A functional inference for multivariate current status data with mismeasured covariate

Abstract

Covariate measurement error problems have been recently studied for current status failure time data but not yet for multivariate current status data. Motivated by the three-hypers dataset from a health survey study, where the failure times for three-hypers (hyperglycemia, hypertension, hyperlipidemia) are subject to current status censoring and the covariate self-reported body mass index may be subject to measurement error, we propose a functional inference method under the proportional odds model for multivariate current status data with mismeasured covariates. The new proposal utilizes the working independence strategy to handle correlated current status observations from the same subject, as well as the conditional score approach to handle mismeasured covariate without specifying the covariate distribution. The asymptotic theory, together with a stable computation procedure combining the Newton–Raphson and self-consistency algorithms, is established for the proposed estimation method. We evaluate the method through simulation studies and illustrate it with three-hypers data.

Appendix

We use the notation ${\mathbb P}_n, P_0,$ and $P$ for the expectations taken under the empirical distribution, the true underlying distribution, and a given model, respectively. For simplicity the assumptions and proofs for theories are presented under the simpler setting where the distribution of $(C_1,\ldots ,C_K)$ is independent of $(X,Z_1,\ldots ,Z_K),$ though the proposed method can allow the dependence case. We consider $H_1,\ldots ,H_K$ are functions in $\mathcal{H},$ the set of right-continuous non-decreasing functions that are uniformly bounded on the study period $[0,\tau ^*].$ Let $\ell ({\varvec{\theta }})=\log L({\varvec{\theta }}).$ The asymptotic theories are based on the following regularity assumptions, which have been similarly made in the studies of univariate current status data (e.g. Huang 1996; van der Vaart 1998; Ma 2009; Wen and Chen 2012).

• (C1) The examination times $C_1,\ldots ,C_K$ possess a common continuous density whose support is an interval $[\tau _1,\tau _2]$ with $0<\tau _1<\tau _2<\tau ^*.$

• (C2) The true parameter $({\varvec{\beta }}_0,\sigma ^2_0)$ lies in the interior of a compact subset $\mathcal B \times \mathcal{Q}$ of ${\mathbb R}^d \times (0,\infty )$; all $H_{10},\ldots , H_{K0}$ are continuously differentiable on $[\tau _1,\tau _2]$ and satisfy $-M<H_{k0}(\tau _1)<H_{k0}(\tau _2)<M$ for $k=1,\ldots ,K.$

• (C3) The distribution of $(X,\mathbf{Z}_1,\ldots ,\mathbf{Z}_K)$ is not concentrated on any proper subspace of ${\mathbb R}^{1+\sum _k d_k}$ and has a bounded support, where $d_k=\dim (\mathbf{Z}_k)$.

• (C4) The functions $\mathbf{g}_1^*,\ldots ,\mathbf{g}_K^*$ given in (6) are differentiable with bounded derivatives on $[\tau _1,\tau _2]$.

• (C5) The information matrix $\mathcal{I}$ defined in (7) is invertible.

Proof of Theorem 1

Let $(\widehat{H}_{1,({\varvec{\beta }},\sigma ^2)}, \ldots , \widehat{H}_{K,({\varvec{\beta }},\sigma ^2)} )$ denote the maximizer of $L_n$ with $({\varvec{\beta }}, \sigma ^2)$ fixed. We first apply Theorem 5.7 of van der Vaart (1998) to establish the consistency of $(\widehat{H}_{1,({\varvec{\beta }}_0,\widehat{\sigma }^2)}, \ldots , \widehat{H}_{K,({\varvec{\beta }}_0,\widehat{\sigma }^2)} )$. Since the class of monotone and uniformly bounded functions is a Donsker class, by Theorem 2.10.6 of van der Vaart and Wellner (1996) and conditions (C1)-(C3), we know that the class $\{ \ell ({\varvec{\beta }}_0,H_1,\ldots ,H_K,$ $\sigma ^2_0) \ | \ H_1,\ldots ,H_K \in \mathcal{H} \}$ is Donsker and hence Glivenko-Cantelli. Further, by Jensen’s inequality, we have

$$\begin{aligned} P_0 \{\ell ({\varvec{\beta }}_0,H_1,\ldots ,H_K,\sigma ^2_0)-\ell ({\varvec{\theta }}_0)\} \le \log (P_0 \left\{ \frac{L({\varvec{\beta }}_0,H_1,\ldots ,H_K,\sigma ^2_0)}{L({\varvec{\theta }}_0)}\right\} ) =0, \end{aligned}$$

and the equality holds only if $H_k=H_{k0}$ on $(\tau _1,\tau _2)$ for all $k=1,\ldots ,K.$ This indicates that

$$\begin{aligned} \sup _{\sum _{k=1}^K\Vert H_k-H_{k0}\Vert _2 >\varepsilon }P_0 \ell ({\varvec{\beta }}_0,H_1,\ldots ,H_K,\sigma ^2_0)< P_0\ell ({\varvec{\theta }}_0). \end{aligned}$$

Furthermore, note that

$$\begin{aligned} {\mathbb P}_n \ell ({\varvec{\theta }}_0)+o_p(1)&= {\mathbb P}_n \ell ({\varvec{\beta }}_0,H_{10},\ldots ,H_{K0},\widehat{\sigma }^2)\\&\le {\mathbb P}_n \ell ({\varvec{\beta }}_0, \widehat{H}_{1,({\varvec{\beta }}_0,\widehat{\sigma }^2)}, \ldots , \widehat{H}_{K,({\varvec{\beta }}_0,\widehat{\sigma }^2)} ,\widehat{\sigma }^2 )\\&= {\mathbb P}_n \ell ({\varvec{\beta }}_0,\widehat{H}_{1,({\varvec{\beta }}_0,\widehat{\sigma }^2)}, \ldots , \widehat{H}_{K,({\varvec{\beta }}_0,\widehat{\sigma }^2)} ,\sigma ^2_0)+o_p(1), \end{aligned}$$

where the inequality follows from the definition of $(\widehat{H}_{1,({\varvec{\beta }},\sigma ^2)}, \ldots , \widehat{H}_{K,({\varvec{\beta }},\sigma ^2)} )$, and two equalities are obtained by the mean value theorem and the consistency of $\widehat{\sigma }^2.$ Therefore, by Theorem 5.7 of citev98, we have $\sum _{k=1}^K \Vert \widehat{H}_{k,({\varvec{\beta }}_0,\widehat{\sigma }^2)}-H_{k0}\Vert _2 {\rightarrow } \ 0,$ and so $\widehat{H}_{k,({\varvec{\beta }}_0,\widehat{\sigma }^2)}(t) \mathop {\rightarrow }\limits ^{P} H_{k0}(t),$ $k=1,\ldots , K,$ for every $t$ in $[\tau _1,\tau _2]$.

By the consistency shown above and the mean value theorem, we have

$$\begin{aligned} {\mathbb P}_n {\varvec{\ell }}_0({\varvec{\beta }}_0,\widehat{H}_{1,({\varvec{\beta }}_0,\widehat{\sigma }^2)}, \ldots , \widehat{H}_{K,({\varvec{\beta }}_0,\widehat{\sigma }^2)},\widehat{\sigma }^2)={\mathbb P}_n {\varvec{\ell }}_0({\varvec{\beta }}_0,H_{10}, \ldots , H_{K0},\sigma ^2_0)+o_p(1). \end{aligned}$$

Since the right hand side of the above display set to zero is an unbiased estimating equation, a consistent sequence of solutions of ${\varvec{\beta }}$ to it exists, indicating the existence of a consistent solution of ${\varvec{\beta }}$ to the CS estimating equation ${\mathbb P}_n {\varvec{\ell }}_0({\varvec{\beta }},\widehat{H}_{1,({\varvec{\beta }},\widehat{\sigma }^2)}, \ldots , \widehat{H}_{K,({\varvec{\beta }},\widehat{\sigma }^2)},\widehat{\sigma }^2)=0.$

Now we shall prove

$$\begin{aligned} \left( \sum _{k=1}^K\Vert \widehat{H}_{k,({\varvec{\beta }},\sigma ^2)}-H_0\Vert _2^2\right) ^{1/2}=O_P(\Vert {\varvec{\beta }}-{\varvec{\beta }}_0\Vert +\Vert \sigma ^2-\sigma ^2_0\Vert +n^{-1/3}), \end{aligned}$$

by verifying the conditions (3.5) and (3.6) in Theorem 3.2 of Murphy and van der Vaart (1999). Here $\Vert \cdot \Vert$ is the Euclidean norm, $\Vert H\Vert _2^2=\int H^2(u)dQ(u),$ and $Q$ denotes the marginal distribution of $C_k.$ The rates of convergence and consistencies of $\widehat{H}_k, k=1,\ldots ,K,$ can then be obtained by the consistency of $(\widehat{\varvec{\beta }}, \widehat{\sigma }^2).$

Given two functions $l$ and $u$, the bracket $[l,u]$ is the set of all functions $f$ with $l\le f \le u$. An $\varepsilon$-bracket in $L_2(P)=\{f: Pf^2<\infty \}$ is a bracket $[l,u]$ with $P(u-l)^2<\varepsilon ^2.$ For a subclass $\mathcal{C}$ of $L^2(P),$ the bracketing number $N_{[ \ ]}(\varepsilon ,\mathcal{C},L_2(P))$ is the minimum number of $\varepsilon$-bracket need to cover $\mathcal C$ (see van der Vaart 1998). Let $\varPsi =\{ \ell ({\varvec{\theta }}) : {\varvec{\theta }}\in \mathcal{B}\times \mathcal{H}^K \times \mathcal{Q}\}.$ Note that each element in $\varPsi$ is uniformly bounded and satisfies $P_0\{ \ell ({\varvec{\theta }}) -\ell ({\varvec{\beta }},H_{10},\ldots ,H_{K0},\sigma ^2) \}^2 \preceq \sum _{k=1}^K \Vert H_k-H_{k0}\Vert _2^2+\Vert {\varvec{\beta }}-{\varvec{\beta }}_0\Vert ^2+\Vert \sigma ^2-\sigma ^2_0\Vert ^2.$ The notation $\preceq$ means smaller than, up to a constant. Lemma 1 below gives the bracketing integral $J(\varDelta ,{\varPsi },L_2(P))$, defined as $\int _0^{\varDelta } \{1+\log N_{[ \ ]}(\varepsilon ,\varPsi ,L_2(P))\}^{1/2}d\varepsilon ,$ is $O(\varDelta ^{1/2})$. It then follows from Lemma 3.3 of Murphy and van der Vaart (1999) that their condition (3.6) is satisfied for $\phi _n(\varDelta )=\varDelta ^{1/2}.$

A similar Taylor series argument used in Lemma 2 below gives $P_0\{\ell ({\varvec{\theta }}_0)-\ell ({\varvec{\beta }},H_{10},\ldots ,H_{K0},\sigma ^2))\}\preceq \Vert {\varvec{\beta }}-{\varvec{\beta }}_0\Vert ^2+\Vert \sigma ^2-\sigma ^2_0\Vert ^2.$ This and Lemma 2 imply

$$\begin{aligned} P_0\{\ell ({\varvec{\theta }})\!-\!\ell ({\varvec{\beta }},H_{10},\ldots ,H_{K0},\sigma ^2)\}\preceq \!-\!\sum _{k=1}^K \Vert H_k\!-\!H_{k0}\Vert _2^2\!+\!\Vert {\varvec{\beta }}\!-\!{\varvec{\beta }}_0\Vert ^2\!+\!\Vert \sigma ^2\!-\!\sigma ^2_0\Vert ^2, \end{aligned}$$

which is condition (3.5) of Murphy and van der Vaart (1999). This completes the proof.$\square$

Lemma 1

$\log N_{[ \ ]}(\varepsilon ,\varPsi ,L_2(P_0)) =O( {1/ \varepsilon })$.

Proof

For fixed $({\varvec{\beta }}, \sigma ^2)$ and $1\le k \le K,$ the functions in $\varPsi$ depend on $H_k$ monotonically for $\varDelta _{k}=1$ and $\varDelta _{k}=0$ separately. Thus, given a $\varepsilon$-bracket $H^{L}_k\le H_k \le H^{U}_k,$ it follows from monotonicity of $\mathcal{E}_{k}$ in $H_k$ that we can get a bracket $(\ell ^L,\ell ^U)$ for $\ell ({\varvec{\theta }})$ where

$$\begin{aligned} \ell ^L&\equiv \log \prod _{k=1}^K \left[ \mathcal{E}_{k} ({\varvec{\beta }},H_1^L,\ldots ,H_K^L,\sigma ^2)^{\varDelta _{k}} \{1-\mathcal{E}_{k} ({\varvec{\beta }},H_1^U,\ldots ,H_K^U,\sigma ^2)\} ^{1-\varDelta _{k}} \right] ;\\ \ell ^U&\equiv \log \prod _{k=1}^K \left[ \mathcal{E}_{k}({\varvec{\beta }},H_1^U,\ldots ,H_K^U,\sigma ^2)^{\varDelta _{k}} \{1-\mathcal{E}_{k}({\varvec{\beta }},H_1^L,\ldots ,H_K^L,\sigma ^2)\} ^{1-\varDelta _{k}} \right] . \end{aligned}$$

Further, by the mean value theorem, we have $|\ell ^L-\ell ^U|^2 \preceq \sum _{k=1}^K (H_k^{U}-H_k^{L})^2 (C_{k}).$ Thus brackets for $H_k$ of $\Vert \cdot \Vert _2$-size $\varepsilon$ can translate into brackets for $\ell ({\varvec{\theta }})$ of $L_2(P_0)$-size proportional to $\varepsilon .$ By Example 19.11 of van der Vaart (1998), we can cover the set of all $H_k$ by $\exp (C/\varepsilon )$ brackets of size $\varepsilon$ for some constant $C.$ Next we allow $\zeta =({\varvec{\beta }}',\sigma ^2)'$ to vary freely as well. Because $\mathcal{B}\times \mathcal{Q}$ is finite-dimensional and $(\partial /\partial \zeta ) \ell ({\varvec{\theta }})(O)$ is uniformly bounded in $({\varvec{\theta }},O),$ this increases the entropy only slightly. Lemma 1 is thus proved.$\square$

Lemma 2

For ${\varvec{\theta }}$ near ${\varvec{\theta }}_0,$ $P_0\{ \ell ({\varvec{\theta }})-\ell ({\varvec{\theta }}_0) \} \preceq -\{\sum _{k=1}^K\Vert H_k-H_{k0}\Vert _2^2+\Vert {\varvec{\beta }}-{\varvec{\beta }}_0\Vert ^2+\Vert \sigma ^2-\sigma ^2_0\Vert ^2\}.$

Proof

Let ${\varvec{\theta }}_1=({\varvec{\beta }},H_{1},\ldots ,H_{K},\sigma ^2_0).$ It suffices to show

$$\begin{aligned}&P_0\{ \ell ({\varvec{\theta }}_1)-\ell ({\varvec{\theta }}_0) \} \preceq -\{\sum _{k=1}^K\Vert H_k-H_{k0}\Vert _2^2+\Vert {\varvec{\beta }}-{\varvec{\beta }}_0\Vert ^2\},\end{aligned}$$

(9)

$$\begin{aligned}&P_0\{ \ell ({\varvec{\theta }})-\ell ({\varvec{\theta }}_1)\} \preceq -\Vert \sigma ^2-\sigma ^2_0\Vert ^2. \end{aligned}$$

(10)

Let $V=\{(C_k,S_k(\beta _{10},\sigma ^2_0),Z_k), k=1,\ldots ,K \}.$ By the Kullback-Leibler inequality, $E_{0}\{ \ell ({\varvec{\theta }}_1)(O)|V \}$ is maximized at $({\varvec{\beta }}_0,H_{10}(C_1),\ldots ,H_{K0}(C_K)).$ So its first derivative is equal to 0 there (this can also be verified directly by the fact that $E_0\{\varDelta _k|V\}=\mathcal{E}_k({\varvec{\theta }}_0)(O)$). Since $(C_k,S_k,Z_k)$’s have bounded support and the parameter spaces are compact, a Taylor’s expansion gives $E_0\{ \ell ({\varvec{\theta }}_1)(O)-\ell ({\varvec{\theta }}_0)(O) |V \} \preceq -\{\sum _{k} (H_k(C_k)-H_{k0}(C_k))^2+\Vert {\varvec{\beta }}-{\varvec{\beta }}_0\Vert ^2\},$ which implies (9).

We now prove (10). Denote $p_{{\varvec{\theta }}}$ the density of observed data $O$ given the model ${\varvec{\theta }}.$ Because the densities $p_{{\varvec{\theta }}_0}/p_{{\varvec{\theta }}_1}$ are uniformly bounded above and below by a positive constant, (10) is equivalent to

$$\begin{aligned} P_{{\varvec{\theta }}_1}\{ \ell ({\varvec{\theta }})-\ell ({\varvec{\theta }}_1) \}\preceq - \Vert \sigma ^2-\sigma ^2_0\Vert ^2. \end{aligned}$$

(11)

Let $V_1=\{(C_k,S_k(\beta _{1},\sigma ^2_0),Z_k), k=1,\ldots ,K \}.$ By the fact that $E_{{\varvec{\theta }}_1}\{\varDelta _k|V_1\}=\mathcal{E}_k({\varvec{\theta }}_1)(O),$ we can show directly that the first derivative of $E_{{\varvec{\theta }}_1}\{ \ell ({\varvec{\theta }})(O)|V_1 \}$ with respect to $\sigma ^2$ at $\sigma ^2=\sigma ^2_0$ is equal to 0. Note that the second derivative of $E_{{\varvec{\theta }}_1}\{ \ell ({\varvec{\theta }})(O)|V_1 \}$ with respect to $\sigma ^2$ is given by $-E_{{\varvec{\theta }}_1}\{\beta _1^4 \sum _k \mathcal{V}_k({\varvec{\theta }})(O)/(4m^2)|V_1 \}.$ Consequently, a Taylor’s expansion around $\sigma ^2=\sigma ^2_0$ can thus yield $E_{{\varvec{\theta }}_1}\{ \ell ({\varvec{\theta }})(O)-\ell ({\varvec{\theta }}_1)(O) |V_1 \} \preceq - \Vert \sigma ^2-\sigma ^2_0\Vert ^2,$ which implies (11). This completes the proof.$\square$

For fixed $1\le k \le K,$ consider a parametric path $H_{k,\varepsilon }$ in $\mathcal H$ through $H_k,$ that is, $H_{k,\varepsilon } \in \mathcal{H}$ and $H_{k,\varepsilon }=H_k$ when $\varepsilon =0.$ Let $\dot{\mathcal{H}}=\{g_k:({\partial }/{\partial \varepsilon })|_{\varepsilon =0} H_{k,\varepsilon }=g_k,k=1,\ldots ,K\}.$ Then the score for $H_k$ along the direction $g_k,$ define by $({\partial }/{\partial \varepsilon })|_{\varepsilon =0} \ell ({\varvec{\beta }},H_1,\ldots ,H_{k-1},H_{k,\varepsilon },H_{k+1},\ldots ,H_K,\sigma ^2),$ has the form

$$\begin{aligned} \ell _k({\varvec{\theta }})[g_k](O)= g(C_{k})\{\varDelta _{k}-\mathcal{E}_{k}({\varvec{\theta }})(O)\}. \end{aligned}$$

Also define $\ell _{0k}({\varvec{\theta }})[g_k]=({\partial }/{\partial \varepsilon }) |_{\varepsilon =0} \ell _0({\varvec{\beta }},H_1,\ldots ,H_{k-1},H_{k,\varepsilon },H_{k+1},\ldots ,H_K,\sigma ^2)$ and $\ell _{k'k}({\varvec{\theta }})[\tilde{g}_{k'},g_k]=({\partial }/{\partial \varepsilon }) |_{\varepsilon =0} \ell _{k'}({\varvec{\beta }},H_1,\ldots ,H_{k-1},H_{k,\varepsilon },H_{k+1},\ldots ,H_K,\sigma ^2)[\tilde{g}_{k'}],$ where $1\le k,k'\le K,$ $g_k$ and $\tilde{g}_k'$ are in $\dot{\mathcal{H}}.$ They have forms

$$\begin{aligned}&\ell _{0k}({\varvec{\theta }})[g_k]=- g_k(C_{k})\{S_{k}-{\varvec{\beta }}_1{\tilde{\sigma }^2}, Z' \}'\mathcal{V}_{k}({\varvec{\theta }})(O),\\&\ell _{kk}({\varvec{\theta }})[\tilde{g}_k, g_k]=- g_k(C_{k}){\tilde{g}}_k(C_{k})\mathcal{V}_{k}({\varvec{\theta }})(O), \end{aligned}$$

and $\ell _{k'k}({\varvec{\theta }})[\tilde{g}_k', g_k]=0$ if $k'\ne k.$ Following semiparametric M-estimator theories (e.g. Korosok 2008), the function ${\varvec{\ell }}^*$ given in Sect. 3 is defined as ${\varvec{\ell }}^*({\varvec{\theta }})= \ell _0({\varvec{\theta }})- \sum _{k=1}^K \ell _k({\varvec{\theta }})[\mathbf g_k^*],$ where $\mathbf{g}_k^*$ is the $d$-dimensional $(d=\dim ({\varvec{\beta }}))$ vector-valued function satisfying

$$\begin{aligned} P_0 \left( \sum _{k=1}^K\ell _{0k}({\varvec{\theta }}_0)[g_k]-\sum _{k'=1}^K\sum _{k=1}^K\ell _{k'k}({\varvec{\theta }}_0)[\mathbf{g}_{k'}^*,g_k]\right) =0, \end{aligned}$$

(12)

for all $g_k$ in $\dot{\mathcal{H}}.$ Fixing one $k$ $(1\le k\le K)$ and setting $g_{k'}=0$ for all $k' \ne k$ in (12), we have

$$\begin{aligned}&\int g_k(u)E[\{S_{k}-\beta _1{\tilde{\sigma }^2}, Z' \}'\mathcal{V}_{k}({\varvec{\theta }})(O)|C_{k}=u ]dQ(u)\\&\quad = \int g_k(u)\mathbf{g}_k^*(u)E[\mathcal{V}_{k}({\varvec{\theta }})(O)|C_{k}=u ]dQ(u), \end{aligned}$$

which implies that $\mathbf{g}_k^*$ is given by (6). Below we establish the asymptotic theory of the CS estimator.

Proof of Theorem 2

We first verify

$$\begin{aligned} \sqrt{n}P_0 {\varvec{\ell }}^*({\varvec{\beta }}_0,\widehat{H}_1,\ldots ,\widehat{H}_K,\sigma ^2_0)=o_p(1). \end{aligned}$$

(13)

Apply a Taylor expansion to ${\varvec{\ell }}^*({\varvec{\beta }}_0, H_1,\ldots ,H_K,\sigma ^2_0)(O)$ at the point $(H_{10}(C_{1}),\ldots ,$ $H_{K0}(C_{K}))$ to get

$$\begin{aligned}&P_0 {\varvec{\ell }}^*({\varvec{\beta }}_0,H_1,\ldots ,H_K,\sigma ^2_0)- P_0 {\varvec{\ell }}^*({\varvec{\theta }}_0) \\ \nonumber&\quad =P_0\left( \sum _{k} \ell _{0k}({\varvec{\theta }}_0)[H_k-H_{k0}]-\sum _{k',k}\ell _{k'k}({\varvec{\theta }}_0)[\mathbf{g}_{k'}^*,H_k-H_{k0}]\right) \\ \nonumber&\qquad +O_p(\sum _{k=1}^K\Vert H_k-H_{k0}\Vert ^2_2). \end{aligned}$$

(14)

Using (12), the fact that $P_0 {\varvec{\ell }}^*({\varvec{\theta }}_0)=0,$ and applying the rates of convergence on $\widehat{H}_k, k=1,\ldots , K,$ to (14), we get (13).

Applying Theorem 2.10.6 of van der Vaart and Wellner (1996), it can be verified with condition (C4) that $\{ {\varvec{\ell }}^*({\varvec{\theta }}) | {\varvec{\theta }}\in \mathcal{B} \times \mathcal{H}^K \times \mathcal{Q}\}$ and $\{ \varphi (\sigma ^2) | \sigma ^2 \in \mathcal{Q} \}$ are uniformly bounded Donsker classes; the proof of which is technical and hence omitted here. Combining this with the consistency of $\widehat{\varvec{\theta }}$ leads to

$$\begin{aligned} \sqrt{n} ({\mathbb P}_n-P_0) \left[ \begin{array}{c} {\varvec{\ell }}^*(\widehat{\varvec{\theta }})-{\varvec{\ell }}^*({\varvec{\theta }}_0)\\ \varphi (\widehat{\sigma }^2)-\varphi (\sigma ^2_0) \end{array} \right] =o_p(1). \end{aligned}$$

Adding (13) to the first row of preceding display and using the facts that $P_0 {\varvec{\ell }}^*({\varvec{\theta }}_0)=0$ and ${\mathbb P}_n {\varvec{\ell }}^*({\widehat{\varvec{\theta }}})={\mathbb P}_n\varphi (\widehat{\sigma }^2)=0,$ it is seen that

$$\begin{aligned} -\sqrt{n}P_0 \left[ \begin{array}{c} {\varvec{\ell }}^*(\widehat{\varvec{\theta }})-{\varvec{\ell }}^*({\varvec{\beta }}_0,{\widehat{H}}_1,\ldots ,\widehat{H}_K,\sigma ^2_0)\\ \varphi (\widehat{\sigma }^2)-\varphi (\sigma ^2_0) \end{array} \right] =\sqrt{n}{\mathbb P}_n \left[ \begin{array}{c} {\varvec{\ell }}^*({\varvec{\theta }}_0)\\ \varphi (\sigma ^2_0) \end{array} \right] +o_p(1). \end{aligned}$$

By the mean value theorem, there exists $(\tilde{{\varvec{\beta }}},\tilde{\sigma ^2})$ lying between $(\widehat{{\varvec{\beta }}},\widehat{\sigma }^2)$ and $({\varvec{\beta }}_0,\sigma ^2_0)$ such that

$$\begin{aligned}&-\sqrt{n} P_0 \left[ \begin{array}{cc} \frac{\partial }{\partial {\varvec{\beta }}} {\varvec{\ell }}^*({\tilde{{\varvec{\beta }}}} ,{\widehat{H}}_1,\ldots ,\widehat{H}_K,\tilde{\sigma ^2}) &{} \frac{\partial }{\partial \sigma ^2 } {\varvec{\ell }}^*({\tilde{{\varvec{\beta }}}},{\widehat{H}}_1,\ldots ,\widehat{H}_K,\tilde{\sigma ^2})\\ 0 &{}\frac{\partial }{\partial \sigma ^2 }\varphi (\tilde{\sigma ^2}) \end{array} \right] \left( \begin{array}{c} {\widehat{\varvec{\beta }}}-{\varvec{\beta }}_0\\ \widehat{\sigma }^2-\sigma ^2_0 \end{array} \right) \\&\quad =\sqrt{n}{\mathbb P}_n \left[ \begin{array}{c} {\varvec{\ell }}^*({\varvec{\theta }}_0)\\ \varphi (\sigma ^2_0) \end{array} \right] +o_p(1). \end{aligned}$$

By the consistency of $(\widehat{\varvec{\beta }},\widehat{\sigma }^2)$ and condition (C5), we have

$$\begin{aligned} \sqrt{n}\left[ \begin{array}{c} {\widehat{\varvec{\beta }}}-{\varvec{\beta }}_0\\ \widehat{\sigma }^2-\sigma ^2_0 \end{array} \right] = \mathcal{I}^{-1}\sqrt{n}{\mathbb P}_n \left[ \begin{array}{c} {\varvec{\ell }}^*({\varvec{\theta }}_0)\\ \varphi (\sigma ^2_0) \end{array} \right] +o_P(1)\mathop {\rightarrow }\limits ^{d} N(0,\mathcal{I}^{-1}{\varSigma }(\mathcal{I}^{-1})'). \end{aligned}$$

This completes the proof.$\square$