Appendix
The following conditions are required for proving the theorems.
-
1.
The observation time points obey a random design in the sense that \(t_i (i=1, \ldots , n)\) are chosen independently from an unknown distribution \(F(\cdot )\) with a density \(f(\cdot )\) on the finit interval. The density function \(f(\cdot )\) of \(t\) is bounded away from zero and continuous on its support set (a,b) with \(-\infty <a<b<\infty \).
-
2.
\(E |Y|^{4}<\infty \) and \(E |X|^{4}<\infty \).
-
3.
The continuous kernel function \(K(\cdot )\) satisfies the following properties:
-
a)
the support of \(K( \cdot )\) is the interval \([-1, 1]\);
-
b)
\(K( \cdot )\) is symmetric about 0;
-
c)
\(\int _{-1}^1 K(u) du =1, \, \, \text{ and} \, \, \int _{-1}^1 |u|K(u)du\ne 0 \).
-
4.
As \(n \rightarrow \infty ,\, \sqrt{n} h^2 \rightarrow 0\) and \( \sqrt{n} { h} \rightarrow \infty \).
-
5.
\(E (\varepsilon ^2| X=x, T=t ) \le c_1\) for some \(c_1\) and all \(x\) and \(t\).
-
6.
\(\beta _{r}(\cdot ,\theta _r)\) is continuously differentiable with respect to \(\theta _r\) in the interior set of \(\Theta _r\) for \(r=0,\ldots ,k\).
Remark 1
For the conditions imposed, we can see that Condition 1) is a typical condition for avoiding the boundary effect when nonparametric smoothing is applied. As to the design of observation time points, see also Huang et al. (2002) for such an assumption. Conditions 2) and 3) are also typical conditions: see Härdle and Mammen (1993). Condition 4) ensures the convergence of the test statistics. This condition means that undersmoothing is applied.
Proof of Theorem 1
Firstly, we prove the asymptotical properties of \(R_{n1}(t)\) and \(T_{n1}\). For \(R_{n1}(t)\), we have
$$\begin{aligned} R_{n1}(t)&= \frac{1}{\sqrt{n}}\sum _{i=1}^n (\hat{Z}_{i1} -Z_i)I(t_i\le t)-\frac{1}{\sqrt{n}}\sum _{i=1}^n (\beta (t_i,\hat{\theta })-\beta (t_i,\theta )) I(t_i\le t)\nonumber \\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n (Z_i-\beta (t_i,\theta )) I(t_i\le t)\nonumber\\&=:I_{n1}-I_{n2}+I_{n3}. \end{aligned}$$
(A.1)
Below we deal with \(I_{n1}\) in (A.1). Noting that
$$\begin{aligned}&\max _{a \le t_i\le b}|\hat{S}^{-1}(t_i)-S^{-1}(t_i)|=O_p(1/\sqrt{nh}+h)\\&\max _{a \le t_i\le b}|\hat{\beta }(t_i)-\beta (t_i)|=O_p(1/\sqrt{nh}+h), \end{aligned}$$
for any subset \((a_1,b_1)\) with \(a<a_1<b_1<b\), see also Zhu and Ng (2003), we have
$$\begin{aligned} \hat{Z}_{i1}-Z_i&= \hat{S}^{-1}(t_i)x_i\hat{y}_{i1}-S^{-1}(t_i)x_i x^\tau _i\beta (t_i) \\&= \hat{S}^{-1}(t_i)x_i(\hat{y}_{i1}-x^\tau _i\hat{\beta }(t_i)) +\hat{S}^{-1}(t_i)x_ix^\tau _i(\hat{\beta }(t_i)-\beta (t_i))\\&+(\hat{S}^{-1}(t_i)-S^{-1}(t_i))x_ix^\tau _i\beta (t_i)\\&= \delta _i(\hat{S}^{-1}(t_i)-S^{-1}(t_i))x_i\epsilon _i+ (1-\delta _i)S^{-1}(t_i)x_ix^\tau _i(\hat{\beta }(t_i)-\beta (t_i))\\& +\delta _iS^{-1}(t_i)x_i\epsilon _i+(\hat{S}^{-1}(t_i)-S^{-1} (t_i))x_ix^\tau _i\beta (t_i)+O_p\left(\frac{1}{nh}+h^2\right). \end{aligned}$$
(A.2)
As a result, we have
$$\begin{aligned} I_{n1}&= \frac{1}{\sqrt{n}}\sum _{i=1}^n \delta _i(\hat{S}^{-1}(t_i)-S^{-1}(t_i)) x_i\epsilon _i I(t_i\le t)\\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n (\hat{S}^{-1}(t_i)-S^{-1}(t_i))x_i x^\tau _i\beta (t_i)I(t_i\le t)+o_p(1)\\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n (1-\delta _i)S^{-1}(t_i)x_ix^\tau _i(\hat{\beta } (t_i)-\beta (t_i))I(t_i\le t)\\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n \delta _i S^{-1}(t_i)x_i\epsilon _i I(t_i\le t)\\& =I_{n1,1}+I_{n1,2}+I_{n1,3}+I_{n1,4}+o_p(1). \end{aligned}$$
(A.3)
We investigate the properties of \(I_{n1,1},\,I_{n1,2},\,I_{n1,3}\) and \(I_{n1,4}\) in (A.3). Denote \(s(t)=\int X X^\tau f_{X,t}(X,t)dX\) and \(\hat{s}(t_i)=n^{-1}\sum _{j=1}^n x_j x^\tau _jK_h(t_i-t_j)\), we can derive that
$$\begin{aligned} \hat{S}^{-1}(t_i)-S^{-1}(t_i)&= \hat{s}^{-1}(t_i)\hat{f}(t_i)-s^{-1}(t_i)f(t_i)\\&= s^{-1}(t_i)\hat{f}(t_i)-s^{-1}(t_i)\hat{s}(t_i)s^{-1}(t_i)f(t_i) +O_p\left(\frac{1}{nh}+h^2\right). \end{aligned}$$
Invoked by this fact, we have
$$\begin{aligned} I_{n1,1}&= \frac{1}{\sqrt{n}}\sum _{i=1}^n \delta _i\Big (s^{-1}(t_i)\hat{f}(t_i)-s^{-1}(t_i)\hat{s}(t_i)s^{-1}(t_i)f(t_i)\Big )x_i\epsilon _i I(t_i\le t)\\&= \frac{1}{\sqrt{n}}\sum _{i=1}^n\delta _is^{-1}(t_i)\frac{1}{n}\sum _{j=1}^nK_h(t_i-t_j)x_i\epsilon _i I(t_i\le t)\\&-\frac{1}{\sqrt{n}}\sum _{i=1}^n\delta _is^{-1}(t_i)\frac{1}{n} \sum _{j=1}^n x_j x^\tau _j K_h(t_i-t_j)s^{-1}(t_i)f(t_i)x_i \epsilon _i I(t_i\le t)+o_p(1)\\&= \frac{1}{\sqrt{n}}\sum _{j=1}^n s^{-1}(t_j)f(t_j)I(t_j\le t) E(\delta X\epsilon |T=t_j)\\&-\frac{1}{\sqrt{n}}\sum _{j=1}^n s^{-1}(t_j)f(t_j)x_j x^\tau _js^{-1} (t_j)f(t_j)I(t_j\le t) E(\delta X\epsilon |T=t_j)+o_p(1)\\& =o_p(1). \end{aligned}$$
(A.4)
The last equation in (A.4) follows because of the fact that \(E(\delta X\epsilon |T)= E(E(\delta X\epsilon |X,T)|T)=E(X\Delta (X,T)E(\epsilon |X,T)|T)=0\).
Similarly, for \(I_{n1,2}\), we can derive
$$\begin{aligned} I_{n1,2}&\!=\frac{1}{\sqrt{n}}\sum _{i=1}^n \Big (s^{-1}(t_i) \hat{f}(t_i)\!-\!s^{-1}(t_i)\hat{s}(t_i)s^{-1}(t_i)f(t_i)\Big ) x_ix^\tau _i\beta (t_i) I(t_i\le t)+o_p(1)\\&= \frac{1}{\sqrt{n}}\sum _{j=1}^n s^{-1}(t_j)f(t_j)E( XX^\tau |T=t_j) \beta (t_j)I(t_j\le t)+o_p(1)\\&-\frac{1}{\sqrt{n}}\sum _{j=1}^n s^{-1}(t_j)f(t_j)x_j x^\tau _j s^{-1}(t_j)f(t_j) E(XX^\tau |T=t_j)\beta (t_j)I(t_j\le t). \end{aligned}$$
Recall that \(S^{-1}(t_j)=s^{-1}(t_j)f(t_j),\,S(t_j)=E( XX^\tau |T=t_j)\) and \(Z_j=S^{-1}(t_j)x_j x^\tau _j \beta (t_j)\), we have
$$\begin{aligned} I_{n1,2}=\frac{1}{\sqrt{n}}\sum _{j=1}^n (\beta (t_j)-Z_j)I(t_j\le t)+o_p(1). \end{aligned}$$
(A.5)
We state the property of \(I_{n1,3}\) in (A.3) as follows. It can be easily derived that
$$\begin{aligned} \hat{\beta }(t_i)-\beta (t_i)&= -S^{-1}_1(t_i) (\hat{S}_1(t_i)-S_1(t_i))S^{-1}_1(t_i)G_1(t_i)\\&+S^{-1}_1(t_i)(\hat{G}_1(t_i)-G_1(t_i))+O_p\left(\frac{1}{nh}+h^2\right). \end{aligned}$$
Moreover, denote \(\hat{s}_1(t_i)=n^{-1}\sum _{j=1}^n \delta _j x_j x^\tau _jK_h(t_i-t_j)\) and \(\hat{g}_1(t_i)=n^{-1}\sum _{j=1}^n \delta _j x_j y_j K_h(t_i-t_j)\), we have
$$\begin{aligned}&\hat{S}_1(t_i)-S_1(t_i)=f^{-1}(t_i)\hat{s}_1(t_i)-f^{-2}(t_i)\hat{f}(t_i) s_1(t_i)+O_p\left(\frac{1}{nh}+h^2\right),\\&\hat{G}_1(t_i)-G_1(t_i)=f^{-1}(t_i)\hat{g}_1(t_i)-f^{-2}(t_i) \hat{f}(t_i)g_1(t_i)+O_p\left(\frac{1}{nh}+h^2\right). \end{aligned}$$
As a result,
$$\begin{aligned} \hat{\beta }(t_i)-\beta (t_i)&= -S^{-1}_1(t_i)\Big (f^{-1} (t_i)\hat{s}_1(t_i)-f^{-2}(t_i) \hat{f}(t_i)s_1(t_i)\Big )S^{-1}_1(t_i)G_1(t_i)\\&+\,S^{-1}_1(t_i)\Big (f^{-1}(t_i)\hat{g}_1(t_i)-f^{-2}(t_i)\hat{f}(t_i)g_1(t_i)\Big )+O_p\left(\frac{1}{nh}+h^2\right). \end{aligned}$$
For \(I_{n1,3}\) in (A.3), based on the above analysis, we have
$$\begin{aligned} I_{n1,3}&= -\frac{1}{\sqrt{n}}\sum _{i=1}^n (1-\delta _i)S^{-1}(t_i)x_ix^\tau _iS^{-1}_1(t_i)f^{-1}(t_i)\hat{s}_1(t_i)S^{-1}_1(t_i)G_1(t_i))I(t_i\le t)\\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n (1-\delta _i)S^{-1}(t_i)x_ix^\tau _iS^{-1}_1(t_i)f^{-2}(t_i)\hat{f}(t_i)s_1(t_i)S^{-1}_1(t_i)G_1(t_i))I(t_i\le t) \\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n (1-\delta _i)S^{-1}(t_i)x_ix^\tau _iS^{-1}_1(t_i)f^{-1}(t_i)\hat{g}_1(t_i))I(t_i\le t)\\&-\frac{1}{\sqrt{n}}\sum _{i=1}^n (1-\delta _i)S^{-1}(t_i)x_ix^\tau _i S^{-1}_1(t_i)f^{-2}(t_i)\hat{f}(t_i)g_1(t_i)I(t_i\le t)+o_p(1). \end{aligned}$$
Recall that \(\beta (t_i)=S^{-1}_1(t_i)G_1(t_i),S_1(t_i)=f^{-1}(t_i)s_1(t_i)\) and \(G_1(t_i)=f^{-1}(t_i)g_1(t_i)\), we obtain
$$\begin{aligned} I_{n1,3}&= \frac{1}{\sqrt{n}}\sum _{i=1}^n \Big \{(1-\delta _i)S^{-1}(t_i) x_i x^\tau _i S^{-1}_1(t_i)f^{-1}(t_i)I(t_i\le t) \\&\times \frac{1}{n}\sum _{j=1}^n \delta _j x_j\epsilon _j K_h(t_i-t_j)\Big \}+o_p(1)\\&= \frac{1}{\sqrt{n}}\sum _{j=1}^n S^{-1}(t_j)E \{(1-\delta )XX^\tau |T=t_j\}S^{-1}_1(t_j)I(t_j\le t) \delta _j x_j\epsilon _j+o_p(1)\\& =\frac{1}{\sqrt{n}}\sum _{j=1}^n \{S^{-1}_1(t_j) -S^{-1}(t_j)\}I(t_j\le t) \delta _j x_j\epsilon _j +o_p(1). \end{aligned}$$
(A.6)
According to (A.3)–(A.6), for \(I_{n1}\), we can have
$$\begin{aligned} I_{n1}&= \frac{1}{\sqrt{n}}\sum _{j=1}^n \{S^{-1}_1(t_j)-S^{-1}(t_j)\} I(t_j\le t) \delta _j x_j\epsilon _j +\frac{1}{\sqrt{n}}\sum _{j=1}^n (\beta (t_j)-Z_j)I(t_j\le t)\\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n \delta _iS^{-1}(t_i)x_i\epsilon _i I(t_i\le t)+o_p(1)\\& =\frac{1}{\sqrt{n}}\sum _{j=1}^n S^{-1}_1(t_j)I(t_j\le t) \delta _j x_j\epsilon _j +\frac{1}{\sqrt{n}}\sum _{j=1}^n (\beta (t_j)-Z_j)I(t_j\le t)+o_p(1). \end{aligned}$$
(A.7)
For \(I_{n2}\), under mild conditions, see Jennrich (1969), we have
$$\begin{aligned} I_{n2}&= E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1}\\&\times \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) (\hat{Z}_{j1}-\beta (t_j,\theta ))+o_p(1)\\&= E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1}\\&\times \left(\frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) (\hat{Z}_{j1}\!-\!Z_j) \!+\!\frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta )(Z_j\!-\!\beta (t_j,\theta ))\right) \!+\!o_p(1). \end{aligned}$$
Here, \(g(t_i,\theta )=grad_\theta (\beta (t_i,\theta ))\). Similar to the derivation for \(I_{n1}\), we have
$$\begin{aligned}&\frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta )(\hat{Z}_{i1}-Z_i)\\&\quad = \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta )(S^{-1}_1(t_j)\delta _j x_j\epsilon _j +(\beta (t_j)-Z_j))\!+o_p(1). \end{aligned}$$
As a result,
$$\begin{aligned} I_{n2}=E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1}\\ \; \times \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) S^{-1}_1(t_j) \delta _j x_j\epsilon _j +o_p(1). \end{aligned}$$
(A.8)
For \(R_{n1}(t)\) in (A.1), based on the asymptotic representation of \(I_{n1}\) in (A.7) and \(I_{n2}\) in (A.8), we can have
$$\begin{aligned} R_{n1}(t)&= -E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1} \frac{1}{\sqrt{n}}\\&\times \sum _{j=1}^n g^\tau (t_j,\theta ) S^{-1}_1(t_j) \delta _j x_j\epsilon _j\\&+\frac{1}{\sqrt{n}}\sum _{j=1}^n S^{-1}_1(t_j)I(t_j\le t) \delta _j x_j\epsilon _j+o_p(1)\\&= \frac{1}{\sqrt{n}}\sum _{j=1}^n L_1(\delta _j,t_j,y_j,x_j;t)+o_p(1). \end{aligned}$$
We notice that the functions \(L_1(\tilde{\delta },\tilde{t},\tilde{y},\tilde{x};t)\) over all indices \((t)\) are functions of a VC class. By VII 21 of Pollard (1984), \(R_{n1}(t)\) converges in distribution to the Gaussian process \(R_1(t)\). By continuous mapping theorem, we prove the limit distribution of \(T_{n1}\) in Theorem 1.
Next, we prove the convergence of \(R_{n2}(t)\) and \(T_{n2}\), it’s easy to prove that
$$\begin{aligned} R_{n2}(t)&= \frac{1}{\sqrt{n}}\sum _{i=1}^n (\hat{Z}_{i2}-Z_i)I(t_i\le t) -\frac{1}{\sqrt{n}}\sum _{i=1}^n (\beta (t_i,\hat{\theta })-\beta (t_i,\theta )) I(t_i\le t)\\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n (Z_i-\beta (t_i,\theta ))I(t_i\le t)\\& =:J_{n1}-J_{n2}+J_{n3}. \end{aligned}$$
(A.9)
We first deal with the term \(J_{n1}\) in (A.9), similar to the derivation for \(\hat{Z}_{i1}-Z_i\) in (A.2), we obtain
$$\begin{aligned} \hat{Z}_{i2}-Z_i&= \frac{\delta _i}{\Delta _t(t_i)} (\hat{S}^{-1}(t_i)-S^{-1}(t_i))x_i\epsilon _i\\&+\left(1-\frac{\delta _i}{\Delta _t(t_i)}\right)S^{-1} (t_i)x_ix^\tau _i(\hat{\beta }(t_i)-\beta (t_i))\\&+\frac{\delta _i}{\hat{\Delta }_t(t_i)}S^{-1} (t_i)x_i\epsilon _i\!+\!(\hat{S}^{-1}(t_i)\!-\!S^{-1}(t_i))x_ix^\tau _i\beta (t_i)+O_p\left(\frac{1}{nh}+h^2\right). \end{aligned}$$
As a result, the term \(J_{n1}\) can be written as follows
$$\begin{aligned} J_{n1}&= \frac{1}{\sqrt{n}}\sum _{i=1}^n \frac{\delta _i}{\Delta _t(t_i)} (\hat{S}^{-1}(t_i)-S^{-1}(t_i))x_i\epsilon _i I(t_i\le t)\\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n (\hat{S}^{-1}(t_i)-S^{-1}(t_i)) x_ix^\tau _i\beta (t_i)I(t_i\le t)\\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n \left(1-\frac{\delta _i}{\Delta _t(t_i)}\right)S^{-1}(t_i)x_ix^\tau _i(\hat{\beta }(t_i)-\beta (t_i))I(t_i\le t)\\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{\delta _i}{\hat{\Delta }_t(t_i)} S^{-1}(t_i)x_i\epsilon _i I(t_i\le t)+o_p(1)\\& =J_{n1,1}+J_{n1,2}+J_{n1,3}+J_{n1,4}+o_p(1) \end{aligned}$$
(A.10)
For \(J_{n1,1}\) in (A.10), similar to the derivation for \(I_{n1,1}\) in (A.4), we have
$$\begin{aligned} J_{n1,1}&= \frac{1}{\sqrt{n}}\sum _{j=1}^n s^{-1} (t_j)f(t_j)I(t_j\le t) E\left(\frac{\delta }{\Delta _t(t)} X\epsilon |T=t_j\right)\\&-\frac{1}{\sqrt{n}}\sum _{j=1}^n s^{-1}(t_j)f(t_j)x_j x^\tau _js^{-1} (t_j)f(t_j)I(t_j\le t) E\left(\frac{\delta }{\Delta _t(t)}X\epsilon |T=t_j\right)+o_p(1)\\& =o_p(1). \end{aligned}$$
(A.11)
Noting that \(J_{n1,2}\) in (A.10) equals to \(I_{n1,2}\) in (A.3) and we have
$$\begin{aligned} J_{n1,2}=\frac{1}{\sqrt{n}}\sum _{j=1}^n (\beta (t_j)-Z_j)I(t_j\le t)+o_p(1). \end{aligned}$$
(A.12)
For the term \(J_{n1,3}\), it can be derived that
$$\begin{aligned} J_{n1,3}&= \frac{1}{\sqrt{n}}\sum _{i=1}^n \Big (\left(1-\frac{\delta _i}{\Delta _t(t_i)}\right) S^{-1}(t_i)x_ix^\tau _iS^{-1}_1(t_i)f^{-1}(t_i)I(t_i\le t) \\&\times \frac{1}{n}\sum _{j=1}^n \delta _j x_j\epsilon _j K_h(t_i-t_j)\Big )+o_p(1)\\&= \frac{1}{\sqrt{n}}\sum _{j=1}^n S^{-1}(t_j)E\left(\left(1-\frac{\delta }{\Delta _t(T)}\right) XX^\tau |T=t_j\right)\\&\times S^{-1}_1(t_j)I(t_j\le t) \delta _j x_j\epsilon _j+o_p(1)\\& =\frac{1}{\sqrt{n}}\sum _{j=1}^n \Big (S^{-1}_1(t_j)\!-\!S^{-1}(t_j)S_2(t_j)S^{-1}_1 (t_j)\Big )I(t_j\le t) \delta _j x_j\epsilon _j\!+\!o_p(1), \end{aligned}$$
(A.13)
where \(S_2(t_j)=E\{({\delta }/{\Delta _t(T)})XX^\tau |T=t_j\}\).
As for \(J_{n1,4}\), it can be proved that
$$\begin{aligned} J_{n1,4}&= \frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{\delta _i}{\Delta _t(t_i)}S^{-1}(t_i)x_i\epsilon _i I(t_i\le t) \\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{\Delta _t(t_i)-\hat{\Delta }_t(t_i)}{\Delta ^2_t(t_i)}\delta _iS^{-1}(t_i)x_i\epsilon _i I(t_i\le t)+o_p(1)\\& =\frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{\delta _i}{\Delta _t(t_i)}S^{-1}(t_i)x_i\epsilon _i I(t_i\le t)+o_p(1), \end{aligned}$$
(A.14)
where the last equation follows based on the following fact
$$\begin{aligned}&\frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{\Delta _t(t_i)-\hat{\Delta }_t(t_i)}{\Delta ^2_t(t_i)}\delta _iS^{-1}(t_i)x_i\epsilon _i I(t_i\le t)\\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{\sum _{j=1}^n (\Delta _t(t_i)-\delta _j) K_h(t_i-t_j)}{nf(t_i)\Delta ^2_t(t_i)}\delta _iS^{-1}(t_i)x_i\epsilon _i I(t_i\le t)+o_p(1)\\&\quad =\frac{1}{\sqrt{n}}\sum _{j=1}^n \frac{(\Delta _t(t_j)-\delta _j)}{\Delta _t(t_j)}S^{-1}(t_j) I(t_j\le t)E(X\epsilon |T=t_j) +o_p(1)\\&\quad =o_p(1), \end{aligned}$$
where the last equation is from \(E(X\epsilon |T)= E(E(X\epsilon |X,T)|T)=E(XE(\epsilon |X,T) |T)=0\).
Combining the Eqs. (A.10)–(A.14), we get
$$\begin{aligned} J_{n1}=\frac{1}{\sqrt{n}}\sum _{j=1}^n \Big [\Big (S^{-1}_1(t_j)-S^{-1}(t_j)S_2(t_j)S^{-1}_1(t_j)+\frac{S^{-1}(t_j)}{\Delta _t(t_j)} \Big )I(t_j\le t) \delta _j x_j\epsilon _j\Big ]\\ \! + \frac{1}{\sqrt{n}}\sum _{j=1}^n (\beta (t_j)-Z_j)I(t_j\le t)+o_p(1). \end{aligned}$$
(A.15)
Now, we turn to prove \(J_{n2}\) in (A.9), and it can be derived that
$$\begin{aligned} J_{n2}&= E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1}\\&\times \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) (\hat{Z}_{j2}-\beta (t_j,\theta ))+o_p(1)\\&= E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1}\\&\times \left(\! \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) (\hat{Z}_{j2}-Z_j)\!+\! \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta )(Z_j\!-\!\beta (t_j,\theta ))\!\right)\! +\!o_p(1). \end{aligned}$$
Noting that
$$\begin{aligned}&\frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta )(\hat{Z}_{i2}-Z_i) \\&\quad =\frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta )\\&\qquad \times \Big [\Big (S^{-1}_1(t_j) -S^{-1}(t_j)S_2(t_j)S^{-1}_1(t_j)+\frac{S^{-1}(t_j)}{\Delta _t(t_j)}\Big ) \delta _j x_j\epsilon _j\Big ]\\&\qquad +\frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta )(\beta (t_j)-Z_j)+o_p(1), \end{aligned}$$
we have
$$\begin{aligned} J_{n2}=E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1}\\ \; \times \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) \Big [\Big (S^{-1}_1(t_j)-S^{-1}(t_j)S_2(t_j)S^{-1}_1(t_j)+\frac{S^{-1} (t_j)}{\Delta _t(t_j)}\Big )\delta _j x_j\epsilon _j\Big ] +o_p(1). \end{aligned}$$
(A.16)
Thus, based on (A.9), (A.15) and (A.16), we have
$$\begin{aligned} R_{n2}(t)&= \frac{1}{\sqrt{n}}\sum _{j=1}^n \Big [\Big (S^{-1}_1(t_j) -S^{-1}(t_j)S_2(t_j)S^{-1}_1(t_j)+\frac{S^{-1}(t_j)}{\Delta _t(t_j)} \Big )I(t_j\le t) \delta _j x_j\epsilon _j\Big ]\\&-E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1}\\&\times \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) \Big [\Big (S^{-1}_1(t_j)-S^{-1}(t_j)S_2(t_j)S^{-1}_1(t_j) +\frac{S^{-1}(t_j)}{\Delta _t(t_j)}\Big ) \delta _j x_j\epsilon _j\Big ]+o_p(1)\\&= \frac{1}{\sqrt{n}}\sum _{j=1}^n L_2(\delta _j,t_j,y_j,x_j;t)+o_p(1), \end{aligned}$$
which converges in distribution to the Gaussian process \(R_2(t)\). By continuous mapping theorem, the asymptotical properties of \(T_{n2}\) is proved in Theorem 1.
Proof of Theorem 2
For \(R_{n1}(t)\) under the alternative hypothesis, we have
$$\begin{aligned} R_{n1}(t)&= \frac{1}{\sqrt{n}}\sum _{i=1}^n (\hat{Z}_{i1} -Z_i)I(t_i\le t)-\frac{1}{\sqrt{n}}\sum _{i=1}^n (\beta (t_i,\hat{\theta })-\beta (t_i,\theta )) I(t_i\le t)\\&+\frac{1}{\sqrt{n}}\sum _{i=1}^n (Z_i-\beta (t_i,\theta )) I(t_i\le t)\\& =:\tilde{I}_{n1}-\tilde{I}_{n2}+\tilde{I}_{n3}. \end{aligned}$$
(A.17)
Note that \(\beta (t)=\beta (t,\theta )+C_n G(t)\) and by a similar derivation as that for \(I_{n1},\,I_{n2}\) and \(I_{n3}\) in (A.1), we have
$$\begin{aligned} \tilde{I}_{n1}+ \tilde{I}_{n3}&= \frac{1}{\sqrt{n}}\sum _{j=1}^n S^{-1}_1(t_j)I(t_j\le t) \delta _j x_j\epsilon _j +\frac{1}{\sqrt{n}}\sum _{j=1}^n (\beta (t_j)-Z_j)I(t_j\le t) \\&+\frac{1}{\sqrt{n}}\sum _{j=1}^n (Z_j-\beta (t_j,\theta ))I(t_j\le t)\\& =\frac{1}{\sqrt{n}}\sum _{j=1}^n S^{-1}_1(t_j)I(t_j\le t) \delta _j x_j\epsilon _j+\frac{C_n}{\sqrt{n}}\sum _{j=1}^n G(t_j)I(t_j\le t). \end{aligned}$$
(A.18)
As for the term \(\tilde{I}_{n2}\), we have
$$\begin{aligned} \tilde{I}_{n2}&= E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1}\qquad \\&\times \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) \Big (S^{-1}_1(t_j)\delta _j x_j\epsilon _j+(\beta (t_j)-Z_j)\Big )\qquad \\&+E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1}qquad\\&\times \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta )(Z_j-\beta (t_j,\theta )) +o_p(1)\qquad \\&= E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1}\qquad \\& \times \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) \Big (S^{-1}_1(t_j)\delta _j x_j\epsilon _j+C_n G(t_j)\Big ) +o_p(1). \end{aligned}$$
(A.19)
Hence, by the Eqs. (A.17)–(A.19), we have
$$\begin{aligned} R_{n1}(t)&\!=-E(g(T,\theta )I(T\!\le \! t))E(g^\tau (T,\theta )g(T,\theta ))^{-1} \frac{1}{\sqrt{n}}\sum _{j=1}^n \! g^\tau (t_j,\theta ) S^{-1}_1(t_j)\delta _j
x_j\epsilon _j \\&-E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1} \frac{C_n}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) G(t_j)\\&+\frac{1}{\sqrt{n}}\sum _{j=1}^n S^{-1}_1(t_j)I(t_j\le t)\delta _j x_j\epsilon _j+\frac{C_n}{\sqrt{n}}\sum _{j=1}^n G(t_j)I(t_j\le t)+o_p(1). \end{aligned}$$
When \(\sqrt{n}C_n\) converges to \(C\) with \(C\ne 0\), we have
$$\begin{aligned}&\frac{C_n}{\sqrt{n}}\sum _{j=1}^n G(t_j)I(t_j\le t) \, \, \mathrm converges to \, \, CE(G(T)I(T\le t)),\\&\frac{C_n}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) G(t_j)\, \, \mathrm converges to \, \, CE(g^\tau (T,\theta ) G(T)). \end{aligned}$$
When \(n^rC_n\) converges to \(C\) with \(C\ne 0\) and \(0<r<1/2\), it’s easily derived that the test statistic converges in probability to infinity. Based on the above analysis for \(R_{n1}(t)\) under the alternative hypothesis, we can get the convergence properties of \(T_{n1}\) according to continuous mapping theorem.
Next, we prove the convergence of \(R_{n2}(t)\) under local alternatives. Similar to the argument for dealing with \(R_{n1}(t)\) under alternative hypothesis, we have
$$\begin{aligned} R_{n2}(t)&= \frac{1}{\sqrt{n}}\sum _{j=1}^n \Big [\Big (S^{-1}_1(t_j)-S^{-1}(t_j)S_2(t_j)S^{-1}_1(t_j)+\frac{S^{-1}(t_i)}{\Delta _t(t_i)}\Big )I(t_j\le t) \delta _j x_j\epsilon _j\Big ]\\&-E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1} \frac{1}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) \Big [\Big (S^{-1}_1(t_j)\\&-S^{-1}(t_j)S_2(t_j)S^{-1}_1(t_j)+\frac{S^{-1}(t_j)}{\Delta _t(t_j)}\Big )I(t_j\le t) \delta _j x_j\epsilon _j\Big ]+\frac{C_n}{\sqrt{n}}\sum _{j=1}^n G(t_j)I(t_j\le t)\\&-E(g(T,\theta )I(T\le t))E(g^\tau (T,\theta )g(T,\theta ))^{-1} \frac{C_n}{\sqrt{n}}\sum _{j=1}^n g^\tau (t_j,\theta ) G(t_j)+o_p(1). \end{aligned}$$
Then the proofs for \(R_{n2}(t)\) and \(T_{n2}\) are finished, and Theorem 2 is proved.
Proof of Theorem 3
For \(\tilde{R}_{nk}\), it can be shown that
$$\begin{aligned} \tilde{R}_{nk}=\frac{1}{\sqrt{n}}\sum _{i=1}^n e_i L_k(\delta _i,t_i,y_i,x_i;t)+o_p(1). \end{aligned}$$
By similar arguments as that for Theorem 1, it can conclude that \(\tilde{R}_{nk}\) converges in distribution to a Gaussian process. It can be shown that \(E(\tilde{R}_{nk})=0\) and \(Cov(\tilde{R}_{nk}(t_1),\tilde{R}_{nk}(t_2))\) converges to \(Cov(R_k(t_1),R_k(t_2)\). By Theorem VII 21 of (Pollard (1984), p. 157), the theorem is proved.
