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A flexible semiparametric transformation model for recurrent event data

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Abstract

In this article, we propose a class of semiparametric transformation models for recurrent event data, in which the baseline mean function is allowed to depend on covariates through an additive model, and some covariate effects are allowed to be time-varying. For inference on the model parameters, estimating equation approaches are developed, and the asymptotic properties of the resulting estimators are established. In addition, a lack-of-fit test is presented to assess the adequacy of the model. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer study is illustrated.

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Acknowledgments

The authors thank the Editor, Professor Mei-Ling Ting Lee, an Associate Editor, and two referees for their insightful comments and suggestions that greatly improved the article. The second author’ research was partly supported by the National Natural Science Foundation of China Grants (No. 11231010, 11171330 and 11021161) and Key Laboratory of RCSDS, CAS (No.2008DP173182).

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Correspondence to Liuquan Sun.

Appendix: Proofs of asymptotic results

Appendix: Proofs of asymptotic results

We will use the same notation defined in the text. Define

$$\begin{aligned} E_{xx}(t)&= n^{-1}\sum _{i=1}^n Y_i(t)\dot{G}\Big \{ X_i(t)'{A}_0(t)e^{{\beta }_0'Z_i(t)} \Big \}e^{{\beta }_0'Z_i(t)}X_i(t)X_i(t)',\\ E_{zx}(t)&= n^{-1}\sum _{i=1}^n Y_i(t)\dot{G}\Big \{X_i(t)'{A}_0(t)e^{{\beta }_0'Z_i(t)}\Big \} e^{{\beta }_0'Z_i(t)}Z_i(t) X_i(t)',\\ E_{xzz}(t)&= n^{-1}\sum _{i=1}^n Y_i(t)\dot{G}\Big \{X_i(t)'{A}_0(t) e^{{\beta }_0'Z_i(t)}\Big \} e^{{\beta }_0'Z_i(t)}X_i(t)' {A}_0(t)Z_i(t)Z_i(t)', \end{aligned}$$

and

$$\begin{aligned} E_{xxz}(t)=n^{-1}\sum _{i=1}^n Y_i(t)\dot{G}\Big \{X_i(t)'{A}_0(t) e^{{\beta }_0'Z_i(t)} \Big \}e^{{\beta }_0'Z_i(t)}X_i(t)' {A}_0(t)X_i(t)Z_i(t)'. \end{aligned}$$

Let \(e_{xx}(t),\, e_{zx}(t),\,e_{xzz}(t)\) and \(e_{xxz}(t)\) denote their limits, respectively. In order to study the asymptotic properties of the proposed estimators, we need the following regularity conditions:

  1. (C1)

    \(P(C \ge \tau )>0\), and \(N(\tau )\) is bounded almost surely.

  2. (C2)

    The covariates \(X(t)\) and \(Z(t)\) are of bounded variation on \([0,\tau ]\).

  3. (C3)

    The weight function \(H(t)\) converges almost surely to a nonrandom and bounded function \(\tilde{H}(t)\) uniformly in \(t\in [0, \tau ]\).

  4. (C4)

    \(A_0(t)\) is right continuous with left-hand limits, and has bounded total variation on \([0, \tau ]\).

  5. (C5)

    \(E[X(t)X(t)']\) is nonsingular uniformly in \(t\in [0, \tau ]\).

  6. (C6)

    The matrix \(B=\int _0^{\tau }[e_{xzz}(t)-e_{zx}(t)e_{xx}(t)^{-1}e_{xxz}(t)]d\tilde{H}(t)\) is nonsingular.

Conditions (C1)–(C4) are standard for regression methods in analyzing recurrent event data (Lin et al. 2001). Condition (C5) is a technical assumption for the existence and uniqueness of the estimator \( \hat{A}(t)\). Note that \(B\) is the limit of \(-n^{-1}\partial U(\beta _0)/ \partial \beta '\). Condition (C6) is needed for the existence and uniqueness of the estimator \(\hat{\beta }\). As discussed in Scheike (2006), the matrix \(B\) is complicated, and it is difficult to find when Condition (C6) is fulfilled. However, when \(X(t) \equiv 1\), model (1) reduces to the transformation model (Lin et al. 2001), and Condition (C6) is fulfilled under the conditions given by Lin et al. (2001) in the first paragraph of their appendix. Also Condition (C6) is fulfilled for the models in the simulation studies, since the algorithm always converges for the situations considered here. In addition, a necessary condition for Condition (C6) to be satisfied is that \(X(t)\) is not proportional to \(Z(t)\).

Proof of Theorem 1

Let \(l^{\infty }[0,\tau ]\) be the set of functions of bounded variation on \([0,\tau ]\), and \(\mathcal{B}=\{\beta : \Vert \beta -\beta _0\Vert \le \varepsilon \}\) for any \(\varepsilon >0\), where \(\Vert \cdot \Vert \) is the Euclidean norm. Define

$$\begin{aligned} U_1(t; A,\beta )=\sum _{i=1}^nY_i(t)X_i(t)\Big [N_i(t)-G\Big \{X_i(t)'A(t)e^{\beta 'Z_i(t)}\Big \}\Big ], \end{aligned}$$

and

$$\begin{aligned} u_1(t; A,\beta )=E\Big (Y_i(t)X_i(t)\Big [N_i(t)-G\Big \{X_i(t)'A(t)e^{\beta 'Z_i(t)}\Big \}\Big ]\Big ). \end{aligned}$$

For given \(\beta \in \mathcal{B}\) and \(A\in ({ l}^{\infty }[0,\tau ])^{p}\), using the Taylor expansion, we have

$$\begin{aligned} u_1(t; A, \beta )=Q(t;\beta )\Big \{A(t)-A_0(t)\Big \}+u_1(t; A_0,\beta ), \end{aligned}$$
(9)

where

$$\begin{aligned} Q(t; \beta )=-E\Big [Y_i(t)\dot{G}\Big \{X_i(t)'A^{*}(t)e^{\beta 'Z_i(t)}\Big \} e^{\beta 'Z_i(t)}X_i(t)X_i(t)'\Big ], \end{aligned}$$

and \(A^{*}(t)\) is on the line segment between \(A(t)\) and \(A_0(t)\). Note that \(E\{X_i(t)X_i(t)'\}\) is nonsingular uniformly in \(t\in [0, \tau ]\) by condition (C5), and \(G\) is strictly increasing. Then it can be checked that \(e_{xx}(t)\) is nonsingular uniformly in \(t\in [0, \tau ],\,Q(t; \beta )\) is nonsingular for given \(\beta \) and all \(t\), and \(\sup _{0 \le t \le \tau }\Vert u_1(t; A_0,\beta )\Vert \le d_1\) for some positive constant \(d_1\). Thus, it follows from (9) that for any \(t\in [0,\tau ]\) and \(\beta \in \mathcal{B}\), there exist a unique \(A(t;\beta )\) that satisfies \(u_1(t; A(t;\beta ),\beta )=0\) and \(A(t;\beta _0)=A_0(t)\). Write

$$\begin{aligned} n^{-1}U_1(t; A, \beta )&= -n^{-1}\sum _{i=1}^nY_i(t)X_i(t)\Big [G\Big \{ X_i(t)'A(t) e^{Z_i(t)'\beta }\Big \}\nonumber \\&-G\Big \{X_i(t)'A_0(t) e^{Z_i(t)'\beta }\Big \}\Big ] \nonumber \\&+\,n^{-1}U_1(t; A_0,\beta ) \nonumber \\&= Q_n(t;\beta )\Big \{A(t)-A_0(t)\Big \}+n^{-1}U_1(t; A_0,\beta ), \end{aligned}$$
(10)

where

$$\begin{aligned} Q_n(t;\beta )=-n^{-1}\sum _{i=1}^nY_i(t)\dot{G}\Big \{X_i(t)'A^{**}(t) e^{\beta 'Z_i(t)}\Big \}e^{\beta 'Z_i(t)}X_i(t)X_i(t)', \end{aligned}$$

and \(A^{**}(t)\) is also on the line segment between \(A(t)\) and \(A_0(t)\). Since any function of bounded variation can be written as the difference of two increasing functions, the processes \(Q_n(t;\beta )\) and \(U_1(t; A_0,\beta )\) can be written as sums or products of monotone functions in \(t\) and all components of \(\beta \). Thus they are manageable (Pollard 1990, p. 38). Using the uniform strong law of large numbers (Pollard 1990, p. 41), it follows that \(Q_n(t;\beta )\) and \(n^{-1}U_1(t; A_0,\beta )\) converge almost surely to nonrandom functions \(Q^{*}(t; \beta )\) and \(u_1(t;A_0,\beta )\) uniformly in \(t\in [0,\tau ]\) and \(\beta \in \mathcal{B}\). Therefore, the nonsingularity of \(Q^{*}(t; \beta )\) and the boundness of \(u_1(t; A_0,\beta )\) ensure that, for given \(\beta ,\,t\in [0,\tau ]\) and all large \(n,\,Q_n(t;\beta )\) is nonsingular and \(\sup _{0 \le t \le \tau }\Vert n^{-1}U_1(t; A_0,\beta )\Vert \le d_2\) for some positive constant \(d_2\). Then by (10), there exist a unique \(\hat{A}(t;\beta )\) that satisfies \(U_1(t;\hat{A}(t;\beta ),\beta )=0\) for any \(t\in [0,\tau ]\) and \(\beta \in \mathcal{B}\). Hence to prove the existence and uniqueness of \(\hat{\beta }\) and \(\hat{A}(t)\), it suffices to show that there exists a unique solution to \(U(\beta )=0\), where \(U(\beta )\) is defined in (3).

Let \( E^{*}_{xx}(t; \beta ),\,E^{*}_{zx}(t; \beta ),\,E^{*}_{xzz}(t; \beta )\) and \(E^{*}_{xxz}(t; \beta )\) be defined as \( E_{xx}(t),\,E_{zx}(t), E_{xzz}(t)\) and \(E_{xxz}(t)\) with \(A_0(t)\) and \(\beta _0\) replaced by \(\hat{A}(t;\beta )\) and \(\beta \), respectively. Differentiation of \(U_1(t; \hat{A}(t; \beta ), \beta )\) with respect to \(\beta '\) gives

$$\begin{aligned}&\sum _{i=1}^nY_i(t)X_i(t)\dot{G}\Big \{X_i(t)'\hat{A}(t;\beta )e^{\beta 'Z_i(t)}\Big \} e^{\beta 'Z_i(t)}\\&\quad \times \left[ X_i(t)'\frac{\partial \hat{A}(t;\beta )}{\partial \beta '} + X_i(t)'\hat{A}(t;\beta )Z_i(t)'\right] =0, \end{aligned}$$

which yields

$$\begin{aligned} \partial \hat{A}(t;\beta )/ \partial \beta '=-E^{*}_{xx}(t; \beta )^{-1}E^{*}_{xxz}(t; \beta ). \end{aligned}$$
(11)

Let \(\hat{B}(\beta )=-n^{-1}\partial U(\beta )/ \partial \beta '\). It follows from (11) that

$$\begin{aligned} \hat{B}(\beta )=\int \limits _0^{\tau }\Big [E^{*}_{xzz}(t; \beta ) -E^{*}_{zx}(t; \beta )E^{*}_{xx}(t; \beta )^{-1}E^{*}_{xxz}(t; \beta )\Big ]dH(t). \end{aligned}$$

Note that

$$\begin{aligned} n^{-1}U_1(t; \hat{A}(t;\beta ),\beta )&= n^{-1}\Big [U_1(t; \hat{A}(t;\beta ),\beta )-U_1(t; A(t;\beta ),\beta )\Big ] \\&+\,n^{-1}U_1(t,A(t;\beta ),\beta ), \end{aligned}$$

and by the uniform strong law of large numbers, we obtain that almost surely uniformly in \(t\in [0,\tau ]\) and \(\beta \in \mathcal{B},\)

$$\begin{aligned} n^{-1}U_1\Big (t ; A(t;\beta ),\beta ) \rightarrow u_1(t; A(t;\beta ),\beta \Big ) = 0. \end{aligned}$$

Then it follows that

$$\begin{aligned} 0=\Vert n^{-1}U_1\Big (t; \hat{A}(t;\beta ),\beta \Big )\Vert \ge c \Vert \hat{A}(t;\beta )-A(t;\beta )\Vert -\varepsilon _n, \end{aligned}$$
(12)

where \(c>0\) which does not depend on \(t\), and

$$\begin{aligned} \varepsilon _n=\sup _{0\le t \le \tau }\Vert n^{-1}U_1\Big (t; A(t;\beta ),\beta \Big )\Vert \rightarrow 0. \end{aligned}$$

Hence (12) implies that \(\hat{A}(t;\beta )\) converges almost surely to \(A(t;\beta )\) in \(t\in [0,\tau ]\) and \(\beta \in \mathcal{B}\). Similarly, \(E^{*}_{xx}(t;\beta ),\,E^{*}_{zx}(t;\beta ),\,E^{*}_{xxz}(t;\beta )\) and \(E^{*}_{xzz}(t;\beta )\) converge almost surely to nonrandom function \(e^{*}_{xx}(t;\beta ),\,e^{*}_{zx}(t;\beta ),\,e^{*}_{xxz}(t;\beta )\) and \(e^{*}_{xzz}(t;\beta )\) uniformly in \(t\) and \(\beta \). Thus, \(\hat{B}(\beta )\) converges almost surely to a nonrandom function \(B(\beta )\) uniformly in \(\beta \), and it is obvious that \(B(\beta _0)=B\), where

$$\begin{aligned} B(\beta )=\int \limits _0^{\tau }\Big [e^{*}_{xzz}(t;\beta )- e^{*}_{zx}(t;\beta )e^{*}_{xx}(t;\beta )^{-1}e^{*}_{xxz}(t;\beta )\Big ]d\tilde{H}(t). \end{aligned}$$

It can also be checked that \( n^{-1}U(\beta _0)\rightarrow 0 \) almost surely, and \(B\) is nonsingular by condition (C6). Therefore, the uniform convergence of \(\hat{B}(\beta )\) and the continuity of \(B(\beta )\) imply that for all large \(n\), there exists a small neighborhood of \(\beta _0\) in which \(\hat{B}(\beta )\) is nonsingular. Hence it follows from the inverse function theorem (Rudin 1976, p. 221) that within a small neighborhood of \(\beta _0\), there exists a unique solution \(\hat{\beta }\) to \(U(\beta )=0\) for all large \(n\). Since this neighborhood of \(\beta _0\) can be arbitrarily small, the preceding proof also implies that \(\hat{\beta }\) is strongly consistent. It then follows from the uniform convergence of \(\hat{A}(t; \beta )\) to \(A(t; \beta )\) that \( \hat{A}(t)\equiv \hat{A}(t; \hat{\beta }) \rightarrow A(t; \beta _0)\equiv A_0(t)\) almost surely uniformly in \(t\in [0, \tau ]\). \(\square \)

Proof of Theorem 2

  1. (i)

    Taking the linear expansion of \(G\{X_i(t)'\hat{A}(t;\beta _0) e^{\beta _0'Z_i(t)} \}\) in \(U(\beta _0)\) at \(A_0(t)\), we obtain

    $$\begin{aligned} U(\beta _0)&= \sum _{i=1}^n\int \limits _0^{\tau }\Big [M_i(t)-Y_i(t)\dot{G}\Big \{X_i(t)'\hat{A}^{*}(t) e^{\beta _0'Z_i(t)}\Big \} e^{\beta _0'Z_i(t)}X_i(t)'\nonumber \\&\times \Big \{\hat{A}(t;\beta _0)-A_0(t) \Big \}\Big ]Z_i(t)dH(t), \end{aligned}$$
    (13)

    where \(\hat{A}^{*}(t)\) lies on the line segment between \(\hat{A}(t; \beta _0)\) and \(A_0(t)\). In a similar manner, the linear expansion of \(U_1(t; \hat{A}(t;\beta ),\beta )\) with \(\beta =\beta _0\) gives

    $$\begin{aligned} \hat{A}(t;\beta _0)-A_0(t)= \tilde{E}_{xx}(t)^{-1} \left[ n^{-1}\sum _{i=1}^n X_i(t)M_i(t)\right] , \end{aligned}$$
    (14)

    where

    $$\begin{aligned} \tilde{E}_{xx}(t)= n^{-1}\sum _{i=1}^n Y_i(t) \dot{G}\Big \{X_i(t)'\hat{A}^{**}(t) e^{{\beta _0}'Z_i(t)} \Big \}e^{{\beta _0}'Z_i(t)}X_i(t)X_i(t)', \end{aligned}$$

    and \(\hat{A}^{**}(t)\) also lies on the line segment between \(\hat{A}(t; \beta _0)\) and \(A_0(t)\). It follows from (13) and (14) that

    $$\begin{aligned} U(\beta _0)=\sum _{i=1}^n \int \limits _0^{\tau } M_i(t) \Big [ Z_i(t) -\tilde{E}_{zx}(t)\tilde{E}_{xx}(t)^{-1} X_i(t)\Big ] dH(t), \end{aligned}$$
    (15)

    where

    $$\begin{aligned} \tilde{E}_{zx}(t)=n^{-1}\sum _{i=1}^n Y_i(t)\dot{G}\Big \{X_i(t)'\hat{A}^{ *}(t) e^{{\beta _0}'Z_i(t)} \Big \}e^{{\beta _0}'Z_i(t)}Z_i(t)X_i(t)'. \end{aligned}$$

    Using the uniform consistency of \(\hat{A}(t; \beta _0)\), and the uniform strong law of large numbers, we obtain that \(\tilde{E}_{xx}(t)\) and \(\tilde{E}_{zx}(t)\) converge almost surely to \(e_{xx}(t)\) and \(e_{zx}(t)\) uniformly in \(t \in [0, \tau ]\), and \(e_{xx}(t)\) is nonsingular uniformly in \(t \in [0,\tau ]\), where \(e_{xx}(t)=e_{xx}^*(t; \beta _0)\) and \(e_{zx}(t)=e_{zx}^*(t; \beta _0)\). Note that \(\sum _{i=1}^n {M}_i(t)=O_p(n^{1/2})\) uniformly in \(t \in [0, \tau ]\) by the functional central limit theorem (Pollard 1990, p. 53). Then uniformly in \(t \in [0, \tau ],\,\)

    $$\begin{aligned} \sum _{i=1}^n {M}_i(t) \Big [\tilde{E}_{zx}(t)\tilde{E}_{xx}(t)^{-1} -e_{zx}(t)e_{xx}(t)^{-1}\Big ]X_i(t) =o_p(n^{1/2}). \end{aligned}$$
    (16)

    Similarly,

    $$\begin{aligned} \Big |\sum _{i=1}^n\int \limits _0^{\tau }{M}_i(t)\Big \{Z_i(t)-e_{zx}(t)e_{xx}(t)^{-1}X_i(t) \Big \} d\Big [{H}-\tilde{H}\Big ](t)\Big |=o_p(n^{1/2}).\quad \quad \end{aligned}$$
    (17)

    Therefore, it follows from (15), (16) and (17) that

    $$\begin{aligned} U(\beta _0)&= \sum _{i=1}^n \int \limits _0^{\tau }{M}_i(t) \Big [Z_i(t)-e_{zx}(t)e_{xx}(t)^{-1} X_i(t)\Big ] d \tilde{H}(t) \nonumber \\&+\sum _{i=1}^n \int \limits _0^{\tau } {M}_i(t) \Big [Z_i(t)-e_{zx}(t)e_{xx}(t)^{-1} X_i(t) \Big ] d[{H}-\tilde{H}](t) \nonumber \\&-\sum _{i=1}^n \int \limits _0^{\tau }{M}_i(t) \Big [\tilde{E}_{zx}(t)\tilde{E}_{xx}(t)^{-1}-e_{zx}(t)e_{xx}(t)^{-1} \Big ]X_i(t)dH(t)\nonumber \\&= \sum _{i=1}^n \xi _i + o_p(n^{1/2}), \end{aligned}$$
    (18)

    where

    $$\begin{aligned} \xi _i=\int \limits _0^{\tau } {M}_i(t)\Big [Z_i(t)- e_{zx}(t)e_{xx}(t)^{-1}X_i(t)\Big ] d\tilde{H}(t). \end{aligned}$$

    By the multivariate central limit theorem, \(n^{-1/2}U(\beta _0)\) is asymptotically normal with mean zero and covariance matrix \(\Sigma = E\{\xi _i ^{\otimes 2}\}\). By the Taylor expansion of \(U(\beta )\) at \(\beta _0\),

    $$\begin{aligned} 0=U(\hat{\beta })=U(\beta _0)-n\hat{B}(\beta ^{*})(\hat{\beta }-\beta _0), \end{aligned}$$

    where \(\beta ^{*}\) is on the line segment between \(\hat{\beta }\) and \(\beta _0\). It then follows from the uniform convergence of \(\hat{B}(\beta )\) and the consistency of \(\hat{\beta }\) that

    $$\begin{aligned} n^{1/2}(\hat{\beta }-\beta _0)=B^{-1}n^{-1/2}U( \beta _0)+o_p(1)=B^{-1}n^{-1/2}\sum _{i=1}^n\xi _i+o_p(1).\qquad \end{aligned}$$
    (19)

    Thus, it follows that \(n^{1/2}(\hat{\beta }-\beta _0)\) is asymptotically zero-mean normal with covariance matrix \(V=B^{-1}\Sigma B^{-1}\), which can be consistently estimated by \(\hat{B}^{-1}\hat{\Sigma }\hat{B}^{-1}\) as defined in Theorem 2(i).

  2. (ii)

    Note that

    $$\begin{aligned} n^{1/2}\Big \{ \hat{A}(t)-A_0(t)\Big \}&= n^{1/2}\Big \{ \hat{A}(t; \beta _0)-A_0(t) \Big \} +n^{1/2}\Big \{ \hat{A}(t; \hat{\beta }) -\hat{A}(t;\beta _0) \Big \} \\&= n^{1/2}\Big \{ \hat{A}(t; \beta _0)-A_0(t) \Big \} - e_{xx}^{-1}e_{xxz}n^{1/2}(\hat{\beta }-\beta _0)+o_p(1), \end{aligned}$$

    where the second equality follows from the Taylor expansion of \(\hat{A}(t; \hat{\beta })\), together with (11) and the convergence of \(E^{*}_{xx}(t; \beta )\) and \(E^{*}_{xxz}(t; \beta )\). Therefore, it follows from (14) and (19) that uniformly in \(t\in [0,\tau ]\),

    $$\begin{aligned} n^{1/2}\Big \{ \hat{A}(t)-A_0(t) \Big \}=n^{-1/2}\sum _{i=1}^n \Phi _i(t) +o_p(1), \end{aligned}$$
    (20)

    where

    $$\begin{aligned} \Phi _i(t)=e_{xx}(t)^{-1}X_i(t)M_i(t)-e_{xx}(t)^{-1}e_{xxz}(t)B^{-1}\xi _i. \end{aligned}$$

    Since \(\Phi _i(t)\,(i=1,\ldots ,n)\) are independent zero-mean random variables for each \(t\), the multivariate central limit theorem implies that \(n^{1/2}\{ \hat{A}(t)-A_0(t) \}\,(0 \le t \le \tau )\) converges in finite-dimensional distribution to a zero-mean Gaussian process. It follows from the functional central limit theorem that the first term of \( \Phi _i(t)\) is tight. The second term is tight because \(n^{-1/2}\sum _{i=1}^n \xi _i\) converges in distribution, and \(e_{xx}(t)\) and \(e_{xxz}(t)\) are deterministic functions. Thus, \(n^{1/2}\{ \hat{A}(t)-A_0(t) \} \) is tight and converges weakly to a zero-mean Gaussian process with covariance function \(\Gamma (s,t)=E\{ \Phi _i(s)\Phi _i(t) \} \) at \((s,t)\), which can be consistently estimated by \(\hat{\Gamma }(s,t)\) defined in Theorem 2(ii).\(\square \)

Proof of (7) in Section 4

Taking the linear expansion of \(G(\cdot )\) and by an argument similar to that in the proof of Theorem 2, we have that uniformly in \(t \in [0, \tau ],\,\)

$$\begin{aligned} \mathcal{F}(t, x, z)&= n^{-1/2} \sum _{i=1}^n I\Big (X_i \le x, Z_i \le z\Big ){M}_i(t) -\Upsilon _1(t, x, z)'n^{1/2}(\hat{\beta }-\beta _0)\\&- \Upsilon _2(t, x,z)'n^{1/2}\Big \{\hat{A}(t)-A_0(t)\Big \}+o_p(1), \end{aligned}$$

where \(\Upsilon _1(t, x,z)\) and \(\Upsilon _2(t, x,z)\) are the limits of \(\hat{\Upsilon }_1(t, x, z)\) and \(\hat{\Upsilon }_2(t, x, z)\), respectively. Hence it follows from (19) and (20) that

$$\begin{aligned} \mathcal{F}(t, x, z)&= n^{-1/2}\sum _{i=1}^n \Big [ I\Big (X_i(t) \le x, Z_i(t) \le z\Big ) M_i(t)-{\Upsilon }_1(t, x, z)'B^{-1}\hat{\xi }_i\\&-{\Upsilon }_2(t, x, z)'{\Phi }_i(t)\Big ]+o_p(1), \end{aligned}$$

which is a sum of i.i.d. zero-mean terms for fixed \(t\). By utilizing the multivariate central limit theorem, \(\mathcal{F}(t, x, z)\) converges in finite-dimensional distributions to a zero-mean Gaussian process. By the same argument as for the tightness of \(n^{1/2}\{\hat{A}(t)-A_0(t)\},\,\mathcal{F}(t, x, z)\) is tight. Thus, \(\mathcal{F}(t, x, z)\) converges weakly to a zero-mean Gaussian process which can be approximated by the zero-mean Gaussian process \( \tilde{\mathcal{F}}(t, x, z)\) given by (7). \(\square \)

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Dong, L., Sun, L. A flexible semiparametric transformation model for recurrent event data. Lifetime Data Anal 21, 20–41 (2015). https://doi.org/10.1007/s10985-013-9285-1

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