EM algorithm for the additive risk mixture cure model with interval-censored data

Abstract

Interval-censored failure time data arise in a number of fields and many authors have recently paid more attention to their analysis. However, regression analysis of interval-censored data under the additive risk model can be challenging in maximizing the complex likelihood, especially when there exists a non-ignorable cure fraction in the population. For the problem, we develop a sieve maximum likelihood estimation approach based on Bernstein polynomials. To relieve the computational burden, an expectation–maximization algorithm by exploiting a Poisson data augmentation is proposed. Under some mild conditions, the asymptotic properties of the proposed estimator are established. The finite sample performance of the proposed method is evaluated by extensive simulations, and is further illustrated through a real data set from the smoking cessation study.

Appendix

In this appendix, we present the detailed proofs for Theorems 1–3, and the calculation of $I(\widehat{\pmb {\theta }})$. Our proof relies heavily on some results in empirical processes. To facilitate our proof for the consistency of our estimators, we need the following lemma, which play an important roles in the proof of our Theorem 1. Lemma 1 establishes the covering number of size $\epsilon$ needed to cover $\mathcal {L}_1=\{ l(\pmb {\theta };\text{ O}):\pmb {\theta }\in \varvec{\Theta }_n\}$.

Whether a given class $\mathcal {L}$ is a Glivenko–Cantelli (GC) or Donsker class depends on the size of the class. A relatively simple way to measure the size of a class $\mathcal {L}$ is to use entropy numbers. The existing results of the known GC or Donsker class are mostly in the parameter or non-parametric framework, while in the semiparametric framework, the size of the class $\mathcal {L}$ can be more clearly reflected by the entropy number. Therefore, the technical analysis on the entropy numbers is necessary.

Lemma 1

The covering number of the function class $\mathcal {L}_1=\{ l(\pmb {\theta };\text{ O}):\pmb {\theta }\in \varvec{\Theta }_n\}$ satisfies

$$\begin{aligned} N(\epsilon ,\mathcal {L}_1,\parallel \cdot \parallel _{\infty })\le CM_n^{(N+1)}\epsilon ^{-(N+p+q+1)}, \end{aligned}$$

where the degree N of Bernstein polynomials satisfies $N=o(n^\nu )$ with $0<\nu <1$ and the size of the sieve space $\varvec{\Theta }_n$ is controlled by $M_n=O(n^a)$ with a constant $a>0$.

Proof of Lemma 1By the Taylor series expansion, for any $\pmb {\theta }_1=\big (\pmb {\alpha }_{1},\pmb {\beta }_{1},\Lambda _{1}(\cdot )\big ),\pmb {\theta }_2=\big (\pmb {\alpha }_{2},\pmb {\beta }_{2},\Lambda _{2}(\cdot )\big )\in \varvec{\Theta }_n$, there exists a large enough constant C such that

$$\begin{aligned} \mid l(\pmb {\theta }_1;\text{ O})-l(\pmb {\theta }_2;\text{ O})\mid \le C\bigl (\Vert \pmb {\alpha }_1-\pmb {\alpha }_2\Vert +\Vert \pmb {\beta }_1-\pmb {\beta }_2\Vert +\Vert \Lambda _{1}(\cdot )-\Lambda _{2}(\cdot )\Vert _{\infty }\bigr ). \end{aligned}$$

For any $\Lambda _1(\cdot ),\Lambda _2(\cdot )\in \mathcal {M}_n,$ let $\pmb {\gamma }_1=(\gamma _{0,1},\ldots ,\gamma _{N,1})^{\intercal }, \pmb {\gamma }_2=(\gamma _{0,2},\ldots ,\gamma _{N,2})^{\intercal }$ be their Bernstein polynomials coefficient vectors, respectively. Then, we have

$$\begin{aligned} \Vert \Lambda _{1}(\cdot )-\Lambda _{2}(\cdot )\Vert _{\infty }= & {} \sup _t\mid \sum _{l=0}^N\gamma _{l,1}b_{l,N}(t)-\sum _{l=0}^N\gamma _{l,2}b_{l,N}(t) \mid \le \max _{0\le l\le N}\mid \gamma _{l,1}-\gamma _{l,2}\mid \\= & {} \Vert \pmb {\gamma }_1-\pmb {\gamma }_2 \Vert _{\infty }. \end{aligned}$$

Therefore one can write that

$$\begin{aligned} \mid l(\pmb {\theta }_1;\text{ O})-l(\pmb {\theta }_2;\text{ O})\mid \le C(\Vert \pmb {\alpha }_1-\pmb {\alpha }_2\Vert +\Vert \pmb {\beta }_1-\pmb {\beta }_2\Vert +\Vert \pmb {\gamma }_1-\pmb {\gamma }_2 \Vert _{\infty }). \end{aligned}$$

Let $\mathcal {B}=\{(\pmb {\alpha }^{\intercal },\pmb {\beta }^{\intercal })^{\intercal } \in \mathbb {R}^{p+q}:\Vert \pmb {\alpha }\Vert +\Vert \pmb {\beta }\Vert \le M \}$ and $\mathcal {C}=\{\pmb {\gamma }\in \mathbb {R}^{N+1}:\sum _{l=0}^N\mid \gamma _l\mid \le M_n \}.$ Combining Problem 18 in Pollard (1984),(p. 40), that is, tensor product of two $\epsilon /2C$ balls in $\mathcal {B}$ and $\mathcal {C}$, respectively, is contained in a $\epsilon$-ball in $\mathcal {L}_1$; tensor products of balls covering $\mathcal {B}$ and balls covering $\mathcal {C}$ produce sets covering the tensor of $\mathcal {B}$ and $\mathcal {C}$ of which $\mathcal {L}_1$ is a subset; Hence the covering number of $\mathcal {L}_1$ is controlled by the covering numbers of $\mathcal {B}$ and $\mathcal {C}$, we obtain that

$$\begin{aligned} N(\epsilon , \mathcal {L}_1, \parallel \cdot \parallel _{\infty }) \le N(\frac{\epsilon }{2C}, \mathcal {B}, \parallel \cdot \parallel ) N(\frac{\epsilon }{2C}, \mathcal {C}, \parallel \cdot \parallel _{\infty }). \end{aligned}$$

Using Lemma 4.1 of Pollard (1990) that presents a method for finding the bounds for the packing numbers of a set and the bounds on the packing numbers grow geometrically, thus the packing numbers of $\mathcal {B}$ and $\mathcal {C}$ that defined in Pollard (1990),(p.10) satisfy

$$\begin{aligned} M(\epsilon , \mathcal {B}, \parallel \cdot \parallel ) \le \biggl (\frac{6M}{\epsilon }\biggr )^{p+q} \text { and } \ M(\epsilon , \mathcal {C}, \parallel \cdot \parallel _{\infty }) \le \biggl (\frac{6M_n}{\epsilon }\biggr )^{N+1}. \end{aligned}$$

Next, by Lemma 9.2 in Györfi et al. (2002) shows that the $L_{(p)}$ covering numbers of the size $\epsilon$ are controlled by the $L_{(p)}$ packing numbers of the size $\epsilon$ where $p\ge 1$, one can obtain that

$$\begin{aligned} N(\epsilon , \mathcal {B}, \parallel \cdot \parallel ) \le \biggl (\frac{6M}{\epsilon }\biggr )^{p+q} \text { and } \ N(\epsilon , \mathcal {C}, \parallel \cdot \parallel _{\infty }) \le \biggl (\frac{6M_n}{\epsilon }\biggr )^{N+1}. \end{aligned}$$

Combining the above results, we have the conclusion that

$$\begin{aligned} N(\epsilon ,\mathcal {L}_1,\parallel \cdot \parallel _{\infty })\le CM_n^{(N+1)}\epsilon ^{-(N+p+q+1)}. \end{aligned}$$

$\square$

Proof of Theorem 1 By adopting the notations of Pollard (1984), we write

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}P_n l=l_n(\pmb {\theta },\mathcal {O})=\frac{1}{n}\sum _{i=1}^{n}l(\pmb {\theta },\text{ O}_i),\\ &{}Pl=E_0 l(\pmb {\theta },\text{ O}), \end{array}\right. } \end{aligned}$$

where $E_0$ denotes the expectation with respect to $\text{ O }$ under true values of parameters.

Proving for the consistency of our estimators requires the following steps.

• (a) : Calculate the covering number of the function class $\mathcal {L}_1=\{ l(\pmb {\theta };\text{ O}):\pmb {\theta }\in \varvec{\Theta }_n\}$ by lemma 1.

• (b) : Prove uniform convergence, i.e. $\sup _{\mathcal {L}_1}\big |P_nl-Pl\big | \rightarrow 0$ a.s.

• (c) : Prove $d(\widehat{\pmb {\theta }}_n,\pmb {\theta }_0)\rightarrow 0$ a.s.

The function class $\mathcal {L}_1=\{ l(\pmb {\theta };\text{ O}):\pmb {\theta }\in \varvec{\Theta }_n\}$ has an envelope G with $Pl^2\le PG^2\le \delta ^2$. Let $\alpha _n=n^{-1/2+\phi _1}(\log n)^{1/2}$ with $\frac{\nu }{2}<\phi _1<\frac{1}{2}$ such that $\alpha _n$ is a non-increasing sequence of positive numbers. For any given $\epsilon >0$, we choose $\epsilon _n=\epsilon \delta ^2\alpha _n$. Then for any $\pmb {\theta }\in \Theta _n$ and a sufficiently large n, we have

$$\begin{aligned} \frac{Var(P_nl)}{(4\epsilon _n)^2} \le \frac{\frac{1}{n}Pl^2}{16\epsilon ^2\delta ^4\alpha _n^2} \le \frac{1}{16\epsilon ^2\delta ^2n^{2\phi _1}\log n} \le \frac{1}{2}. \end{aligned}$$

Let $P_n^{o}$ denote the signed measure that places mass $\pm n^{-1}$ at each of the observations $\{O_1,\ldots ,O_n\}$, with the random ± signs being decided independently of the $O_i$’s. Applying the symmetrization inequality (Pollard (1984), II (30)) and Hoeffding’s inequality (Pollard (1984), II (31)), it follows that

$$\begin{aligned} P\big (\sup _{\mathcal {L}_1}|P_nl-Pl|>8\epsilon _n\big )\le & {} 4P\big (\sup _{\mathcal {L}_1}|P_n^ol|>2\epsilon _n\big ) \\= & {} 4E\Bigl \{I_{ \{ \sup \limits _{\mathcal {L}_1}|P_n^ol|>2\epsilon _n\}} \Bigr \} \\= & {} 4E \Bigl \{E\bigl (I_{ \{ \sup \limits _{\mathcal {L}_1}|P_n^ol|>2\epsilon _n \}}|\mathcal {O}\bigr )\Bigr \} \\= & {} 4E\left\{ P(\sup _{\mathcal {L}_1}|P_n^ol|>2\epsilon _n|\mathcal {O})\right\} \\\le & {} 4E\left\{ 2N(\epsilon _n,\mathcal {L}_1,\parallel \cdot \parallel _{\infty })\exp \left( \frac{-\frac{1}{2}n\epsilon _n^2}{\max \limits _{j}P_nl_j^2}\right) \right\} \\\le & {} 8C M_n^{N+1}\epsilon _n^{-(N+p+q+1)}\exp \Bigl (\frac{-\frac{1}{2}n\epsilon _n^2}{\max \limits _{j}P_nl_j^2}\Bigr ), \end{aligned}$$

where the maximum runs over all functions $\{l_j\} \text { in } \mathcal {L}_1$. Using the law of total probability, we have

$$\begin{aligned}&P\Big (\sup _{\mathcal {L}_1}|P_nl-Pl|>8\epsilon _n\Big ) \nonumber \\&\quad =P\Bigl (\sup _{\mathcal {L}_1}|P_nl-Pl|>8\epsilon _n \Big | \sup _{\mathcal {L}_1}|P_nl^2|\le 64\delta ^2\Bigr ) P\Bigl (\sup _{\mathcal {L}_1}|P_nl^2|\le 64\delta ^2\Bigr ) \nonumber \\&\qquad + P\Bigl (\sup _{\mathcal {L}_1}|P_nl-Pl|>8\epsilon _n \Big | \sup _{\mathcal {L}_1}|P_nl^2|> 64\delta ^2\Bigr ) P\Bigl (\sup _{\mathcal {L}_1}|P_nl^2|> 64\delta ^2\Bigr ) \nonumber \\&\quad \le P\Bigl (\sup _{\mathcal {L}_1}|P_nl-Pl|>8\epsilon _n \Big | \sup _{\mathcal {L}_1}|P_nl^2|\le 64\delta ^2\Bigr ) + P\Bigl (\sup _{\mathcal {L}_1}|P_nl^2|> 64\delta ^2\Bigr ).\nonumber \\ \end{aligned}$$

(12)

Note that $N=o(n^\nu )$, and $\nu /2<\phi _1$, which indicate $N=o(n^{2\phi _1})$. Now we study the first term of the inequality (12),

$$\begin{aligned}&P\Bigl (\sup _{\mathcal {L}_1}|P_nl-Pl|>8\epsilon _n \Big | \sup _{\mathcal {L}_1}|P_nl^2|\le 64\delta ^2\Bigr ) \\&\quad \le 8C M_n^{N+1} \epsilon _n^{-(N+p+q+1)} \exp \Bigl (\frac{-n\epsilon _n^2}{128\delta ^2}\Bigr ) \\&\quad \le 8C \exp \Bigl \{ \bigl ( N+1 \bigr ) a\log n - \bigl ( N+p+q+1 \bigr ) \log {(\epsilon \delta ^2\alpha _n)} - \frac{n\epsilon _n^2}{128\delta ^2} \Bigr \} \\&\quad \le 8C \exp \Bigl \{ \bigl ( N+p+q+1 \bigr ) \bigl [ (a+\frac{1}{2}-\phi _1) \log n - \log {(\epsilon \delta ^2)} \\&\qquad -\frac{1}{2}\log {\log n} \bigr ] - \frac{\epsilon ^2\delta ^2 n^{2\phi _1}\log n}{128} \Bigr \} \\&\quad \le 8C \exp \Bigl \{ n^{2\phi _1}\log n \bigl [ \frac{\bigl ( o(n^{2\phi _1})+p+q+1 \bigr ) (a+\frac{1}{2}-\phi _1)}{n^{2\phi _1}} - \frac{\epsilon ^2\delta ^2}{128} \bigr ] \Bigr \} \\&\quad \le 8C \exp \bigl \{ -\tilde{C} n^{2\phi _1}\log n \bigr \}, \end{aligned}$$

where $\tilde{C}$ is a constant. For the second term of the inequality (12), by Lemma 33 of Pollard (1984) it follows that

$$\begin{aligned} P\left( \sup _{\mathcal {L}_1}\mid P_nl^2 \mid> 64\delta ^2\right)= & {} P\left( \sup _{\mathcal {L}_1}|P_nl^2|^{1/2}> 8\delta \right) \\\le & {} 4E \Bigl \{ N(\delta ,\mathcal {L}_1,P_n) \exp (-n\delta ^2)\wedge 1 \Big \} \\\le & {} 4E \Bigl \{ N(\delta ,\mathcal {L}_1,\parallel \cdot \parallel _{\infty }) \exp (-n\delta ^2)\wedge 1 \Bigr \} \\\le & {} 4 \Bigl \{ C M_n^{N+1} \delta ^{-(N+p+q+1)} \exp (-n\delta ^2)\wedge 1 \Big \} \\\le & {} 4C \exp \Bigl \{ (o(n^\nu )+p+q+1) (a\log n - \log \delta ) -n\delta ^2 \Bigr \}. \end{aligned}$$

This result indicates that the second term converges to zero even faster than the first term. Therefore one can obtain that $\sum _{n=1}^{\infty } P\big (\sup _{\mathcal {L}_1} |P_nl-Pl| > 8\epsilon _n\big ) < \infty$. By adopting the Borel–Contelli lemma, it is easy to obtain that

$$\begin{aligned} \sup _{\mathcal {L}_1}\big |P_nl-Pl\big | \rightarrow 0, \quad a.s. \end{aligned}$$

Note that $P_nl(\pmb {\theta }_0;\text{ O}) \le P_nl(\widehat{\pmb {\theta }}_n;\text{ O})$, then we have

$$\begin{aligned} 0\le & {} Pl(\pmb {\theta }_0;\text{ O})-Pl(\widehat{\pmb {\theta }}_n;\text{ O}) \\= & {} Pl(\pmb {\theta }_0;\text{ O}) - P_nl(\pmb {\theta }_0;\text{ O}) + P_nl(\pmb {\theta }_0;\text{ O}) - P_nl(\widehat{\pmb {\theta }}_n;\text{ O}) + P_nl(\widehat{\pmb {\theta }}_n;\text{ O}) - Pl(\widehat{\pmb {\theta }}_n;\text{ O}) \\\le & {} Pl(\pmb {\theta }_0;\text{ O}) - P_nl(\pmb {\theta }_0;\text{ O}) + P_nl(\widehat{\pmb {\theta }}_n;\text{ O}) - Pl(\widehat{\pmb {\theta }}_n;\text{ O}). \end{aligned}$$

Therefore, we have the conclusion that

$$\begin{aligned}&\&\mid Pl(\pmb {\theta }_0;\text{ O})-Pl(\widehat{\pmb {\theta }}_n;\text{ O})\mid \\\le & {} \mid Pl(\pmb {\theta }_0;\text{ O}) - P_nl(\pmb {\theta }_0;\text{ O})\mid + \mid P_nl(\widehat{\pmb {\theta }}_n;\text{ O}) - Pl(\widehat{\pmb {\theta }}_n;\text{ O})\mid \rightarrow 0, \quad a.s. \end{aligned}$$

Considering the fact that the Kullback–Leibler information is greater than or equal to the square of the Hellinger metric (Xue et al. 2004), one can obtain that

$$\begin{aligned} \mid Pl(\pmb {\theta }_0;\text{ O})-Pl(\widehat{\pmb {\theta }}_n;\text{ O})\mid= & {} E_{\pmb {\theta }_0} \Bigl \{ l(\pmb {\theta }_0;\text{ O}) - l(\widehat{\pmb {\theta }}_n;\text{ O}) \Bigr \} \\\ge & {} \Big \Vert \sqrt{L(\pmb {\theta }_0;\text{ O})} - \sqrt{L(\widehat{\pmb {\theta }}_n;\text{ O})}\Big \Vert _2^2 \\= & {} \Big \Vert \frac{\nabla _{\pmb {\theta }}L(\check{\pmb {\theta }};\text{ O})}{2\sqrt{L(\check{\pmb {\theta }};\text{ O})} } (\pmb {\theta }_0 - \widehat{\pmb {\theta }}_n) \Big \Vert _2^2, \end{aligned}$$

where $\check{\pmb {\theta }}$ is between $\pmb {\theta }_0$ and $\widehat{\pmb {\theta }}_n$, and the derivative of $L(\pmb {\theta };\text{ O})$ with respect to $\pmb {\theta }$ is $\nabla _{\pmb {\theta }}L(\pmb {\theta };\text{ O})$. Note that $\frac{\nabla _{\pmb {\theta }}L(\check{\pmb {\theta }};\text{ O})}{2\sqrt{L(\check{\pmb {\theta }};\text{ O})}}$ is not equal to zero and bounded. Therefore, $d(\widehat{\pmb {\theta }}_n,\pmb {\theta }_0)\rightarrow 0$ almost surely. Note that, $\parallel \widehat{\pmb {\alpha }}_n-\pmb {\alpha }_0\parallel \le d(\widehat{\pmb {\theta }}_n,\pmb {\theta }_0)$, $\parallel \widehat{\pmb {\beta }}_n-\pmb {\beta }_0\parallel \le d(\widehat{\pmb {\theta }}_n,\pmb {\theta }_0)$ and $\parallel \Lambda _{n}-\widehat{\Lambda }_{0}\parallel _2\le d(\widehat{\pmb {\theta }}_n,\pmb {\theta }_0)$. Then we have

$$\begin{aligned} \parallel \widehat{\pmb {\alpha }}_n-\pmb {\alpha }_0\parallel \rightarrow 0,\quad \parallel \widehat{\pmb {\beta }}_n-\pmb {\beta }_0\parallel \rightarrow 0,\quad \parallel \widehat{\Lambda }_n-\Lambda _0\parallel _2\rightarrow 0,\quad a.s. \end{aligned}$$

$\square$

Proof of Theorem 2 ne can derive the convergence rate of $\widehat{\pmb {\theta }}_n$ by verifying the conditions of Theorem 3.2.5 in Van der Vaart and Wellner (1996). First, by the relationship between the Hellinger distance and the Kullback–Leibler information in the proof of theorem 1, we obtain

$$\begin{aligned} Pl(\pmb {\theta }_0;\text{ O})-Pl(\pmb {\theta };\text{ O})\ge Cd^2(\pmb {\theta }_0,\pmb {\theta }),\quad \pmb {\theta }\in \varvec{\Theta }_n. \end{aligned}$$

Second, note from Theorem 1.6.2 of Lorentz (1986) and the conclusions of Osman and Ghosh (2012), there exists the Bernstein polynomials $\Lambda _{0,n}(\cdot )$ such that $\parallel \Lambda _0(\cdot ) - \Lambda _{0,n}(\cdot ) \parallel _{\infty } = O(N^{-r/2})$ with $r\ge 1$. Thus we obtain $d(\pmb {\theta }_0,\pmb {\theta }_{0,n}) = O(n^{-{r\nu }/2})$, where $\pmb {\theta }_{0,n}=(\pmb {\alpha }_{0},\pmb {\beta }_{0},\Lambda _{0,n}(\cdot ))$.

Then, we further explore $P_nl(\widehat{\pmb {\theta }}_n;\text{ O}) - P_nl(\pmb {\theta }_0;\text{ O})$. In the proof of consistency, we know that

$$\begin{aligned} P_nl(\widehat{\pmb {\theta }}_n;\text{ O}) - P_nl(\pmb {\theta }_0;\text{ O})= & {} (P_n-P)\{l(\pmb {\theta }_{0n};\text{ O}) - l(\pmb {\theta }_0;\text{ O})\}+P\{Pl(\pmb {\theta }_{0n};\text{ O}) \\&\quad - Pl(\pmb {\theta }_0;\text{ O})\} \\= & {} I_{1n}+I_{2n}. \end{aligned}$$

We can construct a set of brackets for the class $\mathcal {L}_2=\{l(\vartheta _{0},\Lambda (\cdot ))-l(\vartheta _{0},\Lambda _{0}(\cdot )): \Lambda (\cdot )\in M_{n}\ \text {and} \parallel \Lambda _0(\cdot ) - \Lambda _{0,n}(\cdot ) \parallel _{\infty } \le Cn^{-r\nu /2}\}$ with the $\epsilon$-bracketing number bounded by $(1/\epsilon )^{C(N+1)}$. This yields a finite value bracketing integral. Hence the class $\mathcal {L}_2$ is P-Donsker. Using similar arguments as those in Zhang et al. (2010), we can obtain $I_{1n}=o_p(n^{-r\nu })$ and $I_{2n}\ge -O(n^{-r\nu })$. Thus, we have $P_nl(\widehat{\pmb {\theta }}_n;\text{ O}) - P_nl(\pmb {\theta }_0;\text{ O})\ge -O_p(n^{-r\nu })=O_{p}(n^{-2\min (r\nu /2),(1-\nu )/2})$.

Let $\mathcal {L}_2(\varsigma )=\big \{l(\pmb {\theta };\text{ O})-l(\pmb {\theta }_0;\text{ O}): \pmb {\theta }\in \varvec{\Theta }_n \text { and } d(\pmb {\theta },\pmb {\theta }_0)\le \varsigma \big \}$. Moreover, one can obtain $P\big \{l(\pmb {\theta };\text{ O})-l(\pmb {\theta }_0;\text{ O})\big \}^2 \le Cd^2(\pmb {\theta },\pmb {\theta }_0)\le C\varsigma ^2$ for any $l(\pmb {\theta };\text{ O})-l(\pmb {\theta }_0;\text{ O})\in \mathcal {L}_2(\varsigma )$. Note that $\mathcal {L}_2(\varsigma )$ is uniformly bounded with conditions (C1)–(C3). Obeying the calculating results in Shen and Wong (1994),(p.597), we can establish that for $0<\epsilon <\varsigma$, the bracketing entropy $\log N_{[\,]}(\epsilon ,\mathcal {L}_2(\varsigma ),L_2(P))$ is bounded by $C\tilde{N}\log {(\varsigma /\epsilon })$ with $\tilde{N}=N+1$, where $N_{[\,]}(\epsilon ,\mathcal {L}_2(\varsigma ),L_2(P))$ is the $\epsilon$-bracketing number of $\mathcal {L}_2(\varsigma )$ presented in Definition 2.1.6 of Van der Vaart and Wellner (1996).

Therefore, according to Lemma 3.4.2 in Van der Vaart and Wellner (1996) that the continuity modulus of the empirical process $\sqrt{n}(P_{n}-P)$ gives an upper bounded on the rate, we obtain that

$$\begin{aligned} E_P\parallel n^{1/2}(P_n-P)\parallel _{\mathcal {L}_2(\varsigma )} \le CJ_{[\,]}\big \{\varsigma ,\mathcal {L}_2(\varsigma ),L_2(P) \big \} \biggl \{ 1 + \frac{J_{[\,]}\{\varsigma ,\mathcal {L}_2(\varsigma ),L_2(P) \}}{n^{1/2}\varsigma ^2} \biggr \}, \end{aligned}$$

where $J_{[\,]}\{\varsigma ,\mathcal {L}_2(\varsigma ),L_2(P) \} = \int _0^{\varsigma }\bigl [ 1 +\log N_{[\,]}\{\epsilon ,\mathcal {L}_2(\varsigma ),L_2(P)\} \bigr ]^{1/2}d\epsilon \le C\tilde{N}^{1/2}\varsigma$. This result implies that the function $\phi _n(\varsigma )$ of Theorem 3.2.5 in Van der Vaart and Wellner (1996) can be given by $\phi _n(\varsigma )=\tilde{N}^{1/2}\varsigma +\tilde{N}/n^{1/2}$ and we know that $\phi _{n}(\varsigma )/\varsigma$ is decreasing in $\varsigma$. Note that

$$\begin{aligned} n^{r\nu }\phi _n(1/n^{r\nu /2})=n^{r\nu }\{\tilde{N}^{1/2}n^{-{r\nu }/2}+\tilde{N}n^{-1/2}\} \le n^{1/2}\{n^{(\nu -1)/2+{r\nu }/2}+n^{\nu -1+r\nu }\}. \end{aligned}$$

If $r\nu /2 \le (1-\nu )/2$, then $n^{r\nu }\phi _n(1/n^{{r\nu }/2})\le n^{1/2}$. Thus $r_n$ is given by $r_n=n^{\min \{r\nu /2,(1-\nu )/2\}}$, which leads to $r_n^2\phi _n(1/r_n)\le n^{1/2}$.

Finally, Combining above results that satisfy the conditions of Theorem 3.2.5 in Van der Vaart and Wellner (1996). We have $r_nd(\widehat{\pmb {\theta }}_n,\pmb {\theta }_0)=O_p(1)$; that is, $d(\widehat{\pmb {\theta }}_n,\pmb {\theta }_0)=O_p(n^{-\min \{r\nu /2,(1-\nu )/2\}})$. The proof of Theorem 2 is completed.

Proof of Theorem 3 For any $v^*\in \varvec{\Theta }_0$, by Theorem 1.6.2 in Lorentz (1986) and the conclusions of Osman and Ghosh (2012), there exists $\pi _n v^*\in \varvec{\Theta }_n$ such that $\parallel \pi _n v^*-v^*\parallel =O(n^{-\frac{r\nu }{2}})$. Note that $\delta _n\parallel \pi _n v^*-v^*\parallel =o(n^{-1/2})$ with $r>1$ and $r\nu >1/2$. Let $\varepsilon _{n}$ be any positive sequence with $\varepsilon _{n}=o(n^{-1/2})$ and define $\rho [\pmb {\theta }-\pmb {\theta }_0;\text{ O}]=l(\pmb {\theta };\text{ O})-l(\pmb {\theta }_0;\text{ O})-\dot{l}(\pmb {\theta }_0;\text{ O})[\pmb {\theta }-\pmb {\theta }_0]$. Then by the $P\{\dot{l}(\widehat{\pmb {\theta }}_{0};\text{ O})[\pi _{n}v^{*}]\}=0$. One can obtain that

$$\begin{aligned} \begin{aligned} 0&\le P_n\Bigl \{l(\widehat{\pmb {\theta }}_n;\text{ O}) - l(\widehat{\pmb {\theta }}_n\pm \varepsilon _{n}\pi _{n}v^*;\text{ O})\Bigr \} \\&= P_n\Bigl \{ [ l(\widehat{\pmb {\theta }}_n;\text{ O}) - l(\pmb {\theta }_{0};\text{ O}) ] - [ l(\widehat{\pmb {\theta }}_{n}\pm \varepsilon _{n}\pi _{n}v^{*};\text{ O}) - l(\pmb {\theta }_0;\text{ O}) ] \Bigr \} \\&= \mp \varepsilon _{n}P_n\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*] + P_n\Bigl \{ \rho [\widehat{\pmb {\theta }}_n-\pmb {\theta }_0;\text{ O}] - \rho [\widehat{\pmb {\theta }}_{n}\pm \varepsilon _{n}\pi _{n}v^{*}-\pmb {\theta }_{0};\text{ O}] \Bigr \} \\&= \mp \varepsilon _{n}P_n\dot{l}(\pmb {\theta }_0;\text{ O})[v^*] \mp \varepsilon _{n}P_n\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _{n}v^{*}-v^{*}]\\&\quad + (P_n-P)\Bigl \{\rho [\widehat{\pmb {\theta }}_{n}-\pmb {\theta }_{0};\text{ O}]-\rho [\widehat{\pmb {\theta }}_n\pm \varepsilon _{n}\pi _nv^*-\pmb {\theta }_0;\text{ O}]\Bigr \}\\&\quad + P\Bigl \{ \rho [\widehat{\pmb {\theta }}_{n}-\pmb {\theta }_{0};\text{ O}] - \rho [\widehat{\pmb {\theta }}_{n}\pm \varepsilon _{n}\pi _nv^{*}-\pmb {\theta }_{0};\text{ O}] \Bigr \} \\&:= \mp \varepsilon _{n}P_{n}\dot{l}(\pmb {\theta }_{0};\text{ O})[v^*] + I_1 + I_2 + I_3. \end{aligned} \end{aligned}$$

Next, we will study the asymptotic properties of $I_1, I_2$ and $I_3$.

For $I_1$, by the Chebyshev’s inequality, $P\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*-v^*]=0$ and $\parallel \pi _nv^*-v^*\parallel =o(1)$, we obtain

$$\begin{aligned} P\left( \frac{\mid P_n\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*-v^*] \mid }{n^{-1/2}} \ge \varepsilon \right)\le & {} \frac{Var(P_n\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*-v^*])}{n^{-1}\varepsilon ^2} \\= & {} \frac{ Var(\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*-v^*])}{\varepsilon ^{2}} \\= & {} \frac{P(\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*-v^*]\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*-v^*])}{\varepsilon ^2} \\= & {} \frac{\parallel \pi _nv^*-v^* \parallel ^2}{\varepsilon ^2} \rightarrow 0, \end{aligned}$$

which indicates that $P_n\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*-v^*]=o_p(n^{-1/2})$. Thus, we have $I_1=\varepsilon _{n}\times o_p(n^{-1/2})$.

For $I_2$, by the mean value theorem, we have

$$\begin{aligned} \begin{aligned} I_2&= (P_n-P)\Bigl \{l(\widehat{\pmb {\theta }}_n;\text{ O}) - l(\widehat{\pmb {\theta }}_n\pm \varepsilon _n\pi _nv^*;\text{ O}) \pm \varepsilon _n\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*] \Bigr \}\\&= (P_n-P)\Bigl \{ \dot{l}(\check{\pmb {\theta }};\text{ O})[\mp \varepsilon _n\pi _nv^*] \pm \varepsilon _n\dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*] \Bigr \}\\&= \mp \varepsilon _n(P_n-P)\Bigl \{ \dot{l}(\check{\pmb {\theta }};\text{ O})[\pi _nv^*] - \dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*] \Bigr \}, \end{aligned} \end{aligned}$$

where $\check{\pmb {\theta }}$ is between $\widehat{\pmb {\theta }}_n$ and $\widehat{\pmb {\theta }}_n\pm \varepsilon _n\pi _nv^*$. Consider the function class $\mathcal {L}_3=\{\dot{l}(\pmb {\theta };\text{ O})[\pi _nv^*] :\pmb {\theta }\in \varvec{\Theta }_n \text { and } \parallel \pmb {\theta }-\pmb {\theta }_0\parallel = O(\delta _n)\}$. For any $\dot{l}(\pmb {\theta }_i;\text{ O})[\pi _nv^*]\in \mathcal {L}_3\ (i=1,2)$, we have $\bigl |\dot{l}(\pmb {\theta }_1;\text{ O})[\pi _nv^*] - \dot{l}(\pmb {\theta }_2;\text{ O})[\pi _nv^*]\bigr |\le C\parallel \pmb {\theta }_1-\pmb {\theta }_2\parallel$. Thus it yields that

$$\begin{aligned} N\big (\epsilon ,\mathcal {L}_3,L_2(Q)\big )\le N\big (\epsilon ,\big \{\pmb {\theta }:\pmb {\theta }\in \varvec{\Theta }_n \text { and } \parallel \pmb {\theta }-\pmb {\theta }_0\parallel \le C\delta _n\big \},\parallel \cdot \parallel \big ). \end{aligned}$$

Similar to the proof of Lemma 1, one can obtain that $N(\epsilon ,\mathcal {L}_3,L_2(Q)) \le (\frac{C\delta _n}{\epsilon })^{N+1}$. Then we have the finiteness of the entropy inequality, i.e. $\int _0^{\infty }\sup _{Q}\sqrt{N(\epsilon ,\mathcal {L}_3,L_2(Q))}d\epsilon < \infty$ with $\nu <1/2$ and $r>1$. Note that $\dot{l}(\pmb {\theta };\text{ O})[\pi _nv^*]$ is uniformly bounded under conditions (C1)–(C3). Hence $\mathcal {L}_3$ is a Donsker class by Theorem 2.8.3 of Van der Vaart and Wellner (1996). Applying the relationship between Donsker and asymptotic equicontinuity given by Corollary 2.3.12 of Van der Vaart and Wellner (1996), one can obtain that

$$\begin{aligned} (P_n-P)\Bigl \{\dot{l}(\tilde{\pmb {\theta }};\text{ O})[\pi _nv^*] - \dot{l}(\pmb {\theta }_0;\text{ O})[\pi _nv^*]\Bigr \}=o_p(n^{-1/2}). \end{aligned}$$

Therefore, we have $I_2=\epsilon _n\times o_p(n^{-1/2})$.

For $I_{3}$, note that $P\big \{\ddot{l}(\pmb {\theta }_0;\text{ O})[\widehat{\pmb {\theta }}_n-\pmb {\theta }_0,\widehat{\pmb {\theta }}_n-\pmb {\theta }_0]\big \}=-P\big \{\dot{l}(\pmb {\theta }_0;\text{ O})[\widehat{\pmb {\theta }}_n-\pmb {\theta }_0]\dot{l}(\pmb {\theta }_0;\text{ O})[\widehat{\pmb {\theta }}_n-\pmb {\theta }_0]\big \}.$ For any $\pmb {\theta }\in \big \{\pmb {\theta }:d(\pmb {\theta },\pmb {\theta }_0)=O(\delta _n)\big \}$, $P\big \{\ddot{l}({\pmb {\theta }};\text{ O})[\pmb {\theta }-\pmb {\theta }_0,\pmb {\theta }-\pmb {\theta }_0] - \ddot{l}(\pmb {\theta }_0;\text{ O})[\pmb {\theta }-\pmb {\theta }_0,\pmb {\theta }-\pmb {\theta }_0]\big \} = O(\delta _n^3)$, $\delta _n^3 = o(n^{-1})$ with $2/3r<\nu <1/3$ and $r>2$. Then we have

$$\begin{aligned} P\big (\rho [\widehat{\pmb {\theta }}_n-\pmb {\theta }_0;\text{ O}]\big )= & {} P\bigl \{l(\widehat{\pmb {\theta }}_n;\text{ O}) - l(\pmb {\theta }_0;\text{ O}) - \dot{l}(\pmb {\theta }_0;\text{ O})[\widehat{\pmb {\theta }}_n-\pmb {\theta }_0]\bigr \}\\= & {} \frac{1}{2}P\bigl \{\ddot{l}(\check{\pmb {\theta }};\text{ O})[\widehat{\pmb {\theta }}_n-\pmb {\theta }_0,\widehat{\pmb {\theta }}_n-\pmb {\theta }_0] - \ddot{l}(\pmb {\theta }_0;\text{ O})[\widehat{\pmb {\theta }}_n-\pmb {\theta }_0,\widehat{\pmb {\theta }}_n-\pmb {\theta }_0]\bigr \}\\&\quad + \frac{1}{2}P\bigl \{\ddot{l}(\pmb {\theta }_0;\text{ O})[\widehat{\pmb {\theta }}_n-\pmb {\theta }_0,\widehat{\pmb {\theta }}_n-\pmb {\theta }_0]\bigr \} \\ \\= & {} \varepsilon _n \times o_p(n^{-1/2}) + \frac{1}{2}P\{\ddot{l}(\pmb {\theta }_0;\text{ O})[\widehat{\pmb {\theta }}_n-\pmb {\theta }_0,\widehat{\pmb {\theta }}_n-\pmb {\theta }_0]\} \\= & {} \varepsilon _n \times o_p(n^{-1/2}) - \frac{1}{2}\parallel \widehat{\pmb {\theta }}_n-\pmb {\theta }_0\parallel ^2, \end{aligned}$$

where $\check{\pmb {\theta }}$ is between $\widehat{\pmb {\theta }}_n$ and $\pmb {\theta }_0$. By the facts $\parallel \pi _nv^*\parallel ^2\rightarrow \parallel v^*\parallel ^2<\infty$, Cauchy-Schwarz inequality, and $\delta _n\parallel \pi _n v^*-v^*\parallel =o(n^{-1/2})$, we have

$$\begin{aligned} I_3= & {} - \frac{1}{2}\parallel \widehat{\pmb {\theta }}_n-\pmb {\theta }_0\parallel ^2 + \frac{1}{2}\parallel \widehat{\pmb {\theta }}_n\pm \varepsilon _n\pi _nv^*-\pmb {\theta }_0\parallel ^2 + \varepsilon _n\times o_p(n^{-1/2})\\= & {} \pm \varepsilon _n\langle \widehat{\pmb {\theta }}_n-\pmb {\theta }_0,\pi _nv^*\rangle +\frac{1}{2}\parallel \varepsilon _n\pi _nv^*\parallel ^2 + \epsilon _n\times o_p(n^{-1/2})\\= & {} \pm \varepsilon _n\langle \widehat{\pmb {\theta }}_n-\pmb {\theta }_0,v^*\rangle +\frac{1}{2}\varepsilon _n^2\parallel \pi _nv^*\parallel ^2 + \varepsilon _n\times o_p(n^{-1/2})\\= & {} \pm \varepsilon _n\langle \widehat{\pmb {\theta }}_n-\pmb {\theta }_0,v^*\rangle + \varepsilon _n\times o_p(n^{-1/2}). \end{aligned}$$

Combining the above results, we can obtain

$$\begin{aligned} 0= & {} \le P_n\bigl \{ l(\widehat{\pmb {\theta }}_n;\text{ O}) - l(\widehat{\pmb {\theta }}_n\pm \varepsilon _n\pi _nv^*;\text{ O}) \bigr \} \\= & {} \mp \varepsilon _{n}P_n\dot{l}(\pmb {\theta }_0;\text{ O})[v^*] \pm \varepsilon _n\langle \widehat{\pmb {\theta }}_n-\pmb {\theta }_0,v^*\rangle + \varepsilon _n\times o_p(n^{-1/2}). \end{aligned}$$

Note that $P\dot{l}(\pmb {\theta }_0;\text{ O})[v^*]=0$ and $Var(\dot{l}(\pmb {\theta }_0;\text{ O})[v^*]) = \Vert v^{*}\Vert ^2$. Therefore, by the Central Limits Theorem we obtain

$$\begin{aligned} \sqrt{n}\langle \widehat{\pmb {\theta }}_n-\pmb {\theta }_0,v^*\rangle = \sqrt{n}(P_n-P)\{\dot{l}(\pmb {\theta }_0;\text{ O})[v^*]\} + o_p(1) \rightarrow N(0,\Vert v^*\Vert ^2) \end{aligned}$$

in distribution. Note that $G(\pmb {\theta })-G(\pmb {\theta }_0)=\dot{G}(\pmb {\theta }_0)[\pmb {\theta }-\pmb {\theta }_0]$. By the Riesz representation theorem, there exists $v^{*}\in \mathcal {\bar{V}} \text { such that } \dot{G}(\pmb {\theta }_0)[v]=\langle v,v^*\rangle$ for any $v\in \mathcal {\bar{V}}$ and $\Vert v^*\Vert = \Vert \dot{G}(\pmb {\theta }_0)\Vert$. Then we have

$$\begin{aligned} \sqrt{n}\big \{G(\widehat{\pmb {\theta }}_n)-G(\pmb {\theta }_0)\big \}=\sqrt{n}\langle \widehat{\pmb {\theta }}_n-\pmb {\theta }_0,v^*\rangle +o_{p}(1)\rightarrow N(0,\Vert \dot{G}(\pmb {\theta }_0)\Vert ^{2}) \end{aligned}$$

in distribution, namely, $\sqrt{n} \Bigl \{\text{ b}_{1}^{\intercal }(\widehat{\pmb {\alpha }}_{n}-\pmb {\alpha }_{0})+\text{ b}_{2}^{\intercal }(\widehat{\pmb {\beta }}_{n}-\pmb {\beta }_{0})+ \int _0^{\tau } b_3(t)\big (\widehat{\Lambda }_n(t)-\Lambda _0(t)\big )dt \Bigr \}\xrightarrow {L} N(0,\Vert \dot{G}(\pmb {\theta }_0)\Vert ^2)$. Adopting Theorem 4 in Shen (1997) or the conclusions of Bickel and Kwon (2001), we can establish the semiparametric efficiency of the estimators. The proof is completed.

The calculation of $I(\widehat{\pmb {\theta }})$

One can obtain $Q(\widetilde{\pmb {\theta }}; \widehat{\pmb {\theta }})$ with respect to $\widetilde{\pmb {\theta }}$. Then, the quantities of the first part in $I(\widehat{\pmb {\theta }})$ are given by

$$\begin{aligned} \frac{\partial ^2Q(\widetilde{\pmb {\theta }};\widehat{\pmb {\theta }})}{\partial \alpha _{k}\partial \alpha _{k}^{'}}&= \sum _{i=1}^n Z_{ik}Z_{ik}^{'}\widehat{\pi }(\text{ Z}_{i})\Big \{1-\widehat{\pi }(\text{ Z}_{i}) \Big \} , \\ \frac{\partial ^2Q(\widetilde{\pmb {\theta }};\widehat{\pmb {\theta }})}{\partial \alpha _{k}\partial \eta _{j}^{'}}&=\frac{\partial ^2Q(\widetilde{\pmb {\theta }};\widehat{\pmb {\theta }})}{\partial \eta _{j}\partial \alpha _{k}^{'}} = 0, \\ \frac{\partial ^2Q(\widetilde{\pmb {\theta }};\widehat{\pmb {\theta }})}{\partial \eta _j\partial \eta _{j'}}&= -\eta _j^{-2}\sum _{i=1}^n \Big \{\delta _{1i} E(Y_{ij}|\mathcal {O},\widehat{\pmb {\theta }})\\&\quad +\delta _{2i} E(W_{ij}|\mathcal {O},\widehat{\pmb {\theta }})\Big \}I_{(j'=j)}. \end{aligned}$$

The second part of the $I(\widehat{\pmb {\theta }})$ derived from Eq. (6) is listed as follows

$$\begin{aligned}&Cov\left( \frac{\partial \log {L_c(\widetilde{\pmb {\theta }})}}{\partial \alpha _{k}}, \frac{\partial \log {L_c(\widetilde{\pmb {\theta }})}}{\partial \alpha _{k}^{'}}\right) = \sum _{i=1}^n Z_{ik}Z_{ik}^{'} Var(U_{i}), \\&Cov\left( \frac{\partial \log {L_c(\widetilde{\pmb {\theta }})}}{\partial \alpha _{k}}, \frac{\partial \log {L_c(\widetilde{\pmb {\theta }})}}{\partial \eta _{j}}\right) = - \sum _{i=1}^n Z_{ik}\delta _{3i}\xi _{j}(L_{i}) Var(U_{i}), \\&Cov\left( \frac{\partial \log {L_c(\widetilde{\pmb {\theta }})}}{\partial \eta _j}, \frac{\partial \log {L_c(\widetilde{\pmb {\theta }})}}{\partial \eta _{j'}}\right) = (\eta _j\eta _{j'})^{-1} \sum _{i=1}^n \Big \{\delta _{1i} Cov(Y_{ij},Y_{i{j'}})\\&\quad +\delta _{2i} Cov(Y_{ij},Y_{i{j'}})\Big \}, \end{aligned}$$

where

$$\begin{aligned} Var(U_i)= & {} E(U_{i}^{2})-\{E(U_{i})\}^{2}= E(U_{i})\{1-E(U_{i})\}, \\ Cov(Y_{ij},Y_{ij'})= & {} \frac{\delta _{1i}\widehat{\lambda }_{ij}}{\widehat{c}_{i}^{2}}(\widehat{c}_{i}-\widehat{\lambda }_{ij}+\widehat{c}_{i}\widehat{\lambda }_{ij}), \\ Cov(W_{ij},W_{i{j'}})= & {} \frac{\delta _{2i}\widehat{\omega }_{ij}}{\widehat{d}_{i}^{2}}(\widehat{d}_{i}-\widehat{\omega }_{ij}+\widehat{d}_{i}\widehat{\omega }_{ij}). \end{aligned}$$

Denote $\widehat{c}_{i}=1-\exp {(-\widehat{\lambda }_{i})}$, $\widehat{d}_{i}=1-\exp {(-\widehat{\omega }_{i})}$, then we can obtain $I(\widehat{\pmb {\theta }})$ by Eq. (11) and these closed forms make the estimation of variance-covariance matrix easy to be calculated. $\square$