

484.336

450.740

553.737

560.278

479.782

−0.823 −19.713

701.711

672.759

550.361

35.836

29.837

−3.818

538.632

457.361

502.465

469.660

604.018

599.629

490.307

904.428

792.780

580.511

443.505

484.463

451.080

553.894

560.625

479.783

702.626

673.420

550.374

sqrt(MSE)

−5.225

Std

Bias

Std

Bias

sqrt(MSE)

Kaplan–Meier estimator

Lee and Tsai’s estimator

455.782

497.459

454.390

549.819

554.792

478.634

666.906

612.976

497.892

455.083

497.638

456.367

545.974

566.159

490.888

683.782

649.392

526.103

AES

0.954

0.938

0.942

0.930

0.926

0.929

0.856

0.854

0.881

0.957

0.945

0.956

0.929

0.939

0.956

0.933

0.932

0.935

CR

15.701

23.507

21.138

−10.181

1.101

4.358

−12.684

−18.987

−23.854

−10.796

−16.014

−12.883

−15.800

−18.499

0.680

14.405

22.835

−4.202

Bias

NPMLE

421.121

455.782

417.597

469.893

489.888

422.673

614.458

606.850

509.650

411.697

445.981

413.698

505.719

514.169

435.084

703.195

677.260

524.809

Std

421.414

456.388

418.132

470.004

489.889

422.695

614.588

607.147

510.208

411.839

446.268

413.899

505.966

514.501

435.085

703.343

677.645

524.825

sqrt(MSE)

424.771

451.001

406.461

488.709

503.871

435.396

619.218

613.808

498.870

430.601

457.365

417.697

516.219

521.047

445.071

685.009

655.729

507.769

AES

0.955

0.943

0.933

0.955

0.964

0.946

0.939

0.945

0.934

0.955

0.959

0.960

0.946

0.946

0.956

0.928

0.931

0.936

CR

The AES and the CR are not reported for the Lee and Tsai’s estimator, since the corresponding variance estimator has not been developed. The values of Bias, Std, sqrt(MSE) and AES have been multiplied by 10,000 Bias the empirical bias, Std the empirical standard deviation, sqrt(MSE) the square root of the empirical mean squared error, AES the average estimated standard deviation, CR the empirical coverage rate of the 95 % confidence interval

0.6

1.0

2

2

0.6

1.0

1

0.6

1.0

1

2

0.6

1

2

0.6

0.6

1

1

0.3

0.3

1

0.3

1

1

θ

λ

Table 6 Simulation results when C = +∞ and U is generated from the exponential distribution with P (T > U) = 0.7

Z. Zhao et al.

1.0

1.0

2

2

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

τ

481.920

474.120

491.289

457.478

192.218

−12.100

−23.100

524.993

475.318

500.009

499.220

444.514

−24.800

148.054

57.458

320.370

218.342

106.272

517.978

451.950

2.600

−8.700

464.171

473.206

8.971

−5.500

487.732

440.079

16.974

−1.646

527.098

458.060

491.438

474.768

518.840

545.470

478.778

593.840

544.879

457.041

452.033

517.985

473.238

464.258

488.027

440.082

508.614

526.192

454.476

497.570

456.404

487.073

511.501

459.332

512.668

523.112

462.601

455.201

497.297

455.534

482.019

513.509

463.025

505.214

0.961

0.943

0.944

0.941

0.931

0.936

0.919

0.941

0.942

0.950

0.930

0.948

0.957

0.961

0.960

0.940

0.944

0.948

−25.766

−21.879

−24.034

22.617

27.558

6.531

1.890

2.863

4.028

−7.242

7.841

−2.166

9.506

19.143

−0.956

14.140

−7.799

−32.751

527.004

508.382

15.360

469.845

−9.926

487.632

486.215

−37.142

CR

Bias

AES

Std

Bias

sqrt(MSE)

NPMLE

Kaplan–Meier estimator

429.097

480.151

449.961

447.304

500.385

458.627

481.911

501.590

445.565

431.096

491.710

459.658

438.781

461.533

421.928

511.564

521.589

468.945

Std

429.870

480.650

450.602

447.875

501.143

458.673

481.914

501.598

445.583

431.157

491.773

459.663

438.884

461.930

421.929

511.759

521.647

470.088

sqrt(MSE)

476.058

485.176

441.301

492.088

496.141

444.893

528.818

522.948

457.066

448.312

479.562

443.548

473.189

492.530

448.100

528.501

520.417

458.227

AES

0.970

0.948

0.947

0.973

0.949

0.933

0.972

0.955

0.954

0.950

0.935

0.939

0.960

0.963

0.965

0.956

0.948

0.946

CR

The Lee and Tsai’s estimator is not included, since the additional independent censoring is not incorporated in this approach. The values of Bias, Std, sqrt(MSE) and AES have been multiplied by 10,000 Bias the empirical bias, Std the empirical standard deviation, sqrt(MSE) the square root of the empirical mean squared error, AES the average estimated standard deviation, CR the empirical coverage rate of the 95 % confidence interval

0.6

1.0

2

2

0.6

0.6

2

2

0.3

0.3

2

0.3

2

2

1.0

1.0

1

1

0.6

1.0

1

1

1

0.6

0.6

1

0.3

0.3

1

0.3

1

1

θ

λ

Table 7 Simulation results when P (T > C) = 0.15 and U is generated from the exponential distribution with P (T > U) = 0.3

Semi-Markov model for dependent censoring

123

123

0.6

1.0

1.0

1.0

2

2

2

2

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

0.7

0.5

0.3

τ AES

CR

559.884

32.400

45.700

36.000

286.414

288.481

163.339

658.430

458.891

495.830

460.734

549.552

550.581

486.410

663.681

620.296

496.285

500.169

465.061

−13.100

−11.600

317.395

547.570

448.635

−19.481

−20.200

501.497

577.838

−7.869

−7.043

674.713

636.189

530.033

54.060

15.475

11.840

460.033

497.932

462.139

619.710

621.579

513.103

934.881

835.606

589.100

465.206

500.340

449.089

547.916

577.881

501.559

676.876

636.378

530.165

456.942

497.505

453.857

550.980

555.227

478.177

670.392

612.199

496.722

454.898

497.480

456.502

546.076

565.730

490.966

684.240

650.307

525.895

0.951

0.954

0.951

0.927

0.919

0.917

0.849

0.826

0.873

0.944

0.938

0.965

0.943

0.944

0.931

0.942

0.947

0.928

621.620

−10.036 5.585

31.639

35.270

27.417

16.611

14.533

410.367

434.118

398.346

472.103

495.363

437.106

621.533

501.787

−10.674 4.927

428.946

457.657

400.871

510.310

525.635

451.606

662.199

625.898

504.538

Std

−4.168

−20.281

−22.370

−21.699

−15.676

−10.250

34.744

15.571

15.547

Bias

sqrt(MSE)

Bias

Std

NPMLE

Kaplan–Meier estimator

411.585

435.548

399.288

472.395

495.576

437.142

621.552

621.701

501.900

428.966

458.107

401.494

510.771

525.869

451.723

663.110

626.092

504.778

sqrt(MSE)

425.044

451.011

406.952

490.128

504.191

435.284

618.646

613.743

497.296

430.929

457.652

418.539

515.834

521.541

446.259

685.833

654.501

506.972

AES

0.959

0.956

0.945

0.952

0.954

0.949

0.932

0.943

0.945

0.941

0.957

0.960

0.938

0.944

0.937

0.949

0.955

0.947

CR

The Lee and Tsai’s estimator is not included, since the additional independent censoring is not incorporated in this approach. The values of Bias, Std, sqrt(MSE) and AES have been multiplied by 10,000 Bias the empirical bias, Std the empirical standard deviation, sqrt(MSE) the square root of the empirical mean squared error, AES the average estimated standard deviation, CR the empirical coverage rate of the 95 % confidence interval

0.6

0.6

2

2

0.3

0.3

2

0.3

2

2

1.0

1.0

1

1

0.6

1.0

1

1

0.6

0.6

1

1

0.3

0.3

1

0.3

1

1

θ

λ

Table 8 Simulation results when P (T > C) = 0.15 and U is generated from the exponential distribution with P (T > U) = 0.7

Z. Zhao et al.

Semi-Markov model for dependent censoring

1.0

Survival Function Estimate

0.6 0.4 0.0

0.2

Survival Rate

0.8

NPMLE Kaplan−Meier Esimator Empirical Distribution Pointwise 95% Confidence Interval Based on the NPMLE Approach Pointwise 95% Confidence Interval Based on the Kaplan−Meier Approach

0

20

40

60

80

100

t

Fig. 2 Kaplan–Meier estimate and NPMLE

For the cases with λ = 1 when T was independent of U and the cases with θ = 1 when (X, Δ) was fully observed, the Kaplan–Meier estimator was nearly unbiased and consistent and the corresponding 95 % confidence intervals provided reasonable coverage rates; the proposed NPMLE and the Lee and Tsai’s estimator were also nearly unbiased and consistent, and the coverage rate of the proposed 95 % confidence intervals were close to the nominal level. Nevertheless, the proposed NPMLE had smaller sqrt(MSE) than the Kaplan–Meier estimator. For the cases with λ = 2 and θ < 1, the Kaplan–Meier estimator had larger bias and appeared inconsistent, while the other two estimators performed well. 5.2 An example We illustrate our method with the data from a clinical trial of lung cancer conducted by the Eastern Cooperative Oncology Group, which was initially analyzed by Lagakos and Williams (1978). The data were also used by Lee and Wolfe (1998) and Lee and Tsai (2005) for the illustration of their methods. The complete data can be found in Lee and Tsai (2005). The study consists of 61 patients with inoperable carcinoma of the lung who were treated with the drug cyclophosphamide. Among the 61 patients, 28 patients experienced metastatic disease or a significant increase in the size of their

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1.0

Survival Function Estimate

0.6 0.4 0.0

0.2

Survival Rate

0.8

NPMLE Lee and Tsai’s Estimator Empirical Distribution Pointwise 95% Confidence Interval Based on the NPMLE Approach

0

20

40

60

80

100

t

Fig. 3 Lee and Tsai’s estimator and NPMLE

primary lesion, which are certain types of PD. In addition, Lee and Tsai (2005) did a pseudo second stage sampling, in which they selected 10 of these 28 patients as a random sample and pretended that the death time of the rest of the 18 patients were unknown. For illustration, we also pretended that these 18 patients’ death time were unknown. We treated the 10 selected patients as if they did not leave the study while the rest of the 18 did. Figure 2 shows the NPMLE and its pointwise 95 % confidence intervals along with the Kaplan–Meier estimator and its pointwise 95 % confidence intervals and the empirical distribution based on the complete data. Figure 3 presents the NPMLE and its pointwise 95 % confidence intervals along with the Lee and Tsai’s estimator and the empirical distribution based on the complete data. Figure 2 clearly shows that the Kaplan–Meier estimator is very different from the empirical distribution, and far away from the other estimators. Figures 2 and 3 show that both NPMLE and the Lee and Tsai’s estimator are very close to the empirical distribution. The proposed pointwise 95 % confidence interval contains the Lee and Tsai’s estimator and the empirical distribution.

6 Discussion In oncology studies, in addition to the usual independent censoring, the censoring caused by the PD is a certain type of dependent censoring. In this paper, we have

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adopted the semi-Markov model provided by Lee and Tsai (2005) with an additional independent censoring and studied the NPMLE of the cause specific cumulative hazard functions and the marginal survival function, where we have shown the asymptotic normality of the NPMLE in the space of uniformly bounded functions and established its asymptotic efficiency. We have also developed uniformly consistent estimators for the covariance function based on the information operator. A limitation of the model (1) is that it assumes that the risk of death after disease progression only depends on the time since progression, but not the duration of the progression. The limitation could be addressed by regression type models with the inclusion of the time to progression as a covariate. It is of interest to develop diagnostic methods for the model and to investigate regression type models in future studies. Acknowledgments The research is supported by the National Natural Science Foundation of China (11271081) and the Student Growth Fund Scholarship of School of Management, Fudan University.

Appendix Preliminary Lemmas 1 and 2 provide related Donsker properties. Lemma 1 The classes {(x, δ) → δI {x  t} : t  0} and {(x, δ) → δI {x  t} : t  0} are Donsker classes of functions on R+ × {0, 1}. Proof Combining Lemma 2.6.15 and Lemma 2.6.18(iii, vi) of Vaart and Wellner (1996), it can be shown that {(x, δ) → δI {x  t} : t  0} and {(x, δ) → δI {x  t} : t  0} are VC-classes on R+ × {0, 1}. By Theorem 2.6.8 of Vaart and Wellner (1996), they are Donsker classes, where the measurability conditions could be verified via the denseness of the rational numbers.

Lemma 2 Let η be  a positive real number and Λ be a nondecreasing  cadlag function !t on [0, η]. The class (x, δ) → 0 δI {x  y} Λ (dy) : t ∈ [0, η] is a Donsker class of functions on R+ × {0, 1}. Proof The result is easy to see by combining Example 2.6.21 and Example 2.10.8 of Vaart and Wellner (1996).

Lemma 3 Let η be a positive real number and BV ([0, η]) be the space of all cadlag functions defined on [0, η] whose total variation are bounded by 2. For any (F, G, H) ∈ BV ([0, η]) × BV ([0, η]) × BV ([0, η]) and any t ∈ [0, η], let  φ (F, G, H) [t] = F (t) G (t) −

G (u) H (t − u) F (du) . [0,t]

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For any F0 , G0 , H0 ∈ BV ([0, η]) , φ is Hadamard differentiable at (F0 , G0 , H0 ) with derivative φ  (F0 , G0 , H0 ), where for any t ∈ [0, η],  φ  (β0 ) [β] [t] = F0 (t) G (t) + F (t) G0 (t) − G0 (x) H0 (t − x) F (d x) [0,t]   G0 (x) H (t − x) F0 (d x) − G (x) H0 (t − x) F0 (d x) , − [0,t]

[0,t]

where β = (F, G, H) and β0 = (F0 , G0 , H0 ). Proof Let F0 , G0 , H0 , F, G, H ∈ BV ([0, η]) , {(Fm , Gm , Hm ) : m ∈ N} be a sequence converging to (F, G, H) in BV ([0, η]) × BV ([0, η]) × BV ([0, η]) and {h m : m ∈ N} be a sequence of real numbers converging to 0. Denote β0 = (F0 , G0 , H0 ) and βm = β0 + h m (F, G, H). Note that for any t ∈ [0, η] , φ(βm )[t] = φ(β0 )[t] + h m Γm (t) + h 2m Wm (t), where  Γm (t) = F0 (t) Gm (t) + Fm (t) G0 (t) − G0 (x) H0 (t − x) Fm (d x) [0,t]   G0 (x) Hm (t − x) F0 (d x) − Gm (x) H0 (t − x) F0 (d x) , − [0,t] [0,t]  H0 (t − x) Gm (x) Fm (d x) Wm (t) = Fm (t) Gm (t) − [0,t]   G0 (x) Hm (t − x) Fm (d x) − Gm (x) Hm (t − x) F0 (d x) − [0,t] [0,t]  Gm (x) Hm (t − x) Fm (d x) . − hm [0,t]

For any t ∈ [0, η], let  G0 (x) H0 (t − x) F (d x) Γ0 (t) = F0 (t) G (t) + F (t) G0 (t) − [0,t]   G0 (x) H (t − x) F0 (d x) − G (x) H0 (t − x) F0 (d x) . − [0,t]

[0,t]

Note that for any t ∈ [0, η] , |Wm (t)|  28 + 8h m and |Γm (t) − Γ0 (t)|  6 sup |Fm (s) − F (s)| + 6 sup |Gm (s) − G (s)| s∈[0,η]

s∈[0,η]

+ 4 sup |Hm (s) − H (s)| . s∈[0,η]

Hence, as m → ∞, supt∈[0,η] |Wm (t)| = O(1) and supt∈[0,η] |Γm (t) − Γ0 (t)| = o(1).

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Therefore, φ is Hadamard differentiable at β0 and for any t ∈ [0, η],  G0 (x) H0 (t − x) F (d x) φ  (β0 ) [β] [t] = F0 (t) G (t) + F (t) G0 (t) − [0,t]   G0 (x) H (t − x) F0 (d x) − G (x) H0 (t − x) F0 (d x) , − [0,t]

[0,t]

where β = (F, G, H).



Proof of Theorems 1 to 3 For any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)} and any t  0, define (i, j)

Mn

(i, j)

(t) = Nn

 (t) − [0,t]

(i)

(i, j)

Yn (y) Λ0

(dy) .

Proof Combining Theorem 2.10.6 of Vaart and Wellner (1996) with Lemmas 1 and 2, it can be shown that, as n → ∞,  (0,1) (0,2) (1,2) (0)  (0) (1) (1) n −1/2 Mn , Mn , Mn , Yn − EYn , Yn − EYn weakly converges to a tight zero mean Gaussian process in ∞ ∞ ∞ ∞ ∞

∞ 5 ([0, τ ]) = ([0, τ ]) × ([0, τ ]) × ([0, τ ]) × ([0, τ ]) × ([0, τ ]) .

Combining this with Lemma 3.9.17, Lemma 3.9.25, and Theorem 3.9.4 of Vaart and Wellner (1996), for any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)},   −1   (i, j) (i, j) (i, j) (i) ˆ L0 (y−) sup Λn (t) − Λ0 (t) − n −1 Mn (dy) = o p n −1/2 , t∈[0,τ ]

[0,t]

(10) By the continuity and the linearity of the integral operator, as n → ∞,   (0,1) (0,2) (1,2) − Λ0 , Λˆ n(1,2) − Λ0 n 1/2 Λˆ n(0,1) − Λ0 , Λˆ (0,2) n weakly converges to a tight Gaussian process in ∞ 3 ([0, τ ]) with covariance function U0 . By Lemma 3.9.30 and Theorem 3.9.4 of Vaart and Wellner (1996), as n → ∞,   (0,1) (0,2) ˆ (1,2) (1,2) − S , S − S n 1/2 Sˆ n(0,1) − S0 , Sˆ (0,2) n n 0 0 weakly converges to a tight Gaussian process in ∞ 3 ([0, τ ]) with covariance function V0 .

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Combining Lemma 3 with Theorem 3.9.4 of Vaart and Wellner (1996),

  sup Sˆ n (t) − S0 (t) − φ  (β0 ) βˆn − β0 [t] = o p n −1/2 , t∈[0,τ ]

    (0,1) (0,2) (1,2) (0,1) (0,2) (1,2) where β0 = S0 , S0 , S0 and βˆn = Sˆ n , Sˆ n , Sˆ n . 1/2 ˆ Therefore, as n → ∞, n (Sn − S0 ) weakly converges to a tight zero mean

Gaussian process in ∞ ([0, τ ]) with covariance function Ω0 . Proof of Theorems 4 and 5 Proof To obtain the efficiency, we will verify the conditions of Theorem VIII.3.2 and Theorem VIII.3.3 of Andersen et al. (1993). First, we verify the local asymptotic normality (LAN) Assumption (Assumption VIII.3.1 of Andersen et al. 1993). By the Taylor expansion, it is easy to show that for any h = (h (0,1) , h (0,2) , h (1,2) ) ∈ H, log Rn (h) = Sn(0,1) (h) + Sn(0,2) (h) + Sn(1,2) (h) −

1 h2H + o p (1) , 2

(11)

where for any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)} and any h = (h (0,1) , h (0,2) , h (1,2) ) ∈ H, (i, j) Sn (h)

=n

−1/2

 [0,τ ]

−1 (i, j)  (i, j) h (i, j) (y) 1 − Λ0 {y} Mn (dy) .

Hence, for any m ∈ N+ and any h 1 , . . . , h m ∈ H, (log Rn (h 1 ) , . . . , log Rn (h m )) weakly converges to a Gaussian random vector in Rm with mean 1 − (h 1 H , . . . , h m H ) 2 and covariance matrix ⎛

h 1 , h 1 H . . . ⎜ .. .. ⎝ . . h m , h 1 H · · ·

⎞ h 1 , h m H ⎟ .. ⎠. .

h m , h m H

Next, we verify the Differentiability Assumption (Assumption VIII.3.2 of Andersen et al. 1993).

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For any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)}, any h = (h (0,1) , h (0,2) , h (1,2) ) ∈ H, any t ∈ [0, τ ],    (i, j) (i, j) (i, j) h (i, j) (y) Λ0 (dy) . n 1/2 Λn,h (t) − Λ0 (t) = [0,t]

Note that, for any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)}, any h = (h (0,1) , h (0,2) , h (1,2) ) ∈ H, and any t ∈ [0, τ ], 

(i, j)

[0,t]

h (i, j) (y) Λ0

(i, j)

(dy)  hH Λ0

 −1 (i) . (τ ) L0 (τ −)

Hence, the Differentiability Assumption is verified.   (0,1) ˆ (0,2) ˆ (1,2) ˆ is regular. Next, we show that Λn , Λn , Λn Recall the weak convergence of the log-likelihood ratio log Rn (h). The continuity follows from the Le Cam’s third Lemma and the weak convergence can be shown via similar arguments in the previous subsection. Hence, the regularity follows. It remains to check Equation (8.3.5) of Andersen et al. (1993) for each coordinate. For any t ∈ [0, τ ], define (0,1)

γt

      (0,1) (0,2) (0,2) (1,2) (1,2) = ϕt , 0, 0 , γt = 0, ϕt , 0 , γt = 0, 0, ϕt

where for any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)} and any s ∈ [0, τ ], (i, j)

ϕt

  (i, j) (i) (s) = I {s  t} L0 (s−)−1 1 − Λ0 {y} .

By (10) and (11), for any (i, j) ∈ {(0, 1) , (0, 2) , (1, 2)} and any t ∈ [0, τ ],     1   (i, j) (i, j) (i, j) + h2H = o p n −1/2 . Λˆ n (t) − Λ0 (t) − n −1/2 log Rn γt 2 Therefore, combining all the above results with Theorem VIII.3.2 and  Theorem (0,1) (0,2) (1,2) VIII.3.3 of Andersen et al. (1993), it follows that Λˆ n , Λˆ n , Λˆ n is asymptotically efficient. Combining Theorem VIII.3.4 of Andersen et al. (1993) with Lemma 3, the efficiency of Sˆ n follows.



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