Empirical likelihood test in a posteriori change-point nonlinear model

Abstract

In this paper, in order to test whether changes have occurred in a nonlinear parametric regression, we propose a nonparametric method based on the empirical likelihood. Firstly, we test the null hypothesis of no-change against the alternative of one change in the regression parameters. Under null hypothesis, the consistency and the convergence rate of the regression parameter estimators are proved. The asymptotic distribution of the test statistic under the null hypothesis is obtained, which allows to find the asymptotic critical value. On the other hand, we prove that the proposed test statistic has the asymptotic power equal to 1. These theoretical results allows find a simple test statistic, very useful for applications. The epidemic model, a particular model with two change-points under the alternative hypothesis, is also studied. Numerical studies by Monte Carlo simulations show the performance of the proposed test statistic.

Appendix

The following lemma will be used in the proof of propositions, theorems and of other lemmas.

Lemma 1

Let $\mathbf{X}=(X_1,\ldots ,X_p)$ a random vector (column), with the random variables $X_1,\ldots ,X_p$ not necessarily independent, and $\mathbf M =(m_{ij})_{1 \le i,j \le p}$, such that $\mathbf M = \mathbf{X}\mathbf{X}^t$. If for $j=1,\ldots , p$, we have

$$\begin{aligned} for\,\,all\,\,\eta _j>0,there\,\,exists\,\, \delta _j >0\,\,\,\,\,such\,\,that\,\,\,\,I\!P[|X_j|\ge \delta _j]\le \eta _j, \end{aligned}$$

(51)

then

• (i) $I\!P\big [ \Vert \mathbf{X}\Vert _1 \ge p \max _{1 \le j \le p}\delta _j \big ]\le \max _{1 \le j \le p} \eta _j$,

• (ii) $I\!P\big [ \Vert \mathbf{X}\Vert _2 \ge \sqrt{p} \max _{1 \le j \le p}\delta _j\big ]\le \max _{1 \le j \le p}\eta _j$,

• (iii) $I\!P\big [ \Vert \mathbf M \Vert _1 \ge p \max _{1 \le i,j \le p} \{\delta _i^2,\delta _j^2\}\big ]\le \max _{1 \le i,j \le p} \{\eta _i^2,\eta _j^2\}$,

where $\Vert \mathbf M \Vert _1= \max _{1 \le j \le p} \{ \sum _{i=1}^p |m_{ij}| \}$ is the subordinate norm to the vector norm $\Vert .\Vert _1$.

Proof of Lemma 1

(i):

Using the relation (51), we can write

$$\begin{aligned} I\!P\left[ \Vert \mathbf{X}\Vert _1\ge p \max _{1\le j\le p}\delta _j\right] \le I\!P\left[ p\max _{1\le j\le p}|X_j|\ge p\max _{1\le j\le p}\delta _j\right] \le \max _{1\le j\le p}\eta _j. \end{aligned}$$

(ii):

The relation (51) is equivalent to $I\!P\big [X_j^2\ge \delta _j^2\big ] \le \eta _j$, which implies that

$$\begin{aligned} I\!P\left[ \Vert \mathbf{X}\Vert _2^2\ge p\max _{1\le j\le p}\delta _j^2\right] =I\!P\left[ \max _{1\le j \le p}X_j^2 \ge \max _{1\le j\le p}\delta _j^2\right] \le \max _{1\le j \le p}\eta _j. \end{aligned}$$

(iii):

For $1\le i,j \le p$, we have

$$\begin{aligned} I\!P\left[ |X_i X_j|\ge \max \{\delta _i^2,\delta _j^2\}\right] \le I\!P\left[ \max \{X_i^2,X_j^2\}\ge \max \{\delta _i^2,\delta _j^2\}\right] \le \max \{\eta _i^2,\eta _j^2\}. \end{aligned}$$

Then, $I\!P[|m_{ij}|\ge \max \{\delta _i^2,\delta _j^2\}] \le \max \{\eta _i^2,\eta _j^2\}$. Hence, for each $1\le j \le p$,

$$\begin{aligned} I\!P\left[ \sum _{i=1}^p| m_{ij}|{\ge } p\max \{\delta _i^2,\delta _j^2\}\right] {\le } I\!P\left[ p \max _{1 \le i \le p}|m_{ij} | {\ge } p \max \{ \delta _i^2,\delta _j^2\}\right] \le \max \{\eta _i^2,\eta _j^2\}. \end{aligned}$$

$\square$

Lemma 2

Let the $\eta$-neighbourhood of ${\varvec{\beta }}^\mathbf{0}$, $\mathcal{V}_{\eta }({\varvec{\beta }}^\mathbf{0})= \{ {\varvec{\beta }}\in \varGamma ; \Vert {\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}\Vert _2 \le \eta \}$, with $\eta \rightarrow 0$. Then, under assumptions (A1)–(A4), for all $\epsilon >0$, there exists a positive constant $M>0$, such that, for all ${\varvec{\beta }}\in \mathcal{V}_{\eta }({\varvec{\beta }}^\mathbf{0})$,

$$\begin{aligned} I\!P\big [\Vert \mathbf {g}_i({\varvec{\beta }}) \Vert _1 \ge M \big ] \le \epsilon . \end{aligned}$$

Proof of Lemma 2

In the following, for simplicity, we denote the functions $\mathbf {\overset{.}{f}}(\mathbf{X}_i,{\varvec{\beta }})$ by $\mathbf {\overset{.}{f}}_i({\varvec{\beta }})$, and $\mathbf {\overset{..}{f}}(\mathbf{X}_i,{\varvec{\beta }})$ by $\mathbf {\overset{..}{f}}_i({\varvec{\beta }})$. The Taylor’s expansion up the order 2 of $\mathbf {g}_i({\varvec{\beta }})$ at ${\varvec{\beta }}={\varvec{\beta }}^0$ is

$$\begin{aligned} \mathbf {g}_i({\varvec{\beta }})= & {} \mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0}) \varepsilon _i+\frac{1}{2}\mathbf {M}_{1i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \varepsilon _i -\frac{1}{2}\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}_i^t ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \nonumber \\&-\,\frac{1}{6}\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t \mathbf {M}_{2i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) - \frac{1}{4}\mathbf {M}_{1i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}_i^t ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \nonumber \\&-\,\frac{1}{12}\mathbf {M}_{1i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t \mathbf {M}_{2i}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}), \end{aligned}$$

(52)

where $\mathbf {M}_{1i}= \Big (\frac{\partial ^2 f_i({\varvec{\beta }}^{(1)}_{i,jk})}{\partial \beta _j \partial \beta _k} \Big )_{1\le j,k \le d}$, $\mathbf {M}_{2i}= \Big ( \frac{\partial ^2 f_i( {\varvec{\beta }}^{(2)}_{i,jk})}{\partial \beta _j \partial \beta _ k} \Big )_{1\le j,k \le d}$ and ${\varvec{\beta }}^{(1)}_{i,jk}={\varvec{\beta }}^\mathbf{0}+u_{i,jk}({\varvec{\beta }}- {\varvec{\beta }}^\mathbf{0})$, ${\varvec{\beta }}^{(2)}_{i,jk}={\varvec{\beta }}^\mathbf{0}+v_{i,jk}({\varvec{\beta }}- {\varvec{\beta }}^\mathbf{0})$, with $u_{i,jk}, v_{i,jk} \in [0,1]$.

We note that ${\varvec{\beta }}^{(1)}_{i,jk}$ and ${\varvec{\beta }}^{(2)}_{i,jk}$ are random vectors which depend on $\mathbf{X}_i$.

For $\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0}) \varepsilon _i$, because $\mathbf{X}_i$ and $\varepsilon _i$ are independent, and $I\!E(\varepsilon _i)=0$, we have that $I\!E[\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0}) \varepsilon _i]=0$ and ${\mathbb {V}}\hbox {ar}\,[\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0}) \varepsilon _i]= \sigma ^2 \mathbf{V}$. For the jth component of $\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})$, by the Bienaym-Tchebychev’s inequality, for $1 \le j \le d$, for all $\epsilon _1>0$, we have

$$\begin{aligned} I\!P\left[ |\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j} \varepsilon _i |\ge \epsilon _1 \right] \le \frac{\sigma ^2}{\epsilon _1^2} V_{jj}, \end{aligned}$$

(53)

where $V_{jj}$ is the jth term diagonal of the matrix $\mathbf{V}$.

For all $\epsilon >0$, taking $\epsilon _1= \sigma \sqrt{6V_{jj}/\epsilon }$ in (53), we obtain $I\!P\big [ | \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j} \varepsilon _i | \ge \sigma \sqrt{6V_{jj}/\epsilon } \big ] \le \epsilon / 6$. Applying Lemma 1 (i), we obtain, for all $\epsilon >0$

$$\begin{aligned} I\!P\left[ \Vert \mathbf {\overset{.}{f}}_{i}({\varvec{\beta }}^\mathbf{0})\varepsilon _i\Vert _1\ge {\frac{\sigma d}{\sqrt{\epsilon }}}\max _{1\le j\le d}\sqrt{6V_{jj}} \right] \le \epsilon /6. \end{aligned}$$

(54)

For the second term of the right-hand side of (52), using assumption (A3), we obtain that for $1\le j,k \le d$, for all $\epsilon >0$ there exists $\epsilon _2>0$, such that, $I\!P\big [|\frac{\partial ^2 f_i( {\varvec{\beta }}^{(1)}_{i,jk})}{\partial \beta _j\partial \beta _k} | \ge \epsilon _2\big ] \le \epsilon /6$. By Lemma 1 (iii), we have that for all $\epsilon >0$,

$$\begin{aligned} I\!P\left[ \Vert \mathbf {M}_{1i} \Vert _1 \ge \epsilon _2 \right] \le \frac{\epsilon }{6}. \end{aligned}$$

(55)

Using Bienaymé-Tchebychev’s inequality, and assumption (A1), we obtain that for all $C_1>0$

$$\begin{aligned} I\!P\left[ | \varepsilon _i | > C_1 \right] \le \frac{\sigma ^2}{C_1}. \end{aligned}$$

(56)

Recall that $\Vert {\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}\Vert _2<\eta$, with $\eta \rightarrow 0$. Then, using (55) and (56), we can write that, for all $\epsilon >0$, there exists $\epsilon _2> 0$ such that, $I\!P\big [\Vert \mathbf {M}_{1i}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\varepsilon _i\Vert _1\ge \epsilon _2 \big ]\le I\!P\big [\Vert \mathbf {M}_{1i}\Vert _1|\varepsilon _i|\,\, \Vert {\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}\Vert _1\ge \epsilon _2\big ]\le I\!P\big [\Vert \mathbf {M}_{1i} \Vert _1\ge \epsilon _2/C_1\eta \big ]\le I\!P\big [\Vert \mathbf {M}_{1i}\Vert _1 \ge \epsilon _2 \big ]\le \epsilon /6$. Therefore, for all $\epsilon >0$, there exists $\epsilon _2>0$ such that

$$\begin{aligned} I\!P\left[ \Vert \mathbf {M}_{1i}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\varepsilon _i\Vert _1\ge \epsilon _2\right] \le \frac{\epsilon }{6}. \end{aligned}$$

(57)

We consider now the term $\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}_i^t ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})$ of relation (52). By Markov’s inequality, taking also into account assumption (A4), we obtain for $1 \le j,l \le d$, for all $\epsilon _3>0$, that $I\!P\big [ | \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j} \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _l} | \ge \epsilon _3 \big ] \le I\!E[| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j} \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _l} |]/\epsilon _3$. We choose, for all $\epsilon >0, \epsilon _3= 6 I\!E[| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j} \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _l} | ]/\epsilon$. Then, the last relation becomes $I\!P\big [ | \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j} \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _l} | \ge 6 I\!E[| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j} \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _l} | \big ] / \epsilon \big ] \le \epsilon /6$. Using Lemma 1 (iii), we obtain

$$\begin{aligned} I\!P\left[ \Vert \mathbf {\overset{.}{f}}_{i}({\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}^t_{i}({\varvec{\beta }}^\mathbf{0})\Vert _1\ge \frac{6d}{\epsilon }\max _{1\le j,l \le d}I\!E\left[ \left| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _l}\right| \right] \right] \le \frac{\epsilon }{6}, \end{aligned}$$

relation that involves, since for all $C_2 >0$ we have $\Vert {\varvec{\beta }}-{\varvec{\beta }}_0 \Vert _1 \le C_2 \eta$ for $\eta \rightarrow 0$, that

$$\begin{aligned} I\!P\left[ \Vert \mathbf {\overset{.}{f}}_{i}({\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}^t_{i}({\varvec{\beta }}^\mathbf{0})({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\Vert _1\ge 6d/\epsilon \max _{1\le j,l\le d}I\!E\left[ \left| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _l}\right| \right] \right] \\ \le I\!P\left[ \Vert \mathbf {\overset{.}{f}}_{i}({\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}^t_{i}({\varvec{\beta }}^\mathbf{0})\Vert _1\ge 6d/\epsilon \max _{1\le j,l\le d}I\!E\left[ \left| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _l}\right| \right] \right] \le \epsilon /6. \end{aligned}$$

Then, for all $\epsilon >0$

$$\begin{aligned} I\!P\left[ \Vert \mathbf {\overset{.}{f}}_{i}({\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}^t_{i}({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \Vert _1 \ge \frac{6d}{\epsilon }\max _{1\le j,l\le d} I\!E\left[ \left| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j} \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _l}\right| \right] \right] \le \frac{\epsilon }{6}. \nonumber \\ \end{aligned}$$

(58)

For $\mathbf {M}_{1i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \mathbf {\overset{.}{f}}^t_{i} ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})$ of relation (52), using assumption (A3) and the Markov’s inequality, we obtain for each jth component $\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}$ of the vector $\mathbf {\overset{.}{f}}_{i} ({\varvec{\beta }}^\mathbf{0})$, for all $\epsilon _4>0$, that $I\!P\big [ |\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}| \ge \epsilon _4 \big ] \le I\!E[|\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}|]/\epsilon _4$. We choose, for all $\epsilon >0$, $\epsilon _4=6I\!E[|\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}|]/\epsilon$ and this last relation becomes $I\!P\big [ |\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}| \ge 6I\!E[|\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}|] / \epsilon \big ] \le \epsilon /6$. Applying Lemma 1 (i), for all $\epsilon >0$ we obtain

$$\begin{aligned} I\!P\left[ \Vert \mathbf {\overset{.}{f}}_{i}({\varvec{\beta }}^\mathbf{0})\Vert _1\ge \frac{6d}{\epsilon }\max _{1\le j \le d}I\!E\left[ \left| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}\right| \right] \right] \le \frac{\epsilon }{6}. \end{aligned}$$

(59)

Using assumption (A3), and relations (55), (59), we can write that

$$\begin{aligned}&I\!P\left[ \Vert \mathbf {M}_{1i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \mathbf {\overset{.}{f}}^t_{i} ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \Vert _1 \ge 6d/ \epsilon \max _{1\le j \le d} I\!E\left[ \left| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}\right| \right] \right] \\&\quad \le I\!P\left[ \Vert \mathbf {\overset{.}{f}}^t_{i} ({\varvec{\beta }}^\mathbf{0})\Vert _1 \ge 6d/ \epsilon \max _{1\le j \le d} I\!E\left[ \left| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}\right| \right] \right] \le \epsilon /6. \end{aligned}$$

Therefore, for all $\epsilon >0$,

$$\begin{aligned} I\!P\left[ \Vert \mathbf {M}_{1i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \mathbf {\overset{.}{f}}^t_{i} ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \Vert _1 \ge \frac{ 6d}{\epsilon } \max _{1\le j \le d} I\!E\left[ \left| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}\right| \right] \right] \le \frac{\epsilon }{6}. \end{aligned}$$

(60)

Taking into account assumptions (A3), (A4), by relations (55), (59), we can prove in a similar way as for relation (60) that, for all $\epsilon >0$,

$$\begin{aligned} I\!P\left[ \Vert \mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t \mathbf {M}_{2i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \Vert _1 \ge \frac{ 6d}{\epsilon } \max _{1\le j \le d} I\!E\left[ \left| \frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}\right| \right] \right] \le \frac{\epsilon }{6}. \end{aligned}$$

(61)

For the last term on the right-hand side of (52), using assumption (A3), we have that, for all ${\varvec{\beta }}\in \mathcal{V}_ \eta ({\varvec{\beta }}^\mathbf{0})$, for all $\epsilon >0$, there exists $\epsilon _5>0$, such that $I\!P[\Vert \mathbf {M}_{1i}\Vert _1 \Vert \mathbf {M}_{2i} \Vert _1 \ge \epsilon _5] \le \epsilon /6$. Using this relation, we show similarly, then, for all $\epsilon >0$, there exists $\epsilon _5>0$, such that,

$$\begin{aligned} I\!P\left[ \Vert \mathbf {M}_{1i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t \mathbf {M}_{2i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \Vert _1 \ge \epsilon _5 \right] \le \frac{\epsilon }{6}. \end{aligned}$$

(62)

Choosing

$$\begin{aligned} M=\sup \left\{ \epsilon _2,\epsilon _5,\frac{6 d }{\epsilon } \max _{1\le j,l \le d} \left\{ I\!E\left[ \bigg |\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _l }\bigg |\right] , I\!E\left[ \bigg |\frac{\partial f_i({\varvec{\beta }}^\mathbf{0})}{\partial \beta _j}\bigg |\right] \right\} ,\frac{\sigma d }{\sqrt{\epsilon }}\max _{1\le j \le d} \sqrt{6V_{jj}} \right\} , \end{aligned}$$

and combining (54), (57), (58), (60), (61), (62) together, lemma yields. $\square$

Lemma 3

Under the same assumptions of Theorem 2, we have

$$\begin{aligned} \frac{1}{n \theta _{nk}} \sum _{i \in I} \mathbf {g}_i({\varvec{\beta }})=O_{I\!P}((n \theta _{nk})^{-1/2})+\mathbf{V}_{1n}^0 ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})+o_{I\!P}({\varvec{\beta }}-{\varvec{\beta }}^0). \end{aligned}$$

Proof of Lemma 3

By the Taylor’s expansion up to the order 3 of $\mathbf {g}_i({\varvec{\beta }})$ at ${\varvec{\beta }}={\varvec{\beta }}^\mathbf{0}$, we obtain

$$\begin{aligned} \frac{1}{n\theta _{nk}}\sum _{i\in I}\mathbf {g}_i({\varvec{\beta }})= & {} \frac{1}{n\theta _{nk}}\sum _{i\in I}\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0}) \varepsilon _i +\frac{1}{2n\theta _{nk}}\sum _{i\in I}\mathbf {\overset{..}{f}}_i({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \varepsilon _i \nonumber \\&-\,\frac{1}{2n\theta _{nk}}\sum _{i\in I}\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}_i^t ({\varvec{\beta }}^\mathbf{0})({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\nonumber \\&-\,\frac{1}{6n\theta _{nk}}\sum _{i\in I}\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t\mathbf {M}_{2i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \nonumber \\&-\, \frac{1}{4n\theta _{nk}}\sum _{i\in I} \mathbf {\overset{..}{f}}_i({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}_i^t ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\nonumber \\&-\,\frac{1}{12n\theta _{nk}}\sum _{i\in I} \mathbf {\overset{..}{f}}_i({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t \mathbf {M}_{2i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\nonumber \\&+\,\frac{1}{6n\theta _{nk}}\sum _{i\in I} \mathbf {M}_{i}\varepsilon _i - \frac{1}{12n\theta _{nk}}\sum _{i\in I} \mathbf {M}_{i}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t \mathbf {M}_{2i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}),\nonumber \\ \end{aligned}$$

(63)

with $\mathbf {M}_{2i}$ given by Lemma 2 and

$$\begin{aligned} \mathbf {M}_{i}=\Bigg (\sum _{l=1}^d\sum _{k=1}^d \frac{\partial ^2 \mathbf {\overset{.}{f}}_i({\varvec{\beta }}^{(3)}_{i,kl})}{\partial \beta _k \partial \beta _l} (\beta _k-\beta ^0_k)(\beta _l-\beta ^0_l)\Bigg )_{1\le k,l \le d} \end{aligned}$$

is a vector of dimension $(d \times 1)$, where ${\varvec{\beta }}^{(3)}_{i,kl} ={\varvec{\beta }}^\mathbf{0}+ w_{i,kl}( {\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})$, with $w_{i,kl} \in [0,1]$.

For the first term of the right-hand side of (63), by the central limit theorem, and the fact that $I\!E[\mathbf {g}_i({\varvec{\beta }}^\mathbf{0})]=0$, we have

$$\begin{aligned} (n \theta _{nk})^{-1}\sum _{i \in I}\mathbf {g}_i({\varvec{\beta }}^\mathbf{0})=O_{I\!P}((n \theta _{nk})^{-1/2}). \end{aligned}$$

(64)

For the second term of the right-hand side of (63), by the law of large numbers, the term $(n\theta _{nk})^{-1} \sum _{i\in I}\mathbf {\overset{..}{f}}_i({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \varepsilon _i$ converges almost surely to the expected of $\mathbf {\overset{..}{f}}_i({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \varepsilon _i$ as $n \rightarrow \infty$. Furthermore, since $\varepsilon _i$ is independent of $\mathbf{X}_i$ and $I\!E[\varepsilon _i]=0$, we have

$$\begin{aligned} \frac{1}{n\theta _{nk}}\sum _{i\in I}\mathbf {\overset{..}{f}}_i({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) \varepsilon _i=o_{I\!P}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}). \end{aligned}$$

(65)

For the third term of the right-hand side of (63), by the law of large numbers and assumption (A4), the term $(n\theta _{nk})^{-1} \sum _{i\in I}\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}_i^t ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})$ converges almost surely to the expected value of $\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}_i^t ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})$ as $n \rightarrow \infty$. On the other hand, since $(n\theta _{nk})^{-1} \sum _{i \in I} \mathbf {\overset{..}{f}}_i({\varvec{\beta }}^\mathbf{0}) \varepsilon _i \overset{a.s}{\longrightarrow } 0$, we have

$$\begin{aligned} \frac{1}{n\theta _{nk}}\sum _{i\in I}\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}_i^t ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})=-\mathbf{V}_{1n}^0 ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})(1+o_{I\!P}(1)). \end{aligned}$$

(66)

For the fourth term of the right-hand side of (63), by the law of large numbers, using assumption (A3) and the relation (59), we can write

$$\begin{aligned} (6n\theta _{nk})^{-1}\left\| \sum _{i\in I}\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t\mathbf {M}_{2i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\right\| _1=O_{I\!P}( \Vert {\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}\Vert ^2_2), \end{aligned}$$

which implies

$$\begin{aligned} \frac{1}{6n\theta _{nk}}\sum _{i\in I}\mathbf {\overset{.}{f}}_i({\varvec{\beta }}^\mathbf{0})({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t \mathbf {M}_{2i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})=o_{I\!P}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}). \end{aligned}$$

(67)

In the same way, using assumption (A3) and relation (59), we obtain, for the fifth term on the right-hand side of (63), that

$$\begin{aligned} \frac{1}{4n\theta _{nk}}\sum _{i\in I} \mathbf {\overset{..}{f}}_i({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\mathbf {\overset{.}{f}}_i^t ({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})=o_{I\!P}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}). \end{aligned}$$

(68)

For the sixth term of the right-hand side of (63), using the assumption (A3), we have

$$\begin{aligned} \frac{1}{12n\theta _{nk}}\sum _{i\in I}\mathbf {\overset{..}{f}}_i({\varvec{\beta }}^\mathbf{0}) ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t \mathbf {M}_{2i}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})=o_{I\!P}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}). \end{aligned}$$

(69)

For $1\le j \le d$, and for any fixed $i$, such that $1 \le i \le n \theta _{nk}$, denote by $M_{ij}$ the following random variable designates the jth component of the vector $\mathbf M _i$, such that

$$\begin{aligned} M_{ij}=\sum _{l=1}^d \sum _{k=1}^d \frac{\partial ^3 f_i({\varvec{\beta }}^{(3)}_{i,kl})}{\partial \beta _k \partial \beta _l \partial \beta _j}(\beta _k-\beta ^0_k)(\beta _l-\beta ^0_l). \end{aligned}$$

using assumption (A3), we have with a probability one, $| M_{ij} |\le C_3 \Vert {\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}\Vert _2^2$. Applying Lemma 1 (i), we obtain

$$\begin{aligned} \Vert \mathbf {M}_{i} \Vert _1 \le C_3 \Vert {\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}\Vert _2^2. \end{aligned}$$

(70)

For the term $(6n\theta _{nk})^ {-1} \sum _{i\in I} \mathbf M _i \varepsilon _i$, using relations (56) and (70), we have $(6n\theta _{nk})^{-1}\Vert \sum _{i\in I}\mathbf M _{i}\varepsilon _i\Vert _1 \le (6n\theta _{nk})^{-1}\sum _{i\in I}\Vert \mathbf M _{i}\Vert _1|\varepsilon _i| \le C_4(6n\theta _{nk})^{-1}n\theta _{nk}\Vert {\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}\Vert _2^2 = C_4 \Vert {\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}\Vert _2^2$. Then,

$$\begin{aligned} \frac{1}{6n\theta _{nk}}\sum _{i\in I} \mathbf M _{i}\varepsilon _i=o_{I\!P}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}). \end{aligned}$$

(71)

Finally, for the last term of the right-hand side of (63), using assumption (A3) and relation (70), we obtain with probability 1, $(12n\theta _{nk})^{-1}\Vert \sum _{i\in I} \mathbf M _{i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t \mathbf {M}_{2i} ({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})\Vert _1 \le C_5 \Vert {\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}\Vert _2^2$, which gives,

$$\begin{aligned} \frac{1}{12n\theta _{nk}}\sum _{i\in I} \mathbf M _{i}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0})^t \mathbf {M}_{2i}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}) =o_{I\!P}({\varvec{\beta }}-{\varvec{\beta }}^\mathbf{0}). \end{aligned}$$

(72)

Then, combining relations (64), (65), (66), (67), (68), (69), (71) and (72), we obtain lemma. $\square$

Lemma 4

Under the same assumptions as in Theorem 3, for all $\varrho >0$, there exist two positive constants $B=B(\varrho )$, $T=T(\varrho )$ such that

$$\begin{aligned}&I\!P\left[ {\max }_{\frac{T}{n}\le \theta _{nk}\le 1-\frac{T}{n}}(n\theta _{nk}/\log \log n\theta _{nk})^{{1}/{2}}\left\| \frac{\hat{{\varvec{\lambda }}}(\theta _{nk})}{\min \{\theta _{nk},1-\theta _{nk}\}}\right\| _2>B\right] \le \varrho ,\\&I\!P\left[ {\max }_{\frac{T}{n} \le \theta _{nk} \le 1-\frac{T}{n}} (n\theta _{nk} / \log \log n\theta _{nk})^{{1}/{2}} \left\| \hat{{\varvec{\beta }}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0}\right\| _2> B\right] \le \varrho ,\\&I\!P\left[ n^{-{1}/{2}}{\max }_{\frac{T}{n} \le \theta _{nk} \le 1-\frac{T}{n}} n\theta _{nk} \left\| \frac{\hat{{\varvec{\lambda }}}(\theta _{nk})}{\min \{\theta _{nk}, 1-\theta _{nk}\}} \right\| _2 > B\right] \le \varrho , \\&I\!P\left[ n^{-{1}/{2}}{\max }_{\frac{T}{n} \le \theta _{nk} \le 1-\frac{T}{n}} n\theta _{nk} \left\| \hat{{\varvec{\beta }}}(\theta _{nk}) - {\varvec{\beta }}^\mathbf{0}\right\| _2> B \right] \le \varrho . \end{aligned}$$

Proof of Lemma 4

The proof of this lemma is similar to that of Lemma 1.2.2 of Csörgö and Horváth (1997). $\square$

In order, to prove Lemma 5, we consider

$$\begin{aligned} R_k= n \sigma ^{-2} \theta _{nk}(1-\theta _{nk})\left( \mathbf W _{1n}^0-\mathbf W _{2n}^0\right) ^t\mathbf{V}^{-1} \left( \mathbf W _{1n}^0-\mathbf W _{2n}^0\right) . \end{aligned}$$

Recall that $\mathbf{V}\equiv I\!E[ \mathbf {\overset{.}{f}}(\mathbf{X}_i,{\varvec{\beta }}^\mathbf{0}) \mathbf {\overset{.}{f}}^t(\mathbf{X}_i,{\varvec{\beta }}^\mathbf{0})]$, for all $i=1,\ldots ,n$.

The results of Lemma 5 are similar to that of Theorem 1.1.1 of Csörgö and Horváth (1997).

Lemma 5

Suppose that the assumptions (A1)–(A4) hold. Under the null hypothesis $H_0$, for all $0 \le \alpha < 1/2$ we have

• (i) $n^{\alpha } \max \limits _{\theta _{nk} \in \varTheta _{nk}}[\theta _{nk}(1-\theta _{nk})]^{\alpha } | Z_{nk}(\theta _{nk}, \hat{{\varvec{\lambda }}}(\theta _{nk}), \hat{{\varvec{\beta }}}(\theta _{nk}))-R_k|= O_{I\!P}(1)$.

• (ii) $\max \limits _{\theta _{nk} \in \varTheta _{nk}}[\theta _{nk}(1-\theta _{nk})]| Z_{nk}(\theta _{nk}, \hat{{\varvec{\lambda }}}(\theta _{nk}), \hat{{\varvec{\beta }}}(\theta _{nk}))-R_k| =O_{I\!P}(n^{-1/2}(\log \log n)^{3/2})$.

Proof of Lemma 5

For the score function ${\varvec{\phi }}_{1n}$ of relation (13), the two terms of the right-hand side are replaced by their decomposition obtained by the relations (22) and (25). On the other hand, we have ${\varvec{\phi }}_{1n}(\theta _{nk},\hat{{\varvec{\lambda }}}(\theta _{nk}),\hat{{\varvec{\beta }}}(\theta _{nk}))= \mathbf 0 _d$. Then, we can write

$$\begin{aligned}&\left[ \frac{1}{n\theta _{nk}} \sum \limits _{i \in I}\mathbf {g}_i({\varvec{\beta }}^\mathbf{0})\!+\! \mathbf{V}_{1n}^0(\hat{{\varvec{\beta }}}(\theta _{nk})\!-\!{\varvec{\beta }}^\mathbf{0})\!-\! \frac{1}{n \theta _{nk}^2} \sum \limits _{i \in I}\mathbf {g}_i({\varvec{\beta }}^\mathbf{0})\mathbf {g}^t_i({\varvec{\beta }}^\mathbf{0}) \hat{{\varvec{\lambda }}}(\theta _{nk}) \right] (1+o_{I\!P}(1))\\&\quad -\,\mathbf{V}_{1n}^0 (\mathbf{V}_{2n}^0)^{-1}\left[ \frac{1}{n (1-\theta _{nk})^2}\mathbf{V}_{1n} (\mathbf{V}_{2n}^0)^{-1} \sum \limits _{j \in J}\mathbf {g}_j({\varvec{\beta }}^\mathbf{0})\mathbf {g}^t_j({\varvec{\beta }}^\mathbf{0}) \hat{{\varvec{\lambda }}}(\theta _{nk})\right. \\&\quad \left. +\, \frac{1}{n(1-\theta _{nk})} \sum \limits _{j \in J}\mathbf {g}_j({\varvec{\beta }}^\mathbf{0}) +\mathbf{V}_{2n}^0 (\hat{{\varvec{\beta }}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0}) \right] (1+o_{I\!P}(1))=\mathbf 0 _d. \end{aligned}$$

Hence,

$$\begin{aligned} \hat{{\varvec{\lambda }}}(\theta _{nk})= & {} \Big (\frac{1}{\theta _{nk}} \mathbf {D}_{1n}^0 + \frac{1}{1-\theta _{nk}}(\mathbf{V}_{1n}^0 (\mathbf{V}_{2n}^0)^{-1})(\mathbf{V}_{1n}^0 (\mathbf{V}_{2n}^0)^{-1})^t\mathbf D _{2n}^0\Big )^{-1}\\&\cdot \Big (\frac{1}{n\theta _{nk}} \sum _{i \in I}\mathbf {g}_i({\varvec{\beta }}^\mathbf{0})-\frac{\mathbf{V}_{1n}^0(\mathbf{V}_{2n}^0)^{-1}}{n(1-\theta _{nk})} \sum _{j \in J}\mathbf {g}_j({\varvec{\beta }}^\mathbf{0})\Big )(1+o_{I\!P}(1))\\&+\,o_{I\!P}({\hat{{\varvec{\beta }}}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0}), \end{aligned}$$

with the matrices $\mathbf {D}^0_{1n}$ and $\mathbf {D}^0_{2n}$ given by relation (24).

On the other hand, by the law of large numbers, we have $-\mathbf{V}_{1n}^0 \overset{a.s}{\longrightarrow } \mathbf{V}$ and $-\mathbf{V}_{2n}^0 \overset{a.s}{\longrightarrow } \mathbf{V}$. Then, $\mathbf{V}_{1n}^0 (\mathbf{V}_{2n}^0)^{-1}\overset{a.s}{\longrightarrow } I_d$. Always, by the law of large numbers, $\mathbf {D}_{1n}^0$ and $\mathbf {D}_{2n}^0$ converge almost surely to $\sigma ^2 \mathbf{V}$ as $n\rightarrow \infty$.

By Theorem 2, we proved that $\hat{{\varvec{\lambda }}}(\theta _{nk})=\theta _{nk} O_p((n\theta _{nk})^{-1/2})$. Then, we obtain

$$\begin{aligned} \hat{{\varvec{\lambda }}}(\theta _{nk})= \sigma ^{-2} \theta _{nk} (1-\theta _{nk}) \mathbf{V}^{-1} (\mathbf W _{1n}^0-\mathbf W _{2n}^0) (1+o_{I\!P}(1))+ o_{I\!P}({\hat{{\varvec{\beta }}}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0}).\nonumber \\ \end{aligned}$$

(73)

The limited development of the statistic $Z_{nk}(\theta _{nk}, \hat{{\varvec{\lambda }}}(\theta _{nk}),\hat{{\varvec{\beta }}}(\theta _{nk}))$, specified by the relation (12), in the neighbourhood of $({\varvec{\lambda }},{\varvec{\beta }})=(\mathbf 0 _d,{\varvec{\beta }}^\mathbf{0})$ up to order 2, can be written

$$\begin{aligned}&\Big [\frac{2\hat{{\varvec{\lambda }}}^t(\theta _{nk})}{\theta _{nk}}\sum _{i \in I}\mathbf {g}_i({\varvec{\beta }}^\mathbf{0})-\frac{2\hat{{\varvec{\lambda }}}^t(\theta _{nk})}{1-\theta _{nk}} \mathbf{V}_{1n}^0(\mathbf{V}_{2n}^0)^{-1}\sum _{j\in J}\mathbf {g}_j({\varvec{\beta }}^\mathbf{0})\Big ]\nonumber \\&\quad -\,\Big [\frac{\hat{{\varvec{\lambda }}}^t(\theta _{nk}) }{(1-\theta _{nk})^2}\mathbf{V}_{1n}^0 (\mathbf{V}_{2n}^0)^{-1}\sum _{j \in J} \mathbf {g}_j({\varvec{\beta }}^\mathbf{0})\mathbf {g}^t_j({\varvec{\beta }}^\mathbf{0})\mathbf{V}_{1n}^0 (\mathbf{V}_{2n}^0)^{-1}\hat{{\varvec{\lambda }}}(\theta _{nk})\nonumber \\&\quad +\,\frac{\hat{{\varvec{\lambda }}}^t(\theta _{nk})}{\theta _{nk}^2}\sum _{i\in I}\mathbf {g}_i({\varvec{\beta }}^\mathbf{0})\mathbf {g}^t_i({\varvec{\beta }}^\mathbf{0}) \hat{{\varvec{\lambda }}}(\theta _{nk})\Big ] +\Big [ 2\hat{{\varvec{\lambda }}}^t(\theta _{nk})\Big (\frac{1}{\theta _{nk}} \sum _{i \in I}\mathbf {\overset{.}{g}}_i({\varvec{\beta }}^\mathbf{0})\nonumber \\&\quad -\,\frac{1}{1-\theta _{nk}}\mathbf{V}_{1n}^0(\mathbf{V}_{2n}^0)^{-1}\sum _{j \in J}\mathbf {\overset{.}{g}}_j({\varvec{\beta }}^\mathbf{0})\Big )(\hat{{\varvec{\beta }}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0})\Big ]\nonumber \\&\quad -\,\Big [2\hat{{\varvec{\lambda }}}^t(\theta _{nk})\Big ( \frac{1}{1-\theta _{nk}}\sum _{j\in J}\mathbf {g}_j({\varvec{\beta }}^\mathbf{0})\frac{\partial (\mathbf{V}_{1n}({\varvec{\beta }})(\mathbf{V}_{2n}({\varvec{\beta }}))^{-1})}{\partial {\varvec{\beta }}}\Big ) (\hat{{\varvec{\beta }}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0})\Big ]\nonumber \\&\quad +\,\frac{1}{3 !} \Big [S_1+ 3 S_2 + 3 S_3 +S_4\Big ], \end{aligned}$$

(74)

where

$$\begin{aligned} S_1= & {} \sum _{j=1}^d \sum _{l=1}^d \sum _{k=1}^d \frac{\partial ^3 Z_{nk}(\theta _{nk}, {\varvec{\lambda }}^{(1)}_{jkl}, {\varvec{\beta }}^{(1)}_{jkl})}{\partial \beta _j \partial \beta _k \partial \beta _l } ({\hat{\beta }}_j-\beta ^0_j) ({\hat{\beta }}_k-\beta ^0_k)({\hat{\beta }}_l-\beta ^0_l),\\ S_2= & {} \sum _{j=1}^d \sum _{l=1}^d \sum _{k=1}^d \frac{\partial ^3 Z_{nk}(\theta _{nk}, {\varvec{\lambda }}^{(2)}_{jkl}, {\varvec{\beta }}^{(2)}_{jkl})}{\partial \lambda _j \partial \lambda _k \partial \beta _l } ({\hat{\lambda }}_j) ({\hat{\lambda }}_k)({\hat{\beta }}_l-\beta ^0_l),\\ S_3= & {} \sum _{j=1}^d \sum _{l=1}^d \sum _{k=1}^d \frac{\partial ^3 Z_{nk}(\theta _{nk}, {\varvec{\lambda }}^{(3)}_{jkl}, {\varvec{\beta }}^{(3)}_{jkl})}{\partial \lambda _j \partial \beta _k \partial \beta _l } ({\hat{\lambda }}_j) ({\hat{\beta }}_k-\beta ^0_k)({\hat{\beta }}_l-\beta ^0_l),\\ S_4= & {} \sum _{j=1}^d \sum _{l=1}^d \sum _{k=1}^d \frac{\partial ^3 Z_{nk}(\theta _{nk}, {\varvec{\lambda }}^{(4)}_{jkl}, {\varvec{\beta }}^{(4)}_{jkl})}{\partial \lambda _j \partial \lambda _k \partial \lambda _l } ({\hat{\lambda }}_j) ({\hat{\lambda }}_k)({\hat{\lambda }}_l), \end{aligned}$$

where, for $1 \le j \le d$, ${\hat{\beta }}_j$ is the jth component of ${\hat{{\varvec{\beta }}}} (\theta _{nk})$, and ${\hat{\lambda }}_j$ is the jth component of ${\hat{{\varvec{\lambda }}}} (\theta _{nk})$. In the expression of $S_1$, $S_2$, $S_3$, $S_4$ we have also, for all $1 \le j,k,l \le d$, ${\varvec{\lambda }}^{(a)}_{jkl}= u^{(a)}_{jkl} (\hat{{\varvec{\beta }}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0})$, and ${\varvec{\beta }}^{(a)}_{jkl}= {\varvec{\beta }}^\mathbf{0}+ v^{(a)}_{jkl} (\hat{{\varvec{\beta }}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0})$, with $u^{(a)}_{jkl},v^{(a)}_{jkl} \in [0,1]$ and $a \in \{1,2,3,4\}$.

We note that, the derivative $\partial ( \mathbf{V}_{1n}({\varvec{\beta }})(\mathbf{V}_{2n}({\varvec{\beta }}))^{-1}) / \partial {\varvec{\beta }}$ is considered term by term.

Now, we replace $\hat{{\varvec{\lambda }}}(\theta _{nk})$ in the relation (74) by the value obtained in (73). For the first term of (74), using notations given by relation (30), and the fact that $\mathbf{V}_{1n}^0 (\mathbf{V}_{2n}^0)^{-1}\overset{a.s}{\longrightarrow } I_d$, as $n\rightarrow \infty$, we find that this term is equal to $2n \sigma ^{-2}\theta _{nk}(1-\theta _{nk}) (\mathbf W _{1n}^0-\mathbf W _{2n}^0)^t\mathbf{V}^{-1}(\mathbf W _{1n}^0-\mathbf W _{2n}^0)+o_{I\!P}(\Vert {\hat{{\varvec{\beta }}}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0}\Vert _2)$.

Similarly, for the second term of (74), using notations given by (24), and the fact that $\mathbf {D}_{1n}^0$ and $\mathbf {D}_{2n}^0$ converge to $\sigma ^2 \mathbf{V}$, as $n\rightarrow \infty$, we obtain that this term is equal to $n\sigma ^{-2}\theta _{nk}(1-\theta _{nk})(\mathbf W _{1n}^0-\mathbf W _{2n}^0)^t\mathbf{V}^{-1}(\mathbf W _{1n}^0-\mathbf W _{2n}^0)+o_{I\!P}(\Vert {\hat{{\varvec{\beta }}}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0}\Vert _2)$.

For the third term of (74), we know that, $\mathbf{V}_{1n}^0=(n\theta _{nk})^{-1} \sum _{i\in I}\mathbf {\overset{.}{g}}_i({\varvec{\beta }}^\mathbf{0})$, and $\mathbf{V}_{2n}^0=(n(1-\theta _{nk}))^{-1} \sum _{j\in J}\mathbf {\overset{.}{g}}_j({\varvec{\beta }}^\mathbf{0})$. On the other hand, by the law of large numbers, we have $\mathbf{V}_{1n}^0$ and $\mathbf{V}_{2n}^0$ converge almost surely to $-\mathbf{V}$ as $n\rightarrow \infty$, and $\mathbf{V}_{1n}^0(\mathbf{V}_{2n}^0)^{-1}\overset{a.s}{\longrightarrow }I_d$, which implies that the third term of (74) converge almost surely to zero, as $n\rightarrow \infty$.

By the central limit theorem, we have that $(n(1-\theta _{nk}))^{-1}\sum _{j \in J}\mathbf {g}_j({\varvec{\beta }}^\mathbf{0}) = O_{I\!P}((n(1-\theta _{nk}))^{-1/2})$. Then, the fourth term of (74) is $o_{I\!P}(n \sigma ^{-2}\theta _{nk}(1-\theta _{nk})(\mathbf W _{1n}^0-\mathbf W _{2n}^0)^t\mathbf{V}^{-1}(\mathbf W _{1n}^0-\mathbf W _{2n}^0)$.

For the last term of (74), using assumptions (A2)–(A4) and by an elementary calculations, we prove that this term is $o_{I\!P}(\Vert {\hat{{\varvec{\beta }}}}(\theta _{nk}) -{\varvec{\beta }}^\mathbf{0}\Vert _2)+o_{I\!P}(\Vert {\hat{{\varvec{\lambda }}}}(\theta _{nk}) \Vert _2)+o_{I\!P}(\Vert {\hat{{\varvec{\lambda }}}}(\theta _{nk})\Vert _2 \Vert {\hat{{\varvec{\beta }}}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0}\Vert _2)$. Combining the obtained results, we obtain

$$\begin{aligned}&Z(\theta _{nk}, \hat{{\varvec{\lambda }}}(\theta _{nk}),\hat{{\varvec{\beta }}}(\theta _{nk}))=\\&\quad n \sigma ^{-2} \theta _{nk}(1-\theta _{nk}) (\mathbf W _{1n}^0-\mathbf W _{2n}^0)^t \mathbf{V}^{-1} (\mathbf W _{1n}^0-\mathbf W _{2n}^0) (1+o_{I\!P}(1))\\&\quad +\, o_{I\!P}(\Vert {\hat{{\varvec{\beta }}}}(\theta _{nk})-{\varvec{\beta }}^\mathbf{0}\Vert _2)+o_{I\!P}(\Vert {\hat{{\varvec{\lambda }}}}(\theta _{nk})\Vert _2) +o_{I\!P}(\Vert {\hat{{\varvec{\lambda }}}}(\theta _{nk}) \Vert _2 \Vert {\hat{{\varvec{\beta }}}}(\theta _{nk}) -{\varvec{\beta }}^\mathbf{0}\Vert _2). \end{aligned}$$

This last relation, together with Lemma 4 imply Lemma 5. $\square$