Case-cohort studies for clustered failure time data with a cure fraction

Abstract

In epidemiological studies, the case-cohort design is a widely used method for their outstanding cost-effectiveness. Most of the existing works for the case-cohort design are focused on univariate failure time data. However, clustered failure time data are commonly encountered in epidemiological studies. In this article, we study the marginal nonmixture cure model for clustered failure time data with a cure fraction in the context of case-cohort design. A sieve semiparametric likelihood method is proposed to estimate the parametric and nonparametric components. The proposed method is easy to implement. The resulting estimators are shown to be strongly consistent and asymptotically normal. Simulation studies are carried out to assess the finite sample performance of the proposed method. We also analyze a real dataset from the Prostate, Lung, Colorectal, and Ovarian Cancer Screening Trial to illustrate our method.

Appendix

In this Appendix, we will present the proofs of Theorems 1–3. Let $W_{\cdot j}=\{T_{\cdot j},\delta _{\cdot j},\varvec{Z}_{\cdot j}\}, j=1,\ldots ,K$ denote the data for a generic cluster and $\varvec{W}=\{W_{\cdot 1}, \ldots , W_{\cdot K}\}$. Similarly, we denote $W^{\xi }_{\cdot j}=\{T_{\cdot j},\delta _{\cdot j},\xi _{\cdot j}\varvec{Z}_{\cdot j},\xi _{\cdot j}\}, j=1,\ldots ,K$ as a single observation for a generic cluster under the case-cohort design, and $\varvec{W}^{\xi }=\{W^{\xi }_{\cdot 1}, \ldots , W^{\xi }_{\cdot K}\}$. Furthermore, we define the function class $\mathcal {L}_{n}=\{l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })=\sum _{j=1}^{K}l^{w}(\varvec{\theta };W_{\cdot j}^{\xi }):\varvec{\theta }\in \varvec{\Theta }_{n}\}$. In the whole proofs, let $Pg=\int g(x)\textrm{d}P(x)$, the expectation of g(x) under the distribution P, and $P_{n}g=n^{-1}\sum _{i=1}^{n} g(X_{i})$, the expectation of g(X) under the empirical measure $P_n$. We employ $\widetilde{C}$ to represent a universal positive constant, which may vary from position to position.

For any $\epsilon >0$, the covering number $N(\epsilon , \mathcal {L}_{n}, L_1(P_n))$ is defined as the smallest positive integer $\kappa$, then there exists $\{\varvec{\theta }^{(1)},\ldots ,\varvec{\theta }^{(\kappa )}\}$ such that

$$\begin{aligned} \min _{k\in \{1,\ldots ,\kappa \}}\frac{1}{n}\sum ^{n}_{i=1}|l_{K}^w(\varvec{\theta };\varvec{W}_{i}^{\xi })-l_{K}^w(\varvec{\theta }^{(k)};\varvec{W}_{i}^{\xi })|< \epsilon , \end{aligned}$$

for all $\varvec{\theta }\in \varvec{\Theta }_{n}$, where $k=1,\ldots ,\kappa , \varvec{\theta }^{(k)}=(\varvec{\beta }^{(k)}, F^{(k)})\in \varvec{\Theta }_{n}$. If $\kappa$ does not exist, we define $N(\epsilon ,\mathcal {L}_{n},L_1(P_n))=\infty$.

Lemma 1

Under conditions (C1)–(C3), the covering number of the function class $\mathcal {L}_{n}$ satisfies

$$\begin{aligned} N(\epsilon ,\mathcal {L}_{n},L_1(P_n))\le \widetilde{C}M_{n}^{(m+1)}\epsilon ^{-(m+p+1)}, \end{aligned}$$

where $\widetilde{C}$ is a constant, $m=o(n^{\nu })$ with $0<\nu <1$ and the size of the sieve space $\varvec{\Theta }_{n}$ is controlled by $M_{n}=O(n^{c})$ with a constant $c \in (0,\infty )$.

Proof of Lemma 1

For any $\varvec{\theta }^{1}=(\varvec{\beta }^{1},F^{1}), \varvec{\theta }^{2}=(\varvec{\beta }^{2},F^{2})\in \varvec{\Theta }_{n}$, under conditions (C1)–(C3), there exists a large enough constant $\widetilde{C}$ such that

$$\begin{aligned} \mid l_{K}^{w}(\varvec{\theta }^{1};\varvec{W}^{\xi }) - l_{K}^{w}(\varvec{\theta }^{2};\varvec{W}^{\xi })\mid \le \widetilde{C} (\Vert \varvec{\beta }^{1}-\varvec{\beta }^{2}\Vert +\Vert F^{1}-F^{2}\Vert _{\infty }), \end{aligned}$$

(7)

where $\Vert g\Vert _{\infty }=\sup _t|g(t)|$ for a function g. Denote $\varvec{\gamma }^{j}=(\gamma _{0,j},\ldots ,\gamma _{m,j})^{T}$ as the Bernstein coefficients vector corresponding to $F^{j},j=1,2$. Then, we obtain that

$$\begin{aligned} \Vert F^{1}-F^{2}\Vert _{\infty } ={}&\sup _{t} \Bigg | \sum _{k=0}^{m}\gamma _{k,1}B_{k}(t,m,\tau )- \sum _{k=0}^{m}\gamma _{k,2}B_{k}(t,m,\tau ) \Bigg | \nonumber \\ \le {}&\max _{0\le k\le m}\mid \gamma _{k,1}- \gamma _{k,2}\mid \nonumber \\ := {}&\Vert \varvec{\gamma }^{1} -\varvec{\gamma }^{2}\Vert _{\infty }. \end{aligned}$$

(8)

By plugging (7) into (8), it is easy to show

$$\begin{aligned} \mid l_{K}^{w}(\varvec{\theta }^{1};\varvec{W}^{\xi }) - l_{K}^{w}(\varvec{\theta }^{2};\varvec{W}^{\xi })\mid \le \widetilde{C} (\Vert \varvec{\beta }^{1}-\varvec{\beta }^{2}\Vert +\Vert \varvec{\gamma }^{1} -\varvec{\gamma }^{2}\Vert _{\infty }). \end{aligned}$$

Thus, for any $\varvec{\theta }\in \varvec{\Theta }_{n}$,

$$\begin{aligned} \frac{1}{n}\sum ^{n}_{i=1}\mid l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi }_i) - l_{K}^{w}(\varvec{\theta }^{(k)};\varvec{W}^{\xi }_i)\mid \le \widetilde{C}(\Vert \varvec{\beta }-\varvec{\beta }^{(k)}\Vert +\Vert \varvec{\gamma }-\varvec{\gamma }^{(k)}\Vert _{\infty }), \end{aligned}$$

where $k=1,\ldots ,\kappa$. According to Lemma 2.5 of van de Geer (2000), we can show that $\{\varvec{\beta }\in \mathbb {R}^{p}: \Vert \varvec{\beta }\Vert \le M\}$ is covered by $(10\widetilde{C}M/\epsilon )^{p}$ balls with radius $\epsilon /(2\widetilde{C})$ and $\{\varvec{\gamma }\in R^{m+1},\sum _{k=0}^{m}| \gamma _{k}|\le M_{n}\}$ is covered by $(10\widetilde{C}M_{n}/\epsilon )^{m+1}$ balls with radius $\epsilon /(2\widetilde{C})$. As a consequence, the covering number of the function class $\mathcal {L}_{n}$ satisfies

$$\begin{aligned} N(\epsilon ,\mathcal {L}_{n},L_1(P_n))\le \left( \frac{10\widetilde{C}M}{\epsilon } \right) ^{p}\cdot \left( \frac{10\widetilde{C}M_{n}}{\epsilon } \right) ^{m+1} \le \widetilde{C}M_{n}^{(m+1)}\epsilon ^{-(m+p+1)}. \end{aligned}$$

We finish the proof of Lemma 1. $\square$

Lemma 2

Under conditions (C1)–(C3), we have

$$\begin{aligned} \sup _{\varvec{\theta }\in \varvec{\Theta }_{n}}|P_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-Pl_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })|\rightarrow 0, \end{aligned}$$

almost surely.

Proof of Lemma 2

Under conditions (C1)–(C3), we have the uniform bound of $l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })$. Without loss of generality, we assume $\sup _{\varvec{\theta }\in \varvec{\Theta }}|l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })|\le 1$. Then we have $P(l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi }))^{2}\le P(\sup _{\varvec{\theta }\in \varvec{\Theta }}|l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })|)^{2}\le 1$. Let $\alpha _{n}= n^{-1/2+\iota }(\log n)^{1/2}$ with $\nu /2<\iota <1/2$. It is easy to see that $\{\alpha _{n}\}$ is a nonincreasing sequence of positive numbers. Let $\epsilon _{n}=\epsilon \alpha _{n}$ with $\epsilon >0$. For any $\varvec{\theta }\in \varvec{\Theta }_{n}$ and sufficiently large n, we then have

$$\begin{aligned} \textrm{var} (P_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi }) )/(4\epsilon _{n})^{2}\le \frac{(1/n)P(l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi }))^{2}}{16\epsilon ^{2}\alpha _{n}^{2}} \le \frac{1}{16\epsilon ^{2}n^{2\iota }\log n}\le \frac{1}{2}. \end{aligned}$$

(9)

Denote the observations as $\mathcal {W}^{\xi }=\{\varvec{W}^{\xi }_1, \ldots , \varvec{W}^{\xi }_n\}$. Following Pollard (1984), we denote $P^{0}_{n}$ as the signed measure that places mass $\pm n^{-1}$ at each of the observations $\mathcal {W}^{\xi }$. According to (p.31 Pollard 1984) and the formula (9), we have the following inequality

$$\begin{aligned} P(\sup _{\varvec{\theta }\in \varvec{\Theta }_{n}}|P_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-Pl_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })|>8\epsilon _{n}) \le 4P (\sup _{\varvec{\theta }\in \varvec{\Theta }_{n}} |P^{o}_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi }) |>2\epsilon _{n}). \end{aligned}$$

Given $\mathcal {W}^{\xi }$, by the definition of the covering number, we can choose $\varvec{\theta }^{(1)},\ldots ,\varvec{\theta }^{(\kappa )}$, where $\kappa =N(\epsilon _{n}/2,\mathcal {L}_n,L_{1}(P_{n}))$, such that

$$\begin{aligned} \min _{k \in \{1,\ldots ,\kappa \}}P_{n}|l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }^{(k)};\varvec{W}^{\xi })|< \epsilon _{n}/2, \end{aligned}$$

for all $\varvec{\theta }\in \varvec{\Theta }_{n}$. For each $\varvec{\theta }\in \varvec{\Theta }_{n}$, write $\varvec{\theta }^{*}$ for the $\varvec{\theta }^{(k)}$ at which the minimum is achieved. By some simple calculations, we have the following inequality

$$\begin{aligned} |P_{n}^{o}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-P_{n}^{o}l_{K}^{w}(\varvec{\theta }^{*};\varvec{W}^{\xi }) | \le P_{n}|l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }^{*};\varvec{W}^{\xi })|. \end{aligned}$$

(10)

According to the formula (10), we obtain

$$\begin{aligned} {}&P (\sup _{\varvec{\theta }\in \varvec{\Theta }_{n}} |P^{o}_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi }) |>2\epsilon _{n}|\mathcal {W}^{\xi } )\nonumber \\ {}&\quad \le P (\sup _{\varvec{\theta }\in \varvec{\Theta }_{n}} \{ |P^{o}_{n}l_{K}^{w}(\varvec{\theta }^{*};\varvec{W}^{\xi }) |+ P_{n}|l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }^{*};\varvec{W}^{\xi })| \}>2\epsilon _{n}|\mathcal {W}^{\xi } ) \nonumber \\ {}&\quad \le P (\max _{l\in \{1,\ldots ,\kappa \}} |P^{o}_{n}l_{K}^{w}(\varvec{\theta }^{(k)};\varvec{W}^{\xi }) |>3\epsilon _{n}/2|\mathcal {W}^{\xi } )\nonumber \\ {}&\quad \le N(\epsilon _{n}/2,\mathcal {L}_{n},L_{1}(P_n)) \max _{l\in \{1,\ldots ,\kappa \}}P ( |P^{o}_{n}l_{K}^{w}(\varvec{\theta }^{(k)};\varvec{W}^{\xi }) |>3\epsilon _{n}/2|\mathcal {W}^{\xi }). \end{aligned}$$

(11)

By the definition of the covering number $N(\epsilon _{n}/2,\mathcal {L}_{n},L_{1}(P_n))$, for each $\varvec{\theta }^{(k)}$, there exists $\tilde{\varvec{\theta }}^{(k)} \in \varvec{\Theta }_{n}$ such that $P_{n}|l_{K}^{w}(\varvec{\theta }^{(k)};\varvec{W}^{\xi })-l_{K}^{w}(\tilde{\varvec{\theta }}^{(k)};\varvec{W}^{\xi })|<\epsilon _{n}/2$. Then we have

$$\begin{aligned} P ( |P^{o}_{n}l_{K}^{w}(\varvec{\theta }^{(k)};\varvec{W}^{\xi }) |>3\epsilon _{n}/2|\mathcal {W}^{\xi } ) \le {}&P ( (P_{n} |l_{K}^{w}(\varvec{\theta }^{(k)};\varvec{W}^{\xi })-l_{K}^{w}(\tilde{\varvec{\theta }}^{(k)};\varvec{W}^{\xi })| \\ {}&+|P^{o}_{n}l_{K}^{w}(\tilde{\varvec{\theta }}^{(k)};\varvec{W}^{\xi }) | )>3\epsilon _{n}/2|\mathcal {W}^{\xi } ) \\ \le {}&P ( |P^{o}_{n}l_{K}^{w}(\tilde{\varvec{\theta }}^{(k)};\varvec{W}^{\xi }) | >\epsilon _{n}|\mathcal {W}^{\xi }). \end{aligned}$$

Combining Lemma 2.2.7 of van der Vaart and Wellner (1996) with $|l_{K}^{w}(\tilde{\varvec{\theta }}^{(k)};\varvec{W}^{\xi })|\le 1$, we then have

$$\begin{aligned} P ( |P^{o}_{n}l_{K}^{w}(\tilde{\varvec{\theta }}^{(k)};\varvec{W}^{\xi }) | >\epsilon _{n}|\mathcal {W}^{\xi } ) \le 2\exp (-n\epsilon _{n}^2/2). \end{aligned}$$

(12)

According to Lemma 1, the formulas (11) and (12), we have

$$\begin{aligned} P (\sup _{\varvec{\theta }\in \varvec{\Theta }_{n}} |P^{o}_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi }) |>2\epsilon _{n}|\mathcal {W}^{\xi } ) {}&\quad \le 2N(\epsilon _{n}/2,\mathcal {L}_{n},L_{1}(P_n))\exp (-n\epsilon _{n}^2/2)\\ {}&\quad \le 2\widetilde{C}M_n^{(m+1)}(\epsilon _{n}/2)^{-(p+m+1)}\exp (-n\epsilon _{n}^2/2). \end{aligned}$$

By taking expectations over $\mathcal {W}^{\xi }$, we obtain

$$\begin{aligned} P (\sup _{\varvec{\theta }\in \varvec{\Theta }_{n}} |P^{o}_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })) |>2\epsilon _{n} ) {}&\le 2\widetilde{C}M_n^{(m+1)}(\epsilon _{n}/2)^{-(p+m+1)}\exp (-n\epsilon _{n}^2/2). \end{aligned}$$

According to $M_n=O(n^c)$, $m=o(n^\nu )$ and $\iota > \nu /2$, we can show that

$$\begin{aligned} {}&P (\sup _{\varvec{\theta }\in \varvec{\Theta }_{n}} |P_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-Pl_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi }) |>8\epsilon _{n} ) \le 4P (\sup _{\varvec{\theta }\in \varvec{\Theta }_{n}} |P^{o}_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi }) |>2\epsilon _{n} )\\ {}&\quad \le 8\widetilde{C}M_n^{(m+1)}(\epsilon _{n}/2)^{-(p+m+1)}\exp (-n\epsilon _{n}^2/2)\\ {}&\quad \le 8\widetilde{C}\exp \{(m+1)c\log n-(p+m+1)[\log (\epsilon n^{-1/2+\iota }(\log n)^{1/2})-\log 2]\\ {}&\qquad -n\epsilon ^2n^{-1+2\iota }\log n/2\}\\ {}&\quad \le 8\widetilde{C}\exp \{(p+m+1)[(c+1/2-\iota )\log n-\log \log n/2 -\log \epsilon +\log 2 ]\\ {}&\qquad -\epsilon ^2n^{2\iota }\log n/2\}\\ {}&\quad \le 8\widetilde{C}\exp (-\widetilde{C}n^{2\iota } \log n). \end{aligned}$$

Hence, we obtain that

$$\begin{aligned} \sum _{n=1}^{\infty } P(\sup _{\varvec{\theta }\in \varvec{\Theta }_{n}}|P_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-Pl_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })|>8\epsilon _{n}) <\infty . \end{aligned}$$

By the Borel–Cantelli lemma, we have $\sup _{\varvec{\theta }\in \varvec{\Theta }_{n}}|P_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-Pl_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })|\rightarrow 0$ almost surely. $\square$

Proof of Theorem 1

According to Lemmas 1 and 2, we can show that

$$\begin{aligned} N(\epsilon ,\mathcal {L}_{n},L_1(P_n))\le \widetilde{C}M_{n}^{(m+1)}\epsilon ^{-(m+p+1)}, \end{aligned}$$

and

$$\begin{aligned} \sup _{\varvec{\theta }\in \varvec{\Theta }_{n}}|P_{n}l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-Pl_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })|\rightarrow 0, \end{aligned}$$

(13)

almost surely as $n\rightarrow \infty$. Define $\varvec{\Theta }_{\epsilon }=\{\varvec{\theta }:\textrm{d}(\varvec{\theta },\varvec{\theta }_0)\ge \epsilon ,\varvec{\theta }\in \varvec{\Theta }_n \}$ for $\epsilon > 0$,

$$\begin{aligned} {}&\zeta _{1n}=\sup _{\varvec{\theta }\in \varvec{\Theta }_n}|P_nM_{K}(\varvec{\theta };\varvec{W}^{\xi })-PM_{K}(\varvec{\theta };\varvec{W}^{\xi })|, \end{aligned}$$

and

$$\begin{aligned} {}&\zeta _{2n}=P_nM_{K}(\varvec{\theta }_0;\varvec{W}^{\xi })-PM_{K}(\varvec{\theta }_0;\varvec{W}^{\xi }), \end{aligned}$$

where $M_{K}(\varvec{\theta };\varvec{W}^{\xi })=-l_{K}^w(\varvec{\theta };\varvec{W}^{\xi })$. We have

$$\begin{aligned} \inf _{\varvec{\Theta }_{\epsilon }}PM_{K}(\varvec{\theta };\varvec{W}^{\xi })={}&\inf _{\varvec{\Theta }_{\epsilon }}\{PM_{K}(\varvec{\theta };\varvec{W}^{\xi })-P_nM_{K}(\varvec{\theta };\varvec{W}^{\xi }) +P_nM_{K}(\varvec{\theta };\varvec{W}^{\xi })\}\\ \le {}&\zeta _{1n}+ \inf _{\varvec{\Theta }_{\epsilon }}P_nM_{K}(\varvec{\theta };\varvec{W}^{\xi }). \end{aligned}$$

If $\hat{\varvec{\theta }}_n\in \varvec{\Theta }_{\epsilon }$, we then have

$$\begin{aligned} \inf _{\varvec{\Theta }_{\epsilon }}P_nM_{K}(\varvec{\theta };\varvec{W}^{\xi })=P_nM_{K}(\hat{\varvec{\theta }}_n;\varvec{W}^{\xi }) {}&\le P_nM_{K}(\varvec{\theta }_0;\varvec{W}^{\xi })=\zeta _{2n}+PM_{K}(\varvec{\theta }_0;\varvec{W}^{\xi }). \end{aligned}$$

By identification of the model (1), we obtain that $\delta _\epsilon =\inf _{\varvec{\Theta }_{\epsilon }}PM_{K}(\varvec{\theta };\varvec{W}^{\xi })-PM_{K}(\varvec{\theta }_0;\varvec{W}^{\xi })>0$. By conditions (C1)-(C3), it is easy to show

$$\begin{aligned} \inf _{\varvec{\Theta }_{\epsilon }}PM_{K}(\varvec{\theta };\varvec{W}^{\xi })\le \zeta _{1n}+\zeta _{2n}+PM_{K}(\varvec{\theta }_0;\varvec{W}^{\xi })=\zeta _{n}+PM_{K}(\varvec{\theta }_0;\varvec{W}^{\xi }), \end{aligned}$$

with $\zeta _{n}=\zeta _{1n}+\zeta _{2n}$. Hence, we have $\zeta _{n}\ge \delta _\epsilon$. It is indicated that $\{\textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_0)\ge \epsilon \}\subseteq \{\zeta _n\ge \delta _\epsilon \}$. According to the formula (13) and the strong law of large numbers, we obtain that both $\zeta _{1n}\rightarrow 0$ and $\zeta _{2n}\rightarrow 0$ almost surely as $n \rightarrow \infty$. Thus, $\bigcup ^{\infty }_{k=1}\bigcap ^{\infty }_{n=k} \{\textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_0)\ge \epsilon \} \subseteq \bigcup ^{\infty }_{k=1}\bigcap ^{\infty }_{n=k} \{\zeta _n\ge \delta _\epsilon \}$, which proves that $\textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_0)\rightarrow 0$ almost surely. It is easy to see that $\Vert \hat{\varvec{\beta }}_n- \varvec{\beta }_0\Vert \le \textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_0)$ and $\Vert \hat{F}_n- F_0\Vert _2 \le \textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_0)$. Therefore, we have $\Vert \hat{\varvec{\beta }}_n- \varvec{\beta }_0\Vert \rightarrow 0$ and $\Vert \hat{F}_n- F_0\Vert _2 \rightarrow 0$ almost surely. We complete the proof of Theorem 1. $\square$

Proof of Theorem 2

Now we will obtain the convergence rate of $\hat{\varvec{\theta }}_n$. By Theorem 1.6.2 of Lorentz (1986), there exists a Bernstein polynomial $F_{n0}$ which satisfies $\Vert F_{n0}-F_0\Vert _\infty =O(m^{-r/2})$. For convenience, we define $\varvec{\theta }_{n0}=(\varvec{\beta }_0,F_{n0})$. It is easy to see that $\textrm{d}(\varvec{\theta }_{n0},\varvec{\theta }_0)=O(n^{-r\nu /2})$. For any $\eta >0$, we define the class of functions

$$\begin{aligned} \mathcal {F}_{\eta }=\{l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }_{n0};\varvec{W}^{\xi }):\varvec{\theta }\in \varvec{\Theta }_{n}, \eta /2<\textrm{d}(\varvec{\theta },\varvec{\theta }_{n0})\le \eta \}, \end{aligned}$$

for a given observation $\varvec{W}^{\xi }$. One can easily obtain that $P(l_{K}^{w}(\varvec{\theta }_{0};\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }_{n0};\varvec{W}^{\xi }))\le \widetilde{C}\textrm{d}(\varvec{\theta },\varvec{\theta }_{n0})\le \widetilde{C}n^{-r\nu /2}$. Using the relationship between Hellinger distance and Kullback–Leibler information, we then have

$$\begin{aligned} P(l_{K}^{w}(\varvec{\theta }_{0};\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi }))\ge \widetilde{C}\textrm{d}^2(\varvec{\theta }_0,\varvec{\theta }). \end{aligned}$$

Therefore, for sufficiently large n, it yields that

$$\begin{aligned} P(l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }_{n0};\varvec{W}^{\xi })) ={}&P(l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })- l_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi }))+P(l_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi }) \\ {}&-l_{K}^{w}(\varvec{\theta }_{n0};\varvec{W}^{\xi })) \\ \le {}&-\widetilde{C}\eta ^2+\widetilde{C}n^{-r\nu /2}\\ ={}&-\widetilde{C}\eta ^2, \end{aligned}$$

for any $l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }_{n0};\varvec{W}^{\xi }) \in \mathcal {F}_{\eta }$.

Note that $\mathcal {F}_{\eta }$ is uniformly bounded under conditions (C1)–(C3). Furthermore, by some algebraic manipulations, we obtain $P(l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }_{n0};\varvec{W}^{\xi }))^2\le \widetilde{C}\eta ^2$ for any $l^{w}(\varvec{\theta };W^{\xi })-l^{w}(\varvec{\theta }_{n0};W^{\xi }) \in \mathcal {F}_{\eta }$. By Lemma 3.4.2 of van der Vaart and Wellner (1996), we can prove that

$$\begin{aligned} E_P \Vert n^{1/2}(P_n-P) \Vert _{\mathcal {F}_{\eta }} \le \widetilde{C}J_{[]}\{\eta ,\mathcal {F}_{\eta },L_2(P)\} \left[ 1+\frac{J_{[]}\{\eta ,\mathcal {F}_{\eta },L_2(P)\}}{\eta ^2n^{1/2}} \right] , \end{aligned}$$

where $J_{[]}\{\eta ,\mathcal {F}_{\eta },L_2(P)\}=\int ^{\eta }_0 [1+\log N_{[]}\{\epsilon ,\mathcal {F}_{\eta },L_2(P)\}]^{1/2}\textrm{d}\epsilon$. For $0<\epsilon <\eta$, by (Shen and Wong 1994, p. 597), we have $\log N_{[]}\{\epsilon ,\mathcal {F}_{\eta },L_2(P)\}\le \widetilde{C}N\log (\eta /\epsilon )$ with $N=m+1$. Then we can show that $J_{[]}\{\eta ,\mathcal {F}_{\eta },L_2(P)\} \le \widetilde{C}N^{1/2}\eta$. This yields $\varphi _n(\eta )=N^{1/2}\eta +N/n^{1/2}$. One can easily show that $\varphi _n(\eta )/\eta$ is monotonically decreasing with respect to $\eta$ and $r_n^2\varphi _n(1/r_n)=r_nN^{1/2}+r_n^2N/n^{1/2}\le \widetilde{C}n^{1/2}$, where $r_n=N^{-1/2}n^{1/2}=n^{(1-\nu )/2}$.

Note that $P_n(l_{K}^{w}(\hat{\varvec{\theta }}_n;\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }_{n0};\varvec{W}^{\xi }))\ge 0$ and $\textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_{n0})\le \textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_{0})+\textrm{d}(\varvec{\theta }_0,\varvec{\theta }_{n0})\rightarrow 0$ in probability. Therefore, by Theorem 3.4.1 of van der Vaart and Wellner (1996), we have $n^{(1-\nu )/2}\textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_{n0})=O_p(1)$. Together with $\textrm{d}(\varvec{\theta }_{n0},\varvec{\theta }_{0})=O(n^{-r\nu /2})$, it is easy to show that $\textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_{0})=O_p(n^{-(1-\nu )/2}+n^{-r\nu /2})$. According to $\Vert \hat{\varvec{\beta }}_n- \varvec{\beta }_0\Vert \le \textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_0)$ and $\Vert \hat{F}_n- F_0\Vert _2 \le \textrm{d}(\hat{\varvec{\theta }}_n,\varvec{\theta }_0)$, we have $\Vert \hat{\varvec{\beta }}_n-\varvec{\beta }_0\Vert = O_p(n^{-\min (rv/2,(1-v)/2)})$ and $\Vert \hat{F}_n-F_0\Vert _2 = O_p(n^{-\min (rv/2,(1-v)/2)})$. The proof of Theorem 2 is completed. $\square$

Proof of Theorem 3

First, we will present some necessary notations. The linear span of $\varvec{\Theta }-\varvec{\theta }_0$ is denoted as $\mathcal {V}$, where $\varvec{\Theta }$ is the parameter space and $\varvec{\theta }_0$ is the true value of $\varvec{\theta }$. For convenience, we denote $\varrho _n=n^{-\min \{rv/2,\frac{1-v}{2}\}}$. Following the arguments of van der Vaart (1998, p. 296), for any $\varvec{\theta }\in \{\varvec{\theta }\in \varvec{\Theta }:\Vert \varvec{\theta }-\varvec{\theta }_0\Vert =O(\varrho _n)\}$, the first order directional derivative of $l_{K}^w(\varvec{\theta };\varvec{W}^\xi )$ at the direction $\psi \in \mathcal {V}$ is defined as

$$\begin{aligned} \dot{l}_{K}^w(\varvec{\theta };\varvec{W}^\xi )[\psi ]=\frac{\textrm{d}l_{K}^w(\varvec{\theta }+t\psi ;\varvec{W}^\xi )}{\textrm{d}t}\biggl |_{t=0}. \end{aligned}$$

The second order directional derivative of $l_{K}^w(\varvec{\theta };\varvec{W}^\xi )$ is given by

$$\begin{aligned} \ddot{l}_{K}^w(\varvec{\theta };\varvec{W}^\xi )[\psi ,\tilde{\psi }]= \frac{\textrm{d}^2l_{K}^w(\varvec{\theta }+t\psi +\tilde{t}\tilde{\psi };\varvec{W}^\xi )}{\textrm{d}\tilde{t}\textrm{d}t}\biggl |_{t=0,\tilde{t}=0} =\frac{\textrm{d}\dot{l}_{K}^w(\varvec{\theta }+\tilde{t}\tilde{\psi };\varvec{W}^\xi )}{\textrm{d}\tilde{t}}\biggl |_{\tilde{t}=0}. \end{aligned}$$

The Fisher inner product on the space $\mathcal {V}$ is denoted by $\langle \psi ,\tilde{\psi } \rangle =P(\dot{l}_{K}^w(\varvec{\theta };\varvec{W}^\xi )[\psi ]\dot{l}_{K}^w(\varvec{\theta };\varvec{W}^\xi )[\tilde{\psi }])$. Write a smooth functional of $\varvec{\theta }$ as follows

$$\begin{aligned} \Lambda (\varvec{\theta }) = \varvec{\mu }_1^T\varvec{\beta }+ \int _0^{\tau }\mu _2(t)F(t)dt, \end{aligned}$$

(14)

where $||\varvec{\mu }_1||\le 1$ and $\mu _2(t) \in \mathcal {F}$. The Fisher norm for $\psi \in \mathcal {V}$ is defined as $\Vert \psi \Vert ^{1/2}=\langle \psi ,\psi \rangle$. The closed linear span of $\mathcal {V}$ is denoted as $\overline{\mathcal {V}}$ under the Fisher norm. We can easily find that $(\overline{\mathcal {V}},\Vert \cdot \Vert )$ is a Hilbert space. Similar to $\dot{l}_{K}^w(\varvec{\theta };\varvec{W}^\xi )[\psi ]$, for any $\psi \in \mathcal {V}$, the first directional derivative of $\Lambda (\varvec{\theta })$ at $\varvec{\theta }_0$ is given by

$$\begin{aligned} \dot{\Lambda }(\varvec{\theta }_0)[\psi ]=\frac{\textrm{d}\Lambda (\varvec{\theta }_0+t\psi )}{\textrm{d}t}\biggl |_{t=0}. \end{aligned}$$

Similar to Shen (1997), $\dot{\Lambda }(\varvec{\theta }_0)[\psi ]$ is linear in $\psi$ and

$$\begin{aligned} \Vert \dot{\Lambda }(\varvec{\theta }_0)\Vert =\sup _{\psi \in \overline{\mathcal {V}}:\Vert \psi \Vert >0} \frac{|\dot{\Lambda }(\varvec{\theta }_0)[\psi ]|}{\Vert \psi \Vert }<\infty . \end{aligned}$$

(15)

For any $\psi ^*\in \varvec{\Theta }$, there exists $\psi _n^*\in \varvec{\Theta }_n$ such that $\Vert \psi _n^*-\psi ^*\Vert =O(n^{-\frac{rv}{2}})$. Then we have $\varrho _n\Vert \psi _n^*-\psi ^*\Vert =o(n^{-1/2})$ with $r>1$ and $r\nu >1/2$. For convenience, we define $\rho [\varvec{\theta }-\varvec{\theta }_0;\varvec{W}^{\xi }]=l_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })- \dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\varvec{\theta }-\varvec{\theta }_0]$ and let $\epsilon _n$ be any positive sequence satisfying $\epsilon _n=o(n^{-1/2})$. Some algebra yields that

$$\begin{aligned} 0 \le {}&P_n[l_{K}^{w}(\hat{\varvec{\theta }}_n;\varvec{W}^{\xi })-l_{K}^{w}(\hat{\varvec{\theta }}_n\pm \epsilon _n\psi _n^*;\varvec{W}^{\xi })]\\ ={}&P_n\{[l_{K}^{w}(\hat{\varvec{\theta }}_n;\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })] - [l_{K}^{w}(\hat{\varvec{\theta }}_n\pm \epsilon _n\psi _n^*;\varvec{W}^{\xi })-l_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })]\}\\ ={}&\mp \epsilon _n P_n\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*]+P_n\{\rho [\hat{\varvec{\theta }}_n-\varvec{\theta }_0;\varvec{W}^{\xi }]-\rho [\hat{\varvec{\theta }}_n\pm \epsilon _n\psi _n^*-\varvec{\theta }_0;\varvec{W}^{\xi }]\}\\ ={}&\mp \epsilon _n P_n\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi ^*]\mp \epsilon _n P_n\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*-\psi ^*] +(P_n-P)\\ {}&\cdot \{\rho [\hat{\varvec{\theta }}_n-\varvec{\theta }_0;\varvec{W}^{\xi }]-\rho [\hat{\varvec{\theta }}_n\pm \epsilon _n\psi _n^*-\varvec{\theta }_0;\varvec{W}^{\xi }]\}+P\{\rho [\hat{\varvec{\theta }}_n-\varvec{\theta }_0;\varvec{W}^{\xi }]\\ {}&-\rho [\hat{\varvec{\theta }}_n\pm \epsilon _n\psi _n^*-\varvec{\theta }_0;\varvec{W}^{\xi }]\}\\ :={}&\mp \epsilon _n P_n\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi ^*]+ I_1 + I_2 + I_3. \end{aligned}$$

Next, we will discuss the calculation details of $I_1, I_2$ and $I_3$ respectively. For $I_1$, according to the Chebyshev’s inequality, $\Vert \psi _n^*-\psi ^*\Vert =O(n^{-\frac{rv}{2}})$ and $P\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*-\psi ^*]=0$, it is easy to show that

$$\begin{aligned} {}&P\left( \frac{\mid P_n\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*-\psi ^*] \mid }{n^{-1/2}} \ge \epsilon \right) \\ {}&\quad \le \frac{\textrm{Var}(P_n\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*-\psi ^*])}{n^{-1}\epsilon ^2} \\ {}&\quad = \frac{\textrm{Var}(\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*-\psi ^*])}{\epsilon ^{2}} \\ {}&\quad = \frac{P(\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*-\psi ^*]\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*-\psi ^*])}{\epsilon ^2} \\ {}&\quad = \frac{\Vert \psi _n^*-\psi ^* \Vert ^2}{\epsilon ^2} \\ {}&\quad \rightarrow 0, \end{aligned}$$

which implies that $P_n\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*-\psi ^*]=o_p(n^{-1/2})$ and $I_1=\epsilon _n\times o_p(n^{-1/2})$.

For $I_2$, by some algebraic calculations, we have

$$\begin{aligned} I_2 ={}&(P_n-P)(\rho [\hat{\varvec{\theta }}_n-\varvec{\theta }_0;\varvec{W}^{\xi }]-\rho [\hat{\varvec{\theta }}_n\pm \epsilon _n\psi _n^*-\varvec{\theta }_0;\varvec{W}^{\xi }])\\ ={}&(P_n-P)(\dot{l}_{K}^{w}(\tilde{\varvec{\theta }};\varvec{W}^{\xi })[\mp \epsilon _n\psi _n^*] \pm \epsilon _n\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*])\\ ={}&\mp \epsilon _n (P_n-P)(\dot{l}_{K}^{w}(\tilde{\varvec{\theta }};\varvec{W}^{\xi })[\psi _n^*] -\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*]), \end{aligned}$$

where $\tilde{\varvec{\theta }}$ belongs between $\hat{\varvec{\theta }}_n$ and $\hat{\varvec{\theta }}_n\pm \epsilon _n\psi _n^*$. Denote the function class $\mathcal {L}_3=\{\dot{l}_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })[\psi _n^*]-\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*]:\varvec{\theta }\in \Theta _n \text { and } \Vert \varvec{\theta }-\varvec{\theta }_0\Vert =O(\varrho _n)\}$. For any $\dot{l}_{K}^{w}(\varvec{\theta }_i;\varvec{W}^{\xi })[\psi _n^*]-\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*]\in \mathcal {L}_3$ $(i=1,2)$, we have $\bigl |\dot{l}_{K}^{w}(\varvec{\theta }_1;\varvec{W}^{\xi })[\psi _n^*]-\dot{l}_{K}^{w}(\varvec{\theta }_2;\varvec{W}^{\xi })[\psi _n^*]\bigr |\le \widetilde{C}\Vert \varvec{\theta }_1-\varvec{\theta }_2\Vert$. Thus, it follows that

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

Note that $N(\epsilon ,\mathcal {L}_3,L_2(Q)) \le \widetilde{C}e^{3/\epsilon }$ such that $\int _0^{\infty }\sup _{Q}\sqrt{\log N(\epsilon ,\mathcal {L}_3,L_2(Q))}d\epsilon = \int _0^{\widetilde{C}\varrho _n}\sup _{Q}\sqrt{\log N(\epsilon ,\mathcal {L}_3,L_2(Q))}d\epsilon$ $+ \int _{\widetilde{C}\varrho _n}^{\infty } 0 d\epsilon < \infty$ with $v>1/2$ and $r>1$. Under conditions (C1)-(C3), $\dot{l}_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })[\psi _n^*]$ is uniformly bounded. According to Theorem 2.8.3 of van der Vaart and Wellner (1996), we know that $\mathcal {L}_3$ is a Donsker class. Furthermore, it follows from Corollary 2.3.12 of van der Vaart and Wellner (1996) that

$$\begin{aligned} (P_n-P)(\dot{l}_{K}^{w}(\tilde{\varvec{\theta }};\varvec{W}^{\xi })[\psi _n^*] -\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\psi _n^*]) =o_p(n^{-1/2}). \end{aligned}$$

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

For $I_3$, note that $P(\ddot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\hat{\varvec{\theta }}_n-\varvec{\theta }_0,\hat{\varvec{\theta }}_n-\varvec{\theta }_0])= -P(\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\hat{\varvec{\theta }}_n-\varvec{\theta }_0]\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\hat{\varvec{\theta }}_n-\varvec{\theta }_0])$. For any $\varvec{\theta }\in \{\varvec{\theta }:\textrm{d}(\varvec{\theta }-\varvec{\theta }_0)=O(\varrho _n)\}$, we have $P(\ddot{l}_{K}^{w}(\varvec{\theta };\varvec{W}^{\xi })[\varvec{\theta }-\varvec{\theta }_0,\varvec{\theta }-\varvec{\theta }_0] - \ddot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^{\xi })[\varvec{\theta }-\varvec{\theta }_0,\varvec{\theta }-\varvec{\theta }_0]) = O(\varrho _n^3)$ and $\varrho _n^3 = o(n^{-1})$ with $v<1/3$ and $r>2$. Then, it yields that

$$\begin{aligned} {}&P(\rho [\hat{\varvec{\theta }}_n-\varvec{\theta }_0;\varvec{W}^\xi ]) \\ {}&\quad = P(l_{K}^{w}(\hat{\varvec{\theta }}_n;\varvec{W}^\xi ) - l_{K}^{w}(\varvec{\theta }_0;\varvec{W}^\xi ) - \dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^\xi )[\hat{\varvec{\theta }}_n-\varvec{\theta }_0]) \\ {}&\quad = \frac{1}{2}P(\ddot{l}_{K}^{w}(\tilde{\varvec{\theta }};\varvec{W}^\xi )[\hat{\varvec{\theta }}_n-\varvec{\theta }_0,\hat{\varvec{\theta }}_n-\varvec{\theta }_0] - \ddot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^\xi )[\hat{\varvec{\theta }}_n-\varvec{\theta }_0,\hat{\varvec{\theta }}_n-\varvec{\theta }_0])\\ {}&\qquad \ + \frac{1}{2}P(\ddot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^\xi )[\hat{\varvec{\theta }}_n-\varvec{\theta }_0,\hat{\varvec{\theta }}_n-\varvec{\theta }_0]) \\ {}&\quad = \epsilon _n \times o_p(n^{-1/2}) + \frac{1}{2}P(\ddot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^\xi )[\hat{\varvec{\theta }}_n-\varvec{\theta }_0,\hat{\varvec{\theta }}_n-\varvec{\theta }_0]) \\ {}&\quad = \epsilon _n \times o_p(n^{-1/2}) - \frac{1}{2}\Vert \hat{\varvec{\theta }}_n-\varvec{\theta }_0\Vert ^2, \end{aligned}$$

where $\tilde{\varvec{\theta }}$ is between $\hat{\varvec{\theta }}_n$ and $\varvec{\theta }_0$. Combining the Cauchy–Schwarz inequality, $\Vert \psi _n^*\Vert ^2\rightarrow \Vert \psi ^*\Vert ^2<\infty$ and $\varrho _n\Vert \psi _n^*-\psi ^*\Vert =o(n^{-1/2})$, we have that

$$\begin{aligned} I_3 ={}&- \frac{1}{2}\Vert \hat{\varvec{\theta }}_n-\varvec{\theta }_0\Vert ^2 + \frac{1}{2}\Vert \hat{\varvec{\theta }}_n\pm \epsilon _n\psi _n^*-\varvec{\theta }_0\Vert ^2 + \epsilon _n\times o_p(n^{-1/2})\\ ={}&\pm \epsilon _n \langle \hat{\varvec{\theta }}_n-\varvec{\theta }_0,\psi _n^*\rangle +\frac{1}{2}\Vert \epsilon _n\psi _n^*\Vert ^2 + \epsilon _n\times o_p(n^{-1/2}) \\ ={}&\pm \epsilon _n\langle \hat{\varvec{\theta }}_n-\varvec{\theta }_0,\psi ^*\rangle +\frac{1}{2}\epsilon _n^2\Vert \psi _n^*\Vert ^2 + \epsilon _n\times o_p(n^{-1/2}) \\ ={}&\pm \epsilon _n\langle \hat{\varvec{\theta }}_n-\varvec{\theta }_0,\psi ^*\rangle + \epsilon _n\times o_p(n^{-1/2}). \end{aligned}$$

According to the results of $I_1,I_2$ and $I_3$, we obtain

$$\begin{aligned} 0 \le {}&P_n( l_{K}^{w}(\hat{\varvec{\theta }}_n;\varvec{W}^\xi ) - l_{K}^{w}(\hat{\varvec{\theta }}_n\pm \epsilon _n\psi _n^*;\varvec{W}^\xi ) )\nonumber \\ ={}&\mp \epsilon _nP_n\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^\xi )[\psi ^*] \pm \epsilon _n\langle \hat{\varvec{\theta }}_n-\varvec{\theta }_0,\psi ^*\rangle + \epsilon _n\times o_p(n^{-1/2}). \end{aligned}$$

By some algebraic calculations, we then have $\sqrt{n}\langle \hat{\varvec{\theta }}_n-\varvec{\theta }_0,\psi ^*\rangle = \sqrt{n}(P_n-P)\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^\xi )[\psi ^*] + o_p(1)$ with $P\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^\xi )[\psi ^*]=0$ and $Var(\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^\xi )[\psi ^*]) = \Vert \psi ^*\Vert ^2<\infty$. By the central limit theorem, it yields that $\sqrt{n}\langle \hat{\varvec{\theta }}_n-\varvec{\theta }_0,\psi ^*\rangle \rightarrow N(0,\Vert \psi ^*\Vert ^2)$. By the Riesz representation theorem, there exists $\psi ^*\in {\bar{\mathcal {V}}} \text { such that } \dot{\Lambda }(\varvec{\theta }_0)[\psi ]=\langle \psi ,\psi ^*\rangle$ for any $\psi \in \bar{\mathcal {V}}$ and $\Vert \psi ^*\Vert = \Vert \dot{\Lambda }(\varvec{\theta }_0)\Vert$. Note that $\Lambda (\varvec{\theta })-\Lambda (\varvec{\theta }_0)=\dot{\Lambda }(\varvec{\theta }_0)[\varvec{\theta }-\varvec{\theta }_0]$. Therefore, we have that

$$\begin{aligned} \sqrt{n}(\Lambda (\hat{\varvec{\theta }}_n)-\Lambda (\varvec{\theta }_0))=\sqrt{n}(P_n-P)(\dot{l}_{K}^{w}(\varvec{\theta }_0;\varvec{W}^\xi )[\psi ^*]) + o_p(1) \rightarrow N(0,\Vert \dot{\Lambda }(\varvec{\theta }_0)\Vert ^2), \end{aligned}$$

in distribution. That is $\sqrt{n}[\varvec{\mu }_1^T(\hat{\varvec{\beta }}_n-\varvec{\beta }_0)+ \int _0^{\tau }\mu _2(t)(\hat{F}_n(t)-F_0(t))dt] \rightarrow N(0, \Vert \dot{\Lambda }(\varvec{\theta }_0)\Vert ^2)$ in distribution.

In particular, if we set $\mu _2(\cdot )=0$ in the formula (14), then we have $\Lambda _{\varvec{\beta }}(\varvec{\theta })=\varvec{\mu }_1^T\varvec{\beta }$. The first order directional derivative of $\Lambda _{\varvec{\beta }}(\varvec{\theta })$ is defined as $\dot{\Lambda }_{\varvec{\beta }}(\varvec{\theta }_0)[\psi ]=\frac{\textrm{d}\Lambda _{\varvec{\beta }}(\varvec{\theta }_0+t\psi )}{\textrm{d}t}\Big |_{t=0}$ and

$$\begin{aligned} \Vert \dot{\Lambda }_{\varvec{\beta }}(\varvec{\theta }_0)\Vert =\sup _{\psi \in \overline{\mathcal {V}}:\Vert \psi \Vert >0} \frac{|\dot{\Lambda }_{\varvec{\beta }}(\varvec{\theta }_0)[\psi ]|}{\Vert \psi \Vert }. \end{aligned}$$

(16)

Similarly, it follows that $\sqrt{n}\varvec{\mu }_1^T(\hat{\varvec{\beta }}_n-\varvec{\beta }_0) \rightarrow N(0, \Vert \dot{\Lambda }_{\varvec{\beta }}(\varvec{\theta }_0)\Vert ^2)$ in distribution. The proof of Theorem 3 is completed. $\square$