Empirical likelihood for heteroscedastic partially linear single-index models with growing dimensional data

Abstract

In this paper, we propose a new approach to the empirical likelihood inference for the parameters in heteroscedastic partially linear single-index models. In the growing dimensional setting, it is proved that estimators based on semiparametric efficient score have the asymptotic consistency, and the limit distribution of the empirical log-likelihood ratio statistic for parameters $(\beta ^{\top },\theta ^{\top })^{\top }$ is a normal distribution. Furthermore, we show that the empirical log-likelihood ratio based on the subvector of $\beta$ is an asymptotic chi-square random variable, which can be used to construct the confidence interval or region for the subvector of $\beta$. The proposed method can naturally be applied to deal with pure single-index models and partially linear models with high-dimensional data. The performance of the proposed method is illustrated via a real data application and numerical simulations.

Appendix

To prove the main theorems, we need to give the following set of conditions.

Assumption 1

Let $\mathrm{Var}(X_{i})=\Sigma _{xi}$ and $\mathrm{Var}(Z_{i})=\Sigma _{zi}$, the eigenvalues of $\Sigma _{xi}$ and $\Sigma _{zi}$ satisfy $C_{1}\le \gamma _{1}(\Sigma _{xi})\le \cdots \le \gamma _{p}(\Sigma _{xi})\le C_{2}$ and $C_{1}\le \gamma _{1}(\Sigma _{zi})\le \cdots \le \gamma _{r}(\Sigma _{zi})\le C_{2}$ for some constants $0<C_{1}<C_{2}$, for $i=1\cdots n$. There is a constant $\delta >0$ such that $E(\varepsilon ^{4+\delta }|X,Z)<\infty$.

Assumption 2

There are $v(\cdot )$, $\eta =\eta (X,Z)$, such that $E(\varepsilon ^{2}|X,Z)=v(\eta )$, $0<C_{1}<v(\cdot )<C_{2}<\infty$ for some constants $0<C_{1}<C_{2}$, and The eigenvalues of $\mathrm{Var}(X_{i}|\eta (X_{i}, Z_{i}))$ are bounded away from zero and infinity.

Assumption 3

There exists $v_{1}(X,Z)$ such that

$$\begin{aligned}&\left| \frac{\partial ^{2}E(X|Z^{\top }\theta )}{\partial \theta _{i}\partial \theta _{j}}\right| , \left| \frac{\partial ^{2}E(Z|Z^{\top }\theta )}{\partial \theta _{i}\partial \theta _{j}}\right| , \left| \frac{\partial ^{2}E(w|Z^{\top }\theta )}{\partial \theta _{i}\partial \theta _{j}}\right| , \left| \frac{\partial ^{2}E(wZ|Z^{\top }\theta )}{\partial \theta _{i}\partial \theta _{j}}\right| ,\\&\left| \frac{\partial ^{2}E(wX|Z^{\top }\theta )}{\partial \theta _{i}\partial \theta _{j}}\right|<v_{1}(X,Z),Ev_{1}^{2}<\infty , (i,j=2,\ldots , r). \end{aligned}$$

Further there exists $v_{2}(X,Z)$ such that

$$\begin{aligned} \left| \frac{\partial ^{3}\eta (X,Z)}{\partial \gamma _{i}\partial \gamma _{j}\partial \gamma _{l}}\right|<v_{2}(X,Z), Ev_{2}^{2}<\infty , (i,j,l=1,\ldots , p+r), \end{aligned}$$

where $(X^{\top },Z^{\top })^{\top }=(\gamma _{1},\ldots ,\gamma _{p+r})^{\top }$. Further there exists $v_{3}(X,Z)$ such that

$$\begin{aligned} \left| \frac{\partial ^{4}g(Z^{\top }\theta )}{\partial \theta _{i}\partial \theta _{j}\partial \theta _{k}\partial \theta _{l}}\right| , \left| \frac{\partial ^{4}v(\eta )}{\partial \eta _{i_{1}}\partial \eta _{j_{1}}\partial \eta _{k_{1}}\partial \eta _{l_{1}}}\right|<v_{3}(X,Z), Ev_{3}^{2}<\infty , \end{aligned}$$

where the dimension of $\eta$ is $p_{1}$, and $i,j,k,l=2,\ldots ,r$, $i_{1},j_{1},k_{1},l_{1}=1,\ldots , p_{1}$.

Assumption 4

Assume that the random variable $\eta$ and $Z^{\top }\theta$ have densities $f_{\eta }(\eta )$ and $f_{Z^{\top }\theta }(Z^{\top }\theta )$, satisfying $0<\inf f_{\eta }(\eta )\le \sup f_{\eta }(\eta )<\infty$ and $0<\inf f_{Z^{\top }\theta }(Z^{\top }\theta )\le \sup f_{Z^{\top }\theta }(Z^{\top }\theta )<\infty$. Further there exists $v_{4}(X,Z)$ such that

$$\begin{aligned}&\left| \frac{\partial ^{2}f_{Z^{\top }\theta }(Z^{\top }\theta ))}{\partial \theta _{i}\partial \theta _{j}}\right| , \left| \frac{\partial ^{2}f_{\eta }(\eta )}{\partial \eta _{k}\partial \eta _{l}}\right| \\&\quad<v_{4}(X,Z), Ev_{4}^{2}<\infty , (i,j=2,\ldots , p;~ k,l=1,\ldots , p_{1}). \end{aligned}$$

Assumption 5

The kernel function $K_{h}(\cdot )$ is symmetric and its derivative is continuous with compact support contained in $[-1,1]$.

Assumption 6

The bandwidths $h_{i}$ satisfy $\log ^{2}(n)/(nh_{i})\rightarrow 0$ for $i=1,2,3$. In addition, $nh_{1}^{4}\rightarrow \infty$, $nh_{1}^{8}\rightarrow 0$, $h_{1}^{4}\log ^{2}(n)/h_{i}\rightarrow 0$ and $\log ^{4}(n)/(nh_{1}h_{i})\rightarrow 0$ for $i=1,2,3$, $h_{2}=O(n^{-1/5})$ and $h_{3}=O(n^{-1/5})$.

Assumption 7

$p,r\rightarrow \infty$, $pn^{-1/5}\rightarrow 0$, $rn^{-1/5}\rightarrow 0$, as $n\rightarrow \infty$.

Assumption 8

$E\Vert X\Vert ^{4}<\infty$, $E\Vert Z\Vert ^{4}<\infty$, $E\Vert \varepsilon X\Vert ^{4}<\infty$, $E\Vert \varepsilon Z\Vert ^{4}<\infty$ and $E|\varepsilon |^{4}<\infty$.

Assumption 9

Let

$$\begin{aligned} \xi _{n}(\beta ,\theta )=w\varepsilon \left[ X^{\top }-\frac{E(wX^{\top }|Z^{\top }\theta )}{E(w|Z^{\top }\theta )}, g'(Z^{\top }\theta )\left\{ Z^{\top }-\frac{E(wZ^{\top }|Z^{\top }\theta )}{E(w|Z^{\top }\theta )}\right\} \right] ^{\top }, \end{aligned}$$

and $\xi _{nl}(\beta ,\theta )$ be the l-th component of $\xi _{n}(\beta ,\theta )$, $l=1,\ldots ,p, p+2,\ldots ,p+r$. As $n\rightarrow \infty$, there is a positive constant C such that, $E(\Vert \xi _{n}(\beta ,\theta )/\sqrt{p}\Vert ^{4})<C$, $E(\Vert XX^{\top }\Vert ^{4})<C$, $E(\Vert XZ^{\top }\Vert ^{4})<\infty$ and $E(\Vert ZX^{\top }\Vert ^{4})<C$.

Assumptions 1–6 ensure the function $g(Z_{i}^{\top }\theta )$, $g'(Z_{i}^{\top }\theta )$, $w(X_{i},Z_{i})$, $E\{{\hat{w}}(X,Z)$$|Z_{i}^{\top }\theta \}$, $E\{{\hat{w}}(X,Z)X|Z_{i}^{\top }\theta \}$ and $E\{{\hat{w}}(X,Z)Z_{-1}| Z_{i}^{\top }\theta \}$ are estimated with retained precision and the nonparametric estimation does not affect the asymptotic result of the estimated empirical likelihood ratio, i.e., the estimated empirical likelihood ratio ${\tilde{L}}(\beta ,\theta )$ has the same asymptotic distribution as the ordinary empirical likelihood ratio $L(\beta ,\theta )$. Furthermore, Assumptions 1–6 ensure the existence of the estimator $({\hat{\beta }}^{\top },{\hat{\theta }}^{\top })^{\top }$ for parameters $(\beta ^{\top },\theta ^{\top })^{\top }$. Assumption 7 is a technical condition, and Assumption 8 ensures that there exists an asymptotic variance for the estimator of the growing parameters $(\beta ^{\top },\theta ^{\top })^{\top }$. Assumption 9 controls the tail probability behavior of the estimating equation. Because establishing the asymptotic theoretical results for empirical likelihood approach under the situation with diverging dimensionality on covariates is very challenging, these conditions are not the weakest possible and the bounds in the stochastic analysis are conservative. This is also the case in Ma and Zhu (2013), these strong conditions facilitate technical derivations.

Let ${\tilde{l}}(\lambda ,\beta ,\theta )=n^{-1}\sum _{i=1}^{n}\log \left\{ 1+\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta )\right\}$, $\bar{{\hat{\xi }}}(\beta ,\theta )=n^{-1}\sum _{i=1}^{n}{\hat{\xi }}_{i}(\beta ,\theta )$, $a_{n}=O_{p}\{(p/n)^{1/2}\}$ and C will denote a generic positive constant that may be different in different uses throughout the “Appendix”. In addition, we use the Frobenius norm of a matrix A, defined as $\Vert A\Vert =\{\mathrm {tr}(A^{\top }A)\}^{\frac{1}{2}}$, where $\mathrm {tr}(A)$ denotes the trace ofmatrix A.

Proof of Theorem 2.1

Proof

We first expand

$$\begin{aligned} 0= & {} \frac{1}{\sqrt{n}}A_{2}\sum \limits _{i=1}^{n}\left\{ Y_{i}-X_{i}^{\top }{\hat{\beta }}+{\hat{g}}'\left( Z_{i}^{\top }{\hat{\theta }}\right) \right\} {\hat{w}}_{i}{\hat{g}}' \left( Z_{i}^{\top }{\hat{\theta }}\right) \left\{ Z_{i}-\frac{{\hat{E}}\left( {\hat{w}}Z|Z_{i}^{\top }{\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z_{i}^{\top }{\hat{\theta }}\right) }\right\} \nonumber \\= & {} \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}{\hat{w}}_{i}{\hat{g}}'\left( Z_{i}^{\top }{\hat{\theta }}\right) A_{2}\left\{ Z_{i}-\frac{E\left( wZ|Z_{i}^{\top }\theta _{0}\right) }{E\left( w|Z_{i}^{\top } \theta _{0}\right) }\right\} X_{i}^{\top }\left( \beta _{0}-{\hat{\beta }}\right) \nonumber \\&+\,\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}{\hat{w}}_{i}{\hat{g}}'\left( Z_{i}^{\top }{\hat{\theta }}\right) A_{2}\left\{ \frac{E\left( wZ|Z_{i}^{\top }\theta _{0}\right) }{E\left( w|Z_{i}^{\top }\theta _{0} \right) }-\frac{{\hat{E}}\left( {\hat{w}}Z|Z_{i}^{\top }{\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z_{i}^{\top }{\hat{\theta }}\right) }\right\} X_{i}^{\top }\left( \beta _{0}-{\hat{\beta }}\right) \nonumber \\&+\,\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\left\{ g\left( Z_{i}^{\top }\theta _{0}\right) -{\hat{g}}\left( Z_{i}^{\top }\theta _{0}\right) \right\} {\hat{w}}_{i}{\hat{g}}'\left( Z_{i}^{\top }{\hat{\theta }}\right) A_{2}\left\{ Z_{i}-\frac{E\left( wZ|Z_{i}^{\top }\theta _{0}\right) }{E\left( w|Z_{i}^{\top }\theta _{0}\right) }\right\} \nonumber \\&+\,\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\left\{ g\left( Z_{i}^{\top }\theta _{0}\right) -{\hat{g}}\left( Z_{i}^{\top }\theta _{0}\right) \right\} {\hat{w}}_{i}{\hat{g}}'\left( Z_{i}^{\top }{\hat{\theta }}\right) A_{2}\left\{ \frac{E\left( wZ|Z_{i}^{\top }\theta _{0}\right) }{E\left( w|Z_{i}^{\top }\theta _{0}\right) }\right. \nonumber \\&\left. -\,\frac{{\hat{E}}\left( {\hat{w}}Z|Z_{i}^{\top }{\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z_{i}^{\top }\hat{\theta }\right) }\right\} \nonumber \\&+\,\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\left\{ g\left( Z_{i}^{\top }\theta _{0}\right) -{\hat{g}}\left( Z_{i}^{\top }{\hat{\theta }}\right) \right\} {\hat{w}}_{i}{\hat{g}}'\left( Z_{i}^{\top }{\hat{\theta }}\right) A_{2}\left\{ Z_{i}-\frac{{\hat{E}}\left( {\hat{w}}Z|Z_{i}^{\top }{\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z_{i}^{\top }{\hat{\theta }}\right) }\right\} \nonumber \\&+\,\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\varepsilon _{i}{\hat{w}}_{i}{\hat{g}}'\left( Z_{i}^{\top }{\hat{\theta }}\right) A_{2}\left\{ Z_{i}-\frac{{\hat{E}}\left( {\hat{w}}Z|Z_{i}^{\top } {\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z_{i}^{\top }{\hat{\theta }}\right) }\right\} . \end{aligned}$$

(17)

Similar to the proof of Proposition 2 in Ma and Zhu (2013), we can obtain from the second equation in (3) that

$$\begin{aligned}&A_{2}E\left[ wg'\left( Z^{\top }\theta _{0}\right) \left\{ Z-\frac{E\left( wZ|Z^{\top }\theta _{0}\right) }{E\left( w|Z^{\top }\theta _{0}\right) }\right\} X^{\top }\right] \sqrt{n}\left( {\hat{\beta }}-\beta _{0}\right) \nonumber \\&\qquad +A_{2}E\left[ w\{g'\left( Z^{\top }\theta _{0}\right) \}^{2}\left\{ Z-\frac{E\left( wZ|Z^{\top }\theta _{0}\right) }{E\left( w|Z^{\top }\theta _{0}\right) }\right\} Z^{\top }\right] \sqrt{n}\left( {\hat{\theta }}- \theta _{0}\right) \nonumber \\&\quad =\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\varepsilon _{i}w_{i}g'\left( Z_{i}^{\top }\theta _{0}\right) A_{2}\left\{ Z_{i}-\frac{E\left( wZ|Z_{i}^{\top }\theta _{0}\right) }{E\left( w|Z_{i}^{\top } \theta _{0}\right) }\right\} +o_{p}\left( 1\right) . \end{aligned}$$

(18)

Similarly, from the first equation in (3), we have that

$$\begin{aligned}&A_{1}E\left[ w\left\{ X-\frac{E\left( wX|Z^{\top }\theta _{0}\right) }{E\left( w|Z^{\top }\theta _{0}\right) }\right\} X^{\top }\right] \sqrt{n}\left( {\hat{\beta }}-\beta _{0}\right) \nonumber \\&\qquad +A_{1}E\left[ wg'\left( Z^{\top }\theta _{0}\right) \left\{ X-\frac{E\left( wX|Z^{\top }\theta _{0}\right) }{E\left( w|Z^{\top }\theta _{0}\right) }\right\} Z^{\top }\right] \sqrt{n}\left( {\hat{\theta }}-\theta _{0}\right) \nonumber \\&\quad =\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\varepsilon _{i}w_{i}A_{1}\left\{ X_{i}-\frac{E\left( wX|Z_{i}^{\top }\theta _{0}\right) }{E\left( w|Z_{i}^{\top }\theta _{0}\right) }\right\} +o_{p}\left( 1\right) . \end{aligned}$$

(19)

Combining (17) and (18) implies that

$$\begin{aligned} AV^{1/2}\left( { \begin{array}{*{10}c} {\hat{\beta }}-\beta _{0}\\ {\hat{\theta }}-\theta _{0}\\ \end{array}} \right) {=}AV^{-1/2}\left( { \begin{array}{*{10}c} \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\varepsilon _{i}w_{i}\left\{ X_{i}-\frac{E\left( wX|Z_{i}^{\top }\theta _{0}\right) }{E\left( w|Z_{i}^{\top }\theta _{0}\right) }\right\} \\ \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\varepsilon _{i}w_{i}g'\left( Z_{i}^{\top }\theta _{0}\right) \left\{ Z_{i}-\frac{E\left( wZ|Z_{i}^{\top }\theta _{0}\right) }{E\left( w|Z_{i}^{\top }\theta _{0} \right) }\right\} \\ \end{array}}\right) {+}o_{p}(1). \end{aligned}$$

Applying the Lindeberg–Feller central limit theorem, we can establish

$$\begin{aligned} \sqrt{n}AV^{1/2}\left\{ \left( {\hat{\beta }}^{\top },{\hat{\theta }}^{\top }\right) ^{\top }-\left( \beta _{0}^{\top },\theta _{0}^{\top }\right) ^{\top }\right\} \rightarrow N(0,G) \end{aligned}$$

in distribution, and the proof of Theorem 2.1 is completed. $\square$

Next, we present the following lemmas before proving Theorem 2.2.

Lemma 5.1

Under Assumptions of Theorem 2.2, $\max _{1\le i \le n}\Vert {\hat{\xi }}_{i}(\beta ,\theta )\Vert =o_{p}(n^{1/4}\sqrt{p})$ and $\max _{1\le i \le n}|\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta )|=o_{p}(1)$ for all $\lambda =O_{p}(a_{n})$.

Proof

From Assumptions 8 and 9, for any $\epsilon >0$,

$$\begin{aligned} P\left\{ \max _{1\le i \le n}\Vert \xi _{i}\left( \beta ,\theta \right) \Vert \le n^{1/4}\sqrt{p}\epsilon \right\}\le & {} \sum _{i=1}^{n}P\left\{ \Vert \xi _{i}\left( \beta ,\theta \right) \Vert \le n^{1/4}\sqrt{p}\epsilon \right\} \nonumber \\\le & {} \frac{1}{np^{2}\epsilon ^{4}}\sum _{i=1}^{n}E\Vert \xi _{i}\left( \beta ,\theta \right) \Vert ^{4}\nonumber \\= & {} \frac{1}{\epsilon ^{k}}E\Vert \xi _{1}\left( \beta ,\theta \right) /\sqrt{p}\Vert ^{4}. \end{aligned}$$

(20)

By Cauchy–Schwarz inequality, $\Vert \xi _{1}(\beta ,\theta )/\sqrt{p}\Vert ^{4}\le 1/p\sum _{l=1}^{p+r}|\xi _{1l}(\beta ,\theta )|^{4}$, where $\xi _{1l}(\beta ,\theta )$ are the lth component of $\xi _{1}(\beta ,\theta )$. According to (20), we have

$$\begin{aligned} \max _{1\le i \le n}\Vert \xi _{i}(\beta ,\theta )\Vert =o_{p}\left( n^{1/4}\sqrt{p}\right) . \end{aligned}$$

Similar to the proof of (17) and (18) above, it is easy to check that

$$\begin{aligned} \Vert {\hat{\xi }}_{i}(\beta ,\theta )\Vert =\Vert \xi _{i}(\beta ,\theta )\Vert +O_{p}(p). \end{aligned}$$

Then, by Assumption 7, we have

$$\begin{aligned} \Vert {\hat{\xi }}_{i}\left( \beta ,\theta \right) \Vert =o_{p}\left( n^{1/4}\sqrt{p}\right) +O_{p}\left( p\right) =o_{p}\left( n^{1/4}\sqrt{p}\right) , \end{aligned}$$

and for all $\lambda =O_{p}(a_{n})$,

$$\begin{aligned} \max _{1\le i \le n}|\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta )|=o_{p}(1). \end{aligned}$$

The proof of Lemma 5.1 is completed. $\square$

Lemma 5.2

Under Assumptions of Theorem 2.2, $\Vert S_{n}-V\Vert =O_{p}(p/\sqrt{n})$, where $S_{n}=1/n\sum _{i=1}^{n}{\hat{\xi }}_{i}(\beta ,\theta ){\hat{\xi }}_{i}(\beta ,\theta )^{\top }$.

Proof

Similar to the proof of Lemma 5.4 in Chen et al. (2009), we have $tr\{(S_{n}-V)^{\otimes 2}\}=O_{p}(p^{2}/n)$. Therefore, by the definition of Frobenius norm, $\Vert S_{n}-V\Vert =\{tr[(S_{n}-V)^{\top }(S_{n}-V)]\}^{1/2}=O_{p}(p/\sqrt{n})$. $\square$

Lemma 5.3

Under Assumptions of Theorem 2.2, $\Vert \lambda \Vert =O_{p}(a_{n})$, where $\lambda$ is the root of (8).

Proof

According to (8), $\lambda \in {\mathbb {R}}^{p+r}$ satisfies

$$\begin{aligned} 0=\frac{1}{n}\sum _{i=1}^{n}\frac{{\hat{\xi }}_{i}(\beta ,\theta )}{1+\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta )}=:\psi (\lambda ). \end{aligned}$$

Let $\lambda =\rho \alpha$, where $\rho \ge 0$, $\alpha \in {\mathbb {R}}^{p+r}$ and $\Vert \alpha \Vert =1$. Substituting $1/(1+\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta ))=1-\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta )/(1+\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta ))$ into $\alpha ^{\top }\psi (\lambda )=0$, we have

$$\begin{aligned} |\alpha ^{\top }\bar{{\hat{\xi }}}_{i}(\beta ,\theta )|\ge \frac{\rho }{1+\rho \max \limits _{1\le i \le n}\Vert \xi _{i}(\beta ,\theta )\Vert }\alpha ^{\top }S_{n}\alpha , \end{aligned}$$

where $S_{n}=\frac{1}{n}\sum \limits _{i=1}^{n}{\hat{\xi }}_{i}(\beta ,\theta ){\hat{\xi }}_{i}(\beta ,\theta )^{\top }$. Because of

$$\begin{aligned} 0< 1+\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta )\le 1+\rho \max \limits _{1\le i \le n}\Vert \xi _{i}(\beta ,\theta )\Vert , \end{aligned}$$

we have

$$\begin{aligned} \rho [\alpha ^{\top }S_{n}\alpha -\alpha ^{\top }{\hat{\xi }}_{i}(\beta ,\theta )\max \limits _{1\le i \le n}\Vert \xi _{i}(\beta ,\theta )\Vert ]\le \left| \alpha ^{\top }\bar{{\hat{\xi }}}_{i}(\beta ,\theta )\right| . \end{aligned}$$

(21)

Because $|\alpha ^{\top }\bar{{\hat{\xi }}}_{i}(\beta ,\theta )|\le \Vert \bar{{\hat{\xi }}}_{i}(\beta ,\theta )\Vert =O_{p}(\sqrt{p/n})$ and Lemma 5.1, then

$$\begin{aligned} \max \limits _{1\le i \le n}\Vert \xi _{i}(\beta ,\theta )\Vert \left| \alpha ^{\top }\bar{{\hat{\xi }}}_{i}(\beta ,\theta )\right| =o_{p}(1). \end{aligned}$$

(22)

By combining (21) and (22), we have

$$\begin{aligned} |\rho [\alpha ^{\top }S_{n}\alpha +o_{p(1)}]|=O_{p}(\sqrt{p/n}). \end{aligned}$$

According to Lemma 5.2, for a constant $C_{1}>0$, $P(\alpha ^{\top }S_{n}\alpha \ge \frac{1}{2}C_{1})\rightarrow 1$ as $n\rightarrow \infty$. Hence, $\rho =O_{p}(\sqrt{p/n})$, that is $\Vert \lambda \Vert =\rho =O_{p}(\sqrt{p/n})$, and the proof of Lemma 5.3 is completed. $\square$

Lemma 5.4

Under Assumptions of Theorem 2.2, as $n\rightarrow \infty$,

$$\begin{aligned}&\left\{ 2\left( p+r-1\right) \right\} ^{-1}\left\{ \left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) ^{\top }V^{-1}\left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n} {\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) \right. \\&\quad \left. -\left( p+r-1\right) \right\} {\mathop {\rightarrow }\limits ^{L}} N\left( 0,1\right) \!. \end{aligned}$$

Proof

The proof entails applying the martingale central limit theorem as given in Hall and Hyde (1980), and is omitted. $\square$

Lemma 5.5

Under Assumptions of Theorem 2.2,

$$\begin{aligned} \left\{ \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right\} ^{\top }\left( S_{n}^{-1}-V^{-1}\right) \left\{ \frac{1}{\sqrt{n}}\sum _{i=1}^{n} {\hat{\xi }}_{i}\left( \beta ,\theta \right) \right\} =o_{p}\left( \sqrt{p}\right) . \end{aligned}$$

Proof

Let $D_{n}=V^{-1/2}S_{n}V^{-1/2}-I_{p+r}$, where $I_{p+r}$ is the $p+r$ dimensional identity matrix.

$$\begin{aligned} S_{n}^{-1}-V^{-1}= & {} V^{-1/2}\left( V^{1/2}S_{n}^{-1}V^{1/2}-I_{p+r}\right) V^{-1/2}\\= & {} V^{-1/2}\left\{ -D_{n}+D_{n}^{2}+D_{n}^{2}\left( V^{1/2}S_{n}^{-1}V^{1/2}-I_{p+r}\right) \right\} V^{-1/2}. \end{aligned}$$

It is easy to check that

$$\begin{aligned} tr\left( S_{n}-V\right)= & {} tr\left( V^{1/2}\left( V^{-1/2}S_{n}V^{-1/2}-I_{p+r}\right) V^{1/2}\right) \\= & {} tr\left( D_{n}VD_{n}V\right) \ge \gamma _{1}^{2}\left( V\right) tr\left( D_{n}^{2}\right) , \end{aligned}$$

where $\gamma _{1}(V)$ is the smallest eigenvalue of V. Similar to the proof of Lemma 5.4 in Chen et al. (2009), we have

$$\begin{aligned} tr\left( D_{n}^{2}\right) \le tr\left\{ \left( S_{n}-V\right) ^{2}\right\} =O_{p}\left( p^{2}/n\right) . \end{aligned}$$

Then

$$\begin{aligned} tr\left( S_{n}^{-1}-V^{-1}\right) ^{2}\le & {} 2tr\left\{ V^{-2}\left( -D_{n}+D_{n}^{2}\right) ^{2}\right\} +2tr\left\{ D_{n}^{4}\left( S_{n}^{-1}-V^{-1}\right) ^{2}\right\} \\\le & {} 2tr\left\{ V^{-2}\left( -D_{n}+D_{n}^{2}\right) ^{2}\right\} \\&+2\left\{ tr\left( D_{n}^{2}\right) \right\} ^{2}tr\left\{ \left( S_{n}^{-1}-V^{-1}\right) ^{2}\right\} \\= & {} 2tr\left\{ V^{-2}\left( -D_{n}+D_{n}^{2}\right) ^{2}\right\} +o_{p}\left( tr\left\{ \left( S_{n}^{-1}-V^{-1}\right) ^{2}\right\} \right) \\= & {} o_{p}\left( p^{2}/n\right) . \end{aligned}$$

Because $\Vert \frac{1}{n}\sum _{i=1}^{n}{\hat{\xi }}_{i}(\beta ,\theta )\Vert =O_{p}(\sqrt{p/n})$, we can obtain

$$\begin{aligned}&\left\{ \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right\} ^{\top }\left( S_{n}^{-1}-V^{-1}\right) \left\{ \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right\} \\&\quad \le n\Vert \frac{1}{n}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \Vert ^{2}\sqrt{tr\left( S_{n}^{-1}-V^{-1}\right) ^{2}}=o_{p}\left( \sqrt{p}\right) . \end{aligned}$$

$\square$

Proof of Theorem 2.2

Proof

Put $W_{i}=\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta ), i=1,\ldots ,n$. By expanding Eq. (8), we obtain

$$\begin{aligned} 0=\sum _{i=1}^{n}\frac{{\hat{\xi }}_{i}\left( \beta ,\theta \right) }{1+\lambda ^{\top }{\hat{\xi }}_{i}\left( \beta ,\theta \right) } =\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) -\sum _{i=1}^{n}\left\{ {\hat{\xi }}_{i}\left( \beta ,\theta \right) {\hat{\xi }}_{i}\left( \beta ,\theta \right) ^{\top }\right\} \lambda +R_{n},\qquad \end{aligned}$$

(23)

where $R_{n}=\sum _{i=1}^{n}\frac{{\hat{\xi }}_{i}(\beta ,\theta )(\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta ))^{2}}{(1+\vartheta _{i})^{3}}$ and $|\vartheta _{i}|\le |\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta )|$. By Lemma 5.1, we have $\max _{1\le i\le n}|\vartheta _{i}|=o_{p}(1)$. Hence $R_{n}=R_{n1}\{1+o_{p}(1)\}$, where

$$\begin{aligned} R_{n1}=\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \left( \lambda ^{\top }{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) ^{2}. \end{aligned}$$

Apply Lemmas 5.1 and 5.3, we obtain

$$\begin{aligned} \Vert n^{-1}R_{n}\Vert \le C\Vert \lambda \Vert ^{2}\max _{1\le i \le n}\Vert {\hat{\xi }}_{i}(\beta ,\theta )\Vert n^{-1}\sum _{i=1}^{n}\Vert {\hat{\xi }}_{i}(\beta ,\theta )\Vert ^{2}=o_{p}(a_{n}). \end{aligned}$$

(24)

By (23), we have

$$\begin{aligned} \lambda =\left\{ \sum _{i=1}^{n}{\hat{\xi }}_{i}(\beta ,\theta ){\hat{\xi }}_{i}(\beta ,\theta )^{\top }\right\} ^{-1}\sum _{i=1}^{n}{\hat{\xi }}_{i}(\beta ,\theta )+ \left\{ \sum _{i=1}^{n}{\hat{\xi }}_{i}(\beta ,\theta ){\hat{\xi }}_{i}(\beta ,\theta )^{\top }\right\} ^{-1}R_{n}. \end{aligned}$$

Applying Taylor’s expansion, for some $\zeta _{i}$ such that $|\zeta _{i}|\le |\lambda ^{\top }{\hat{\xi }}_{i}(\beta ,\theta )|$, we obtain

$$\begin{aligned} \log \left( 1+\lambda ^{\top }{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) =\lambda ^{\top }{\hat{\xi }}_{i}\left( \beta ,\theta \right) -\frac{\left\{ \lambda ^{\top }{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right\} ^{2}}{2}+ \frac{\left\{ \lambda ^{\top }{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right\} ^{3}}{3\left( 1+\zeta _{i}\right) ^{4}}. \end{aligned}$$

Therefore,

$$\begin{aligned} {\tilde{l}}\left( \beta ,\theta \right)= & {} \left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) ^{\top }\left\{ \frac{1}{n}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) {\hat{\xi }}_{i}\left( \beta ,\theta \right) ^{\top }\right\} ^{-1}\left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) \nonumber \\&-\frac{1}{n}R_{n}^{\top }\left\{ \frac{1}{n}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) {\hat{\xi }}_{i}\left( \beta ,\theta \right) ^{\top }\right\} ^{-1}R_{n}+\sum _{i=1}^{n}\frac{2\left\{ \lambda ^{\top } {\hat{\xi }}_{i}\left( \beta ,\theta \right) \right\} ^{3}}{3\left( 1+\zeta _{i}\right) ^{4}}\nonumber \\= & {} \left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) ^{\top }V^{-1}\left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) \nonumber \\&+\left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) ^{\top }\left[ \left\{ \frac{1}{n}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) {\hat{\xi }}_{i}\left( \beta ,\theta \right) ^{\top }\right\} ^{-1} -V^{-1}\right] \nonumber \\&\times \left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) -\frac{1}{n}R_{n}^{\top }\left\{ \frac{1}{n}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) {\hat{\xi }}_{i}\left( \beta ,\theta \right) ^{\top }\right\} ^{-1}R_{n}\nonumber \\&+\frac{2}{3}\sum _{i=1}^{n} \left\{ \lambda ^{\top }{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right\} ^{3}\left\{ 1+o_{p}\left( 1\right) \right\} . \end{aligned}$$

(25)

By Lemma 5.5, we have

$$\begin{aligned}&\left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) ^{\top }\left[ \left\{ \frac{1}{n}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) {\hat{\xi }}_{i}\left( \beta ,\theta \right) ^{\top }\right\} ^{-1}-V^{-1}\right] \nonumber \\&\times \left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) =o_{p}\left( 1\right) . \end{aligned}$$

(26)

By Lemmas 5.1–5.3 and (24), we can obtain

$$\begin{aligned} \frac{1}{n}R_{n}^{\top }\left\{ \frac{1}{n}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) {\hat{\xi }}_{i}\left( \beta ,\theta \right) ^{\top }\right\} ^{-1}R_{n}=o_{p}\left( 1\right) , \end{aligned}$$

(27)

and

$$\begin{aligned} \frac{2}{3}\sum _{i=1}^{n} \left\{ \lambda ^{\top }{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right\} ^{3}\left\{ 1+o_{p}\left( 1\right) \right\} =o_{p}\left( \sqrt{p}\right) . \end{aligned}$$

(28)

It follows from (25)–(28) that

$$\begin{aligned} {\tilde{l}}\left( \beta ,\theta \right) =\left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n}{\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) ^{\top }V^{-1}\left( \frac{1}{\sqrt{n}}\sum _{i=1}^{n} {\hat{\xi }}_{i}\left( \beta ,\theta \right) \right) +o_{p}\left( \sqrt{p}\right) . \end{aligned}$$

Hence the theorem follows from Lemmas 5.4 and 5.5, and the proof of Theorem 2.2 is completed. $\square$

Proof of Theorem 2.3

Proof

We first prove that $\max _{1\le i \le n}\Vert \hat{{\tilde{\xi }}}_{i}(\beta ^{(1)})\Vert =o_{p}(n^{1/2})$. It can be shown that

$$\begin{aligned} \hat{{\tilde{\xi }}}_{i}\left( \beta \right)= & {} {\hat{w}}_{i}\left\{ Y_{i}-X_{i}^{\left( 1\right) \top }\beta ^{\left( 1\right) }-X_{i}^{\left( 2\right) \top }{\hat{\beta }}^{\left( 2\right) }-{\hat{g}}\left( Z_{i}^{\top }{\hat{\theta }}\right) \right\} \left\{ X_{i}^{\left( 1\right) }-\frac{{\hat{E}}\left( {\hat{w}}X^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z_{i}^{\top }{\hat{\theta }}\right) }\right\} \nonumber \\= & {} \left\{ w_{i}\left( 1+o_{p}\left( 1\right) \right) \right\} \left\{ \varepsilon _{i}+X_{i}^{\top }\left( \beta -{\hat{\beta }}\right) +X_{i}^{\left( 1\right) \top }\left( {\hat{\beta }}^{\left( 1\right) }-\beta ^{\left( 1\right) }\right) +\left( g\left( Z_{i}^{\top }{\hat{\theta }}\right) \right. \right. \nonumber \\&\left. \left. -{\hat{g}}\left( Z_{i}^{\top }{\hat{\theta }}\right) \right) \right\} \nonumber \\&\times \left\{ \left( X_{i}^{\left( 1\right) }-\frac{E\left( wX^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{E\left( w|Z_{i}^{\top }{\hat{\theta }}\right) }\right) +\left( \frac{E\left( wX^{\left( 1\right) } |Z_{i}^{\top }{\hat{\theta }}\right) }{E\left( w|Z^{\top }{\hat{\theta }}\right) }-\frac{{\hat{E}}\left( {\hat{w}}X^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z_{i}^{\top } {\hat{\theta }}\right) }\right) \right\} \nonumber \\= & {} M_{i1}+M_{i2}+M_{i3}+M_{i4}+M_{i5}+M_{i6}+M_{i7}+M_{i8}, \end{aligned}$$

(29)

where

$$\begin{aligned} M_{i1}= & {} \{w_{i}\left( 1+o_{p}\left( 1\right) \right) \}\varepsilon _{i}\left\{ X_{i}^{\left( 1\right) }-\frac{E\left( wX^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{E\left( w|Z_{i}^{\top }{\hat{\theta }}\right) } \right\} ,\\ M_{i2}= & {} \{w_{i}\left( 1+o_{p}\left( 1\right) \right) \}\varepsilon _{i}\left\{ \frac{E\left( wX^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{E\left( w|Z^{\top }{\hat{\theta }}\right) }-\frac{{\hat{E}} \left( {\hat{w}}X^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z^{\top }{\hat{\theta }}\right) }\right\} ,\\ M_{i3}= & {} \{w_{i}\left( 1+o_{p}\left( 1\right) \right) \}X_{i}^{\top }\left( \beta -{\hat{\beta }}\right) \left\{ X_{i}^{\left( 1\right) }-\frac{E\left( wX^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{E\left( w|Z_{i}^{\top } {\hat{\theta }}\right) }\right\} ,\\ M_{i4}= & {} \{w_{i}\left( 1+o_{p}\left( 1\right) \right) \}X_{i}^{\top }\left( \beta -{\hat{\beta }}\right) \left\{ \frac{E\left( wX^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{E\left( w|Z^{\top }{\hat{\theta }}\right) } -\frac{{\hat{E}}\left( {\hat{w}}X^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z^{\top }{\hat{\theta }}\right) }\right\} ,\\ M_{i5}= & {} \{w_{i}\left( 1+o_{p}\left( 1\right) \right) \}X_{i}^{\left( 1\right) \top }\left( {\hat{\beta }}^{\left( 1\right) }-\beta ^{\left( 1\right) }\right) \left\{ X_{i}^{\left( 1\right) }-\frac{E\left( wX^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{E\left( w|Z_{i}^{\top }{\hat{\theta }}\right) }\right\} ,\\ M_{i6}= & {} \{w_{i}\left( 1+o_{p}\left( 1\right) \right) \}X_{i}^{\left( 1\right) \top }\left( {\hat{\beta }}^{\left( 1\right) }\right. \\&\left. -\beta ^{\left( 1\right) }\right) \left\{ \frac{E\left( wX^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{E\left( w|Z_{i} {\hat{\theta }}\right) }-\frac{{\hat{E}}\left( {\hat{w}}X^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z^{\top }{\hat{\theta }}\right) }\right\} ,\\ M_{i7}= & {} \{w_{i}\left( 1+o_{p}\left( 1\right) \right) \}\left( g\left( z_{i}^{\top }{\hat{\theta }}\right) -{\hat{g}}\left( Z_{i}^{\top }{\hat{\theta }}\right) \right) \left\{ X_{i}^{\left( 1\right) }-\frac{E\left( wX^{\left( 1\right) }|Z_{i}^{\top } {\hat{\theta }}\right) }{E\left( w|Z_{i}^{\top }{\hat{\theta }}\right) }\right\} ,\\ M_{i8}= & {} \{w_{i}\left( 1+o_{p}\left( 1\right) \right) \}\left( g\left( z_{i}^{\top }{\hat{\theta }}\right) \right. \\&\left. -{\hat{g}}\left( Z_{i}^{\top }{\hat{\theta }}\right) \right) \left\{ \frac{E\left( wX^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{E\left( w| Z_{i}^{\top }{\hat{\theta }}\right) }-\frac{{\hat{E}}\left( {\hat{w}}X^{\left( 1\right) }|Z_{i}^{\top }{\hat{\theta }}\right) }{{\hat{E}}\left( {\hat{w}}|Z^{\top }{\hat{\theta }}\right) }\right\} . \end{aligned}$$

By (29), we can obtain that

$$\begin{aligned} \max _{1\le i \le n}\Vert \hat{{\tilde{\xi }}}_{i}(\beta ^{(1)})\Vert\le & {} \max _{1\le i \le n}\Vert M_{i1}\Vert +\max _{1\le i \le n}\Vert M_{i2}\Vert +\max _{1\le i \le n}\Vert M_{i3}\Vert +\max _{1\le i \le n}\Vert M_{i4}\Vert \\+ & {} \max _{1\le i \le n}\Vert M_{i5}\Vert +\max _{1\le i \le n}\Vert M_{i6}\Vert +\max _{1\le i \le n}\Vert M_{i7}\Vert +\max _{1\le i \le n}\Vert M_{i8}\Vert . \end{aligned}$$

Similar to the proof of Proposition 2 in Ma and Zhu (2013), it is easy to show that

$$\begin{aligned} \max _{1\le i \le n}\Vert M_{l1}\Vert =o_{p}(n^{1/2}),\quad l=1,\ldots ,8. \end{aligned}$$

Therefore, we have $\max _{1\le i \le n}\Vert \hat{{\tilde{\xi }}}_{i}(\beta ^{(1)})\Vert =o_{p}(n^{1/2})$. In addition, from the proof of Theorem 3.1 in Li and Wang (2003), as $n\rightarrow \infty$, we can also show that

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\hat{{\tilde{\xi }}}_{i}\left( \beta ^{\left( 1\right) }\right) {\mathop {\longrightarrow }\limits ^{L}} N\left( 0,V_{1}\left( \beta ^{\left( 1\right) }\right) \right) , \end{aligned}$$

(30)

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}\hat{{\tilde{\xi }}}_{i}\left( \beta ^{\left( 1\right) }\right) \hat{{\tilde{\xi }}}_{i}^{\top }\left( \beta ^{\left( 1\right) }\right) {\mathop {\longrightarrow }\limits ^{p}} V_{1}\left( \beta ^{\left( 1\right) }\right) , \end{aligned}$$

(31)

where

$$\begin{aligned} V_{1}\left( \beta ^{\left( 1\right) }\right) =E\left\{ wX^{\left( 1\right) }X^{\left( 1\right) \top }-\frac{E\left( wX^{\left( 1\right) }|Z^{\top }{\hat{\theta }}\right) E\left( wX^{\left( 1\right) T}|Z^{\top }{\hat{\theta }}\right) }{E\left( w|Z^{\top }{\hat{\theta }}\right) }\right\} \end{aligned}$$

and ${\mathop {\rightarrow }\limits ^{p}}$ stands for convergence in probability. By $\max _{1\le i \le n}\Vert \hat{{\tilde{\xi }}}_{i}(\beta ^{(1)})\Vert =o_{p}(n^{1/2})$ and Talor expansion to (10), we can obtain that

$$\begin{aligned} {\tilde{l}}\left( \beta ^{\left( 1\right) }\right) =2\sum _{i=1}^{n}\lambda ^{\left( 1\right) \top }\hat{{\tilde{\xi }}}_{i}\left( \beta ^{\left( 1\right) }\right) -\sum _{i=1}^{n}\left\{ \lambda ^{\left( 1\right) \top }\hat{{\tilde{\xi }}}_{i} \left( \beta ^{\left( 1\right) }\right) \right\} ^{2}+o_{p}\left( 1\right) . \end{aligned}$$

(32)

Similar to the proof of Theorem 17 in Owen (1990), we have

$$\begin{aligned}&\sum _{i=1}^{n}\left\{ \lambda ^{\left( 1\right) \top }\hat{{\tilde{\xi }}}_{i}\left( \beta ^{\left( 1\right) }\right) \right\} ^{2}=\sum _{i=1}^{n}\lambda ^{\left( 1\right) \top }\hat{{\tilde{\xi }}}_{i}\left( \beta ^{\left( 1\right) }\right) +o_{p}\left( 1\right) , \end{aligned}$$

(33)

$$\begin{aligned}&\lambda ^{\left( 1\right) }=\left\{ \sum _{i=1}^{n}\hat{{\tilde{\xi }}}_{i}\left( \beta ^{\left( 1\right) }\right) \hat{{\tilde{\xi }}}_{i}\left( \beta ^{\left( 1\right) }\right) ^{\top }\right\} ^{-1}\sum _{i=1}^{n}\hat{{\tilde{\xi }}}_{i} \left( \beta ^{\left( 1\right) }\right) +o_{p}\left( n^{-1/2}\right) . \end{aligned}$$

(34)

Combining (32)–(34) implies that

$$\begin{aligned}&{\tilde{l}}\left( \beta ^{\left( 1\right) }\right) \\&\quad =\left\{ \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\hat{{\tilde{\xi }}}_{i}\left( \beta ^{\left( 1\right) }\right) \right\} ^{\top }\left\{ \frac{1}{n}\sum _{i=1}^{n}\hat{{\tilde{\xi }}}_{i} \left( \beta ^{\left( 1\right) }\right) \hat{{\tilde{\xi }}}_{i}^{\top }\left( \beta ^{\left( 1\right) }\right) \right\} ^{-1}\left\{ \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\hat{{\tilde{\xi }}}_{i}\left( \beta ^{\left( 1\right) }\right) \right\} \\&\qquad +o_{p}\left( 1\right) . \end{aligned}$$

Therefore, together with (30) and (31), we can show that ${\tilde{l}}(\beta ^{(1)}){\mathop {\rightarrow }\limits ^{L}} \chi _{k}^{2}$, and the proof is completed. $\square$

The partially linear model or the single-index model is a special case of the partially linear single-index model. We can prove Theorems 3.1 and 3.2 by using the same arguments in the proofs of Theorems 2.1–2.3, hence their proofs are omitted.