Estimating the survival function based on the semi-Markov model for dependent censoring

Abstract

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

Appendix

1.1 Preliminary

Lemmas 1 and 2 provide related Donsker properties.

Lemma 1

The classes $\left\{ \left( {x},\delta \right) \!\mapsto \!\delta \mathrm {I}\left\{ {x}\!\leqslant {t}\!\right\} :~{t}\geqslant {0}\right\}$ and $\left\{ \left( {x},\delta \right) \!\mapsto \!\delta \mathrm {I}\left\{ {x}\!\geqslant {t}\!\right\} :~{t}\geqslant {0}\right\}$ are Donsker classes of functions on $\mathbb {R}_{+}\times \left\{ {0},{1}\right\}$.

Proof

Combining Lemma 2.6.15 and Lemma 2.6.18(iii, vi) of Vaart and Wellner (1996), it can be shown that $\left\{ \left( {x},\delta \right) \mapsto \delta \mathrm {I}\left\{ {x}\leqslant {t}\right\} :~{t}\geqslant {0}\right\}$ and $\left\{ \left( {x},\delta \right) \mapsto \delta \mathrm {I}\left\{ {x}\geqslant {t}\right\} :{t}\geqslant {0}\right\}$ are VC-classes on $\mathbb {R}_{+}\times \left\{ {0},{1}\right\}$. By Theorem 2.6.8 of Vaart and Wellner (1996), they are Donsker classes, where the measurability conditions could be verified via the denseness of the rational numbers.$\square$

Lemma 2

Let $\eta$ be a positive real number and $\varLambda$ be a nondecreasing cadlag function on $\left[ {0},\eta \right]$. The class $\left\{ \left( {x},\delta \right) \mapsto \int _{0}^{t}\delta \mathrm {I}\left\{ {x}\geqslant {y}\right\} ~\varLambda \left( d{y}\right) :~{t}\in \left[ {0},\eta \right] \right\}$ is a Donsker class of functions on $\mathbb {R}_{+}\times \left\{ {0},{1}\right\}$.

Proof

The result is easy to see by combining Example 2.6.21 and Example 2.10.8 of Vaart and Wellner (1996). $\square$

Lemma 3

Let $\eta$ be a positive real number and $\mathcal {BV}\left( \left[ {0},\eta \right] \right)$ be the space of all cadlag functions defined on $\left[ {0},\eta \right]$ whose total variation are bounded by ${2}$. For any $(\mathsf {F},\mathsf {G},\mathsf {H})\in \mathcal {BV}\left( \left[ {0},\eta \right] \right) \times \mathcal {BV}\left( \left[ {0},\eta \right] \right) \times \mathcal {BV}\left( \left[ {0},\eta \right] \right)$ and any ${t}\in \left[ {0},\eta \right]$, let

$$\begin{aligned} \phi \left( \mathsf {F},\mathsf {G},\mathsf {H}\right) \left[ {t}\right] =\mathsf {F}\left( {t}\right) \mathsf {G}\left( {t}\right) -\int _{\left[ {0},{t}\right] }\mathsf {G}\left( {u}\right) \mathsf {H}\left( {t}-{u}\right) \;\mathsf {F}\left( d{u}\right) . \end{aligned}$$

For any $\mathsf {F}_{0},\mathsf {G}_{0},\mathsf {H}_{0}\in \mathcal {BV}\left( \left[ {0},\eta \right] \right) , \phi$ is Hadamard differentiable at $(\mathsf {F}_{0},\mathsf {G}_{0},\mathsf {H}_{0})$ with derivative $\phi ^{\prime }(\mathsf {F}_{0},\mathsf {G}_{0},\mathsf {H}_{0})$, where for any ${t}\in \left[ {0},\eta \right]$,

$$\begin{aligned} \phi ^{\prime }\left( \beta _{0}\right) \left[ \beta \right] \left[ {t}\right]= & {} \mathsf {F}_{0}\left( {t}\right) \mathsf {G}\left( {t}\right) +\mathsf {F}\left( {t}\right) \mathsf {G}_{0}\left( {t}\right) -\int _{\left[ {0},{t}\right] }\mathsf {G}_{0}\left( {x}\right) \mathsf {H}_{0}\left( {t}-{x}\right) \mathsf {F}\left( dx\right) \\&-\,\int _{\left[ {0},{t}\right] }\mathsf {G}_{0}\left( {x}\right) \mathsf {H}\left( {t}-{x}\right) \mathsf {F}_{0}\left( dx\right) -\int _{\left[ {0},{t}\right] }\mathsf {G}\left( {x}\right) \mathsf {H}_{0}\left( {t}-{x}\right) \mathsf {F}_{0}\left( dx\right) , \end{aligned}$$

where $\beta =(\mathsf {F},\mathsf {G},\mathsf {H})$ and $\beta _{0}=(\mathsf {F}_{0},\mathsf {G}_{0},\mathsf {H}_{0})$.

Proof

Let $\mathsf {F}_{0},\mathsf {G}_{0},\mathsf {H}_{0},\mathsf {F},\mathsf {G},\mathsf {H}\in \mathcal {BV}\left( \left[ {0},\eta \right] \right) , \{(\mathsf {F}_{m},\mathsf {G}_{m},\mathsf {H}_{m}):\;{m}\in \mathbb {N}\}$ be a sequence converging to $(\mathsf {F},\mathsf {G},\mathsf {H})$ in $\mathcal {BV}\left( \left[ {0},\eta \right] \right) \times \mathcal {BV}\left( \left[ {0},\eta \right] \right) \times \mathcal {BV}\left( \left[ {0},\eta \right] \right)$ and $\{{h}_{m}:\;{m}\in \mathbb {N}\}$ be a sequence of real numbers converging to $0$.

Denote $\beta _{0}=(\mathsf {F}_{0},\mathsf {G}_{0},\mathsf {H}_{0})$ and $\beta _{m}=\beta _{0}+{h}_{m}(\mathsf {F},\mathsf {G},\mathsf {H})$.

Note that for any ${t}\in \left[ {0},\eta \right] , \phi (\beta _{m})[t]=\phi (\beta _{0})[t]+{h}_{m}\varGamma _{m}({t})+{h}_{m}^{2}\mathsf {W}_{m}({t})$, where

$$\begin{aligned} \varGamma _{m}\left( {t}\right)= & {} \mathsf {F}_{0}\left( {t}\right) \mathsf {G}_{m}\left( {t}\right) +\mathsf {F}_{m}\left( {t}\right) \mathsf {G}_{0}\left( {t}\right) -\int _{\left[ {0},{t}\right] }\mathsf {G}_{0}\left( {x}\right) \mathsf {H}_{0}\left( {t}-{x}\right) \mathsf {F}_{m}\left( dx\right) \\&-\,\int _{\left[ {0},{t}\right] }\mathsf {G}_{0}\left( {x}\right) \mathsf {H}_{m}\left( {t}-{x}\right) \mathsf {F}_{0}\left( dx\right) -\int _{\left[ {0},{t}\right] }\mathsf {G}_{m}\left( {x}\right) \mathsf {H}_{0}\left( {t}-{x}\right) \mathsf {F}_{0}\left( dx\right) ,\\ \mathsf {W}_{m}\left( {t}\right)= & {} \mathsf {F}_{m}\left( {t}\right) \mathsf {G}_{m}\left( {t}\right) -\int _{\left[ {0},{t}\right] }\mathsf {H}_{0}\left( {t}-{x}\right) \mathsf {G}_{m}\left( {x}\right) \mathsf {F}_{m}\left( dx\right) \\&-\,\int _{\left[ {0},{t}\right] }\mathsf {G}_{0}\left( {x}\right) \mathsf {H}_{m}\left( {t}-{x}\right) \mathsf {F}_{m}\left( dx\right) -\int _{\left[ {0},{t}\right] }\mathsf {G}_{m}\left( {x}\right) \mathsf {H}_{m}\left( {t}-{x}\right) \mathsf {F}_{0}\left( dx\right) \\&-\,{h}_{m}\int _{\left[ {0},{t}\right] }\mathsf {G}_{m}\left( {x}\right) \mathsf {H}_{m}\left( {t}-{x}\right) \mathsf {F}_{m}\left( dx\right) . \end{aligned}$$

For any ${t}\in \left[ {0},\eta \right]$, let

$$\begin{aligned} \varGamma _{0}\left( {t}\right)= & {} \mathsf {F}_{0}\left( {t}\right) \mathsf {G}\left( {t}\right) +\mathsf {F}\left( {t}\right) \mathsf {G}_{0}\left( {t}\right) -\int _{\left[ {0},{t}\right] }\mathsf {G}_{0}\left( {x}\right) \mathsf {H}_{0}\left( {t}-{x}\right) \mathsf {F}\left( dx\right) \\&-\,\int _{\left[ {0},{t}\right] }\mathsf {G}_{0}\left( {x}\right) \mathsf {H}\left( {t}-{x}\right) \mathsf {F}_{0}\left( dx\right) -\int _{\left[ {0},{t}\right] }\mathsf {G}\left( {x}\right) \mathsf {H}_{0}\left( {t}-{x}\right) \mathsf {F}_{0}\left( dx\right) . \end{aligned}$$

Note that for any ${t}\in \left[ {0},\eta \right] , |\mathsf {W}_{m}({t})|\leqslant {28}+{8}{h}_{m}$ and

$$\begin{aligned} \left| \varGamma _{m}\left( {t}\right) -\varGamma _{0}\left( {t}\right) \right|\leqslant & {} {6}\sup _{{s}\in \left[ {0},\eta \right] }\left| \mathsf {F}_{m}\left( {s}\right) -\mathsf {F}\left( {s}\right) \right| +{6}\sup _{{s}\in \left[ {0},\eta \right] }\left| \mathsf {G}_{m}\left( {s}\right) -\mathsf {G}\left( {s}\right) \right| \\&+\,{4}\sup _{{s}\in \left[ {0},\eta \right] }\left| \mathsf {H}_{m}\left( {s}\right) -\mathsf {H}\left( {s}\right) \right| . \end{aligned}$$

Hence, as ${m}\rightarrow \infty , \sup _{{t}\in \left[ {0},\eta \right] }|\mathsf {W}_{m}({t})|={O}(1)$ and $\sup _{{t}\in \left[ {0},\eta \right] }|\varGamma _{m}({t})-\varGamma _{0}({t})|={o}(1)$.

Therefore, $\phi$ is Hadamard differentiable at $\beta _{0}$ and for any ${t}\in \left[ {0},\eta \right]$,

$$\begin{aligned} \phi ^{\prime }\left( \beta _{0}\right) \left[ \beta \right] \left[ {t}\right]= & {} \mathsf {F}_{0}\left( {t}\right) \mathsf {G}\left( {t}\right) +\mathsf {F}\left( {t}\right) \mathsf {G}_{0}\left( {t}\right) -\int _{\left[ {0},{t}\right] }\mathsf {G}_{0}\left( {x}\right) \mathsf {H}_{0}\left( {t}-{x}\right) \mathsf {F}\left( dx\right) \\&-\,\int _{\left[ {0},{t}\right] }\mathsf {G}_{0}\left( {x}\right) \mathsf {H}\left( {t}-{x}\right) \mathsf {F}_{0}\left( dx\right) -\int _{\left[ {0},{t}\right] }\mathsf {G}\left( {x}\right) \mathsf {H}_{0}\left( {t}-{x}\right) \mathsf {F}_{0}\left( dx\right) , \end{aligned}$$

where $\beta =(\mathsf {F},\mathsf {G},\mathsf {H})$.$\square$

1.2 Proof of Theorems 1 to 3

For any $\left( {i},{j}\right) \in \left\{ \left( {0},{1}\right) ,\left( {0},{2}\right) ,\left( {1},{2}\right) \right\}$ and any ${t}\geqslant {0}$, define

$$\begin{aligned} \overline{\mathsf {M}}_{n}^{\left( {i},{j}\right) }\left( {t}\right) =\overline{\mathsf {N}}_{n}^{\left( {i},{j}\right) }\left( {t}\right) -\int _{\left[ {0},{t}\right] }\overline{\mathsf {Y}}_{n}^{\left( {i}\right) }\left( {y}\right) \;\varLambda _{0}^{\left( {i},{j}\right) }\left( d{y}\right) . \end{aligned}$$

Proof

Combining Theorem 2.10.6 of Vaart and Wellner (1996) with Lemmas 1 and 2, it can be shown that, as ${n}\rightarrow \infty$,

$$\begin{aligned} {n}^{-{1}/{2}}\left( \overline{\mathsf {M}}_{n}^{\left( {0},{1}\right) },\overline{\mathsf {M}}_{n}^{\left( {0},{2}\right) },\overline{\mathsf {M}}_{n}^{\left( {1},{2}\right) },\overline{\mathsf {Y}}_{n}^{\left( {0}\right) }-\mathsf {E}\overline{\mathsf {Y}}_{n}^{\left( {0}\right) },\overline{\mathsf {Y}}_{n}^{\left( {1}\right) }-\mathsf {E}\overline{\mathsf {Y}}_{n}^{\left( {1}\right) }\right) \end{aligned}$$

weakly converges to a tight zero mean Gaussian process in

$$\begin{aligned} \ell _{5}^{\infty }\left( \left[ {0},\tau \right] \right) =\ell ^{\infty }\left( \left[ {0},\tau \right] \right) \times \ell ^{\infty }\left( \left[ {0},\tau \right] \right) \times \ell ^{\infty }\left( \left[ {0},\tau \right] \right) \times \ell ^{\infty }\left( \left[ {0},\tau \right] \right) \times \ell ^{\infty }\left( \left[ {0},\tau \right] \right) . \end{aligned}$$

Combining this with Lemma 3.9.17, Lemma 3.9.25, and Theorem 3.9.4 of Vaart and Wellner (1996), for any $\left( {i},{j}\right) \in \left\{ \left( {0},{1}\right) ,\left( {0},{2}\right) ,\left( {1},{2}\right) \right\}$,

$$\begin{aligned} \sup _{{t}\in \left[ {0},\tau \right] }\left| \hat{\varLambda }_{n}^{\left( {i},{j}\right) }\left( {t}\right) -\varLambda _{0}^{\left( {i},{j}\right) }\left( {t}\right) -\int _{\left[ {0},{t}\right] }\left( \mathsf {L}_{0}^{\left( {i}\right) }\left( {y}-\right) \right) ^{-1}\;{n}^{-1}\overline{\mathsf {M}}_{n}^{\left( {i},{j}\right) }\left( d{y}\right) \right| ={o}_{p}\left( {n}^{-{1}/{2}}\right) ,\nonumber \\ \end{aligned}$$

(10)

By the continuity and the linearity of the integral operator, as ${n}\rightarrow \infty$,

$$\begin{aligned} {n}^{{1}/{2}}\left( \hat{\varLambda }_{n}^{\left( {0},{1}\right) }-\varLambda _{0}^{\left( {0},{1}\right) },\hat{\varLambda }_{n}^{\left( {0},{2}\right) }-\varLambda _{0}^{\left( {0},{2}\right) },\hat{\varLambda }_{n}^{\left( {1},{2}\right) }-\varLambda _{0}^{\left( {1},{2}\right) }\right) \end{aligned}$$

weakly converges to a tight Gaussian process in $\ell _{3}^{\infty }\left( \left[ {0},\tau \right] \right)$ with covariance function $\mathcal {U}_{0}$.

By Lemma 3.9.30 and Theorem 3.9.4 of Vaart and Wellner (1996), as ${n}\rightarrow \infty$,

$$\begin{aligned} {n}^{{1}/{2}}\left( \hat{\mathsf {S}}_{n}^{\left( {0},{1}\right) }-\mathsf {S}_{0}^{\left( {0},{1}\right) },\hat{\mathsf {S}}_{n}^{\left( {0},{2}\right) }-\mathsf {S}_{0}^{\left( {0},{2}\right) },\hat{\mathsf {S}}_{n}^{\left( {1},{2}\right) }-\mathsf {S}_{0}^{\left( {1},{2}\right) }\right) \end{aligned}$$

weakly converges to a tight Gaussian process in $\ell _{3}^{\infty }([{0},\tau ])$ with covariance function $\mathcal {V}_{0}$.

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

$$\begin{aligned} \sup \limits _{t\in \left[ {0},\tau \right] }\left| \hat{\mathsf {S}}_{n}\left( {t}\right) -\mathsf {S}_{0}\left( {t}\right) -\phi ^{\prime }\left( \beta _{0}\right) \left[ \hat{\beta }_{n}-\beta _{0}\right] \left[ {t}\right] \right| ={o}_{p}\left( {n}^{-1/2}\right) , \end{aligned}$$

where $\beta _{0}=\left( \mathsf {S}_{0}^{\left( {0},{1}\right) },\mathsf {S}_{0}^{\left( {0},{2}\right) },\mathsf {S}_{0}^{\left( {1},{2}\right) }\right)$ and $\hat{\beta }_{n}=\left( \hat{\mathsf {S}}_{n}^{\left( {0},{1}\right) },\hat{\mathsf {S}}_{n}^{\left( {0},{2}\right) },\hat{\mathsf {S}}_{n}^{\left( {1},{2}\right) }\right)$.

Therefore, as ${n}\rightarrow \infty , {n}^{{1}/{2}}(\hat{\mathsf {S}}_{n}-\mathsf {S}_{0})$ weakly converges to a tight zero mean Gaussian process in $\ell ^{\infty }\left( \left[ {0},\tau \right] \right)$ with covariance function $\varOmega _{0}$.$\square$

1.3 Proof of Theorems 4 and 5

Proof

To obtain the efficiency, we will verify the conditions of Theorem VIII.3.2 and Theorem VIII.3.3 of Andersen et al. (1993).

First, we verify the local asymptotic normality (LAN) Assumption (Assumption VIII.3.1 of Andersen et al. 1993).

By the Taylor expansion, it is easy to show that for any ${h}=({h}^{\left( {0},{1}\right) },{h}^{\left( {0},{2}\right) },{h}^{\left( {1},{2}\right) })\in \mathbb {H}$,

$$\begin{aligned} \log \mathcal {R}_{n}\left( {h}\right) =\mathcal {S}_{n}^{\left( {0},{1}\right) }\left( {h}\right) +\mathcal {S}_{n}^{\left( {0},{2}\right) }\left( {h}\right) +\mathcal {S}_{n}^{\left( {1},{2}\right) }\left( {h}\right) -\dfrac{1}{2}\left\| {h}\right\| _{\mathbb {H}}^{2}+{o}_{p}\left( {1}\right) , \end{aligned}$$

(11)

where for any $\left( {i},{j}\right) \in \left\{ \left( {0},{1}\right) ,\left( {0},{2}\right) ,\left( {1},{2}\right) \right\}$ and any ${h}=({h}^{\left( {0},{1}\right) },{h}^{\left( {0},{2}\right) },{h}^{\left( {1},{2}\right) })\in \mathbb {H}$,

$$\begin{aligned} \mathcal {S}_{n}^{\left( {i},{j}\right) }\left( {h}\right) ={n}^{-{1}/{2}}\int _{\left[ {0},\tau \right] }{h}^{\left( {i},{j}\right) }\left( {y}\right) \left( {1}-\varLambda _{0}^{\left( {i},{j}\right) }\left\{ {y}\right\} \right) ^{-1}\;\overline{\mathsf {M}}_{n}^{\left( {i},{j}\right) }\left( d{y}\right) . \end{aligned}$$

Hence, for any ${m}\in \mathbb {N}^{+}$ and any ${h}_{1},\ldots ,{h}_{m}\in \mathbb {H}$,

$$\begin{aligned} \left( \log \mathcal {R}_{n}\left( {h}_{1}\right) ,\ldots ,\log \mathcal {R}_{n}\left( {h}_{m}\right) \right) \end{aligned}$$

weakly converges to a Gaussian random vector in $\mathbb {R}^{m}$ with mean

$$\begin{aligned} -\dfrac{1}{2}\left( \left\| {h}_{1}\right\| _{\mathbb {H}},\ldots ,\left\| {h}_{m}\right\| _{\mathbb {H}}\right) \end{aligned}$$

and covariance matrix

$$\begin{aligned} \begin{pmatrix} \left<{h}_{1},{h}_{1}\right>_{\mathbb {H}}&{}\quad \ldots &{}\quad \left<{h}_{1},{h}_{m}\right>_{\mathbb {H}}\\ \vdots &{}\ddots &{}\vdots \\ \left<{h}_{m},{h}_{1}\right>_{\mathbb {H}}&{}\quad \cdots &{}\quad \left<{h}_{m},{h}_{m}\right>_{\mathbb {H}}\\ \end{pmatrix}. \end{aligned}$$

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

For any $\left( {i},{j}\right) \in \left\{ \left( {0},{1}\right) ,\left( {0},{2}\right) ,\left( {1},{2}\right) \right\}$, any ${h}=({h}^{\left( {0},{1}\right) },{h}^{\left( {0},{2}\right) },{h}^{\left( {1},{2}\right) })\in \mathbb {H}$, any ${t}\in \left[ {0},\tau \right]$,

$$\begin{aligned} {n}^{{1}/{2}}\left( \varLambda _{n,h}^{\left( {i},{j}\right) }\left( {t}\right) -\varLambda _{0}^{\left( {i},{j}\right) }\left( {t}\right) \right) =\int _{\left[ {0},{t}\right] }{h}^{\left( {i},{j}\right) }\left( {y}\right) \;\varLambda _{0}^{\left( {i},{j}\right) }\left( d{y}\right) . \end{aligned}$$

Note that, for any $\left( {i},{j}\right) \in \left\{ \left( {0},{1}\right) ,\left( {0},{2}\right) ,\left( {1},{2}\right) \right\}$, any ${h}=({h}^{\left( {0},{1}\right) },{h}^{\left( {0},{2}\right) },{h}^{\left( {1},{2}\right) })\in \mathbb {H}$, and any ${t}\in \left[ {0},\tau \right]$,

$$\begin{aligned} \left| \int _{\left[ {0},{t}\right] }{h}^{\left( {i},{j}\right) }\left( {y}\right) \;\varLambda _{0}^{\left( {i},{j}\right) }\left( d{y}\right) \right| \leqslant \left\| {h}\right\| _{\mathbb {H}}\varLambda _{0}^{\left( {i},{j}\right) }\left( \tau \right) \left( \mathsf {L}_{0}^{\left( {i}\right) }\left( \tau -\right) \right) ^{-1}. \end{aligned}$$

Hence, the Differentiability Assumption is verified.

Next, we show that $\left( \hat{\varLambda }_{n}^{\left( {0},{1}\right) },\hat{\varLambda }_{n}^{\left( {0},{2}\right) },\hat{\varLambda }_{n}^{\left( {1},{2}\right) }\right)$ is regular.

Recall the weak convergence of the log-likelihood ratio $\log \mathcal {R}_{n}\left( {h}\right)$. The continuity follows from the Le Cam’s third Lemma and the weak convergence can be shown via similar arguments in the previous subsection. Hence, the regularity follows.

It remains to check Equation (8.3.5) of Andersen et al. (1993) for each coordinate.

For any ${t}\in \left[ {0},\tau \right]$, define

$$\begin{aligned} \gamma _{t}^{\left( {0},{1}\right) }=\left( \varphi _{t}^{\left( {0},{1}\right) },{0},{0}\right) ,\; \gamma _{t}^{\left( {0},{2}\right) }=\left( {0},\varphi _{t}^{\left( {0},{2}\right) },{0}\right) ,\; \gamma _{t}^{\left( {1},{2}\right) }=\left( {0},{0},\varphi _{t}^{\left( {1},{2}\right) }\right) \end{aligned}$$

where for any $\left( {i},{j}\right) \in \left\{ \left( {0},{1}\right) ,\left( {0},{2}\right) ,\left( {1},{2}\right) \right\}$ and any ${s}\in \left[ {0},\tau \right]$,

$$\begin{aligned} \varphi _{t}^{\left( {i},{j}\right) }\left( {s}\right) =\mathrm {I}\left\{ {s}\leqslant {t}\right\} \mathsf {L}_{0}^{\left( {i}\right) }\left( {s}-\right) ^{-1}\left( {1}-\varLambda _{0}^{\left( {i},{j}\right) }\left\{ {y}\right\} \right) . \end{aligned}$$

By (10) and (11), for any $\left( {i},{j}\right) \in \left\{ \left( {0},{1}\right) ,\left( {0},{2}\right) ,\left( {1},{2}\right) \right\}$ and any ${t}\in \left[ {0},\tau \right]$,

$$\begin{aligned} \hat{\varLambda }_{n}^{\left( {i},{j}\right) }\left( {t}\right) -\varLambda _{0}^{\left( {i},{j}\right) }\left( {t}\right) -{n}^{-{1}/{2}}\left( \log \mathcal {R}_{n}\left( \gamma _{t}^{\left( {i},{j}\right) }\right) +\dfrac{1}{2}\left\| {h}\right\| _{\mathbb {H}}^{2}\right) ={o}_{p}\left( {n}^{-{1}/{2}}\right) . \end{aligned}$$

Therefore, combining all the above results with Theorem VIII.3.2 and Theorem VIII.3.3 of Andersen et al. (1993), it follows that $\left( \hat{\varLambda }_{n}^{\left( {0},{1}\right) },\hat{\varLambda }_{n}^{\left( {0},{2}\right) },\hat{\varLambda }_{n}^{\left( {1},{2}\right) }\right)$ is asymptotically efficient.

Combining Theorem VIII.3.4 of Andersen et al. (1993) with Lemma 3, the efficiency of $\hat{\mathsf {S}}_{n}$ follows.$\square$