Empirical likelihood inference for generalized additive partially linear models

Abstract

Generalized additive partially linear models enjoy the simplicity of GLMs and the flexibility of GAMs because they combine both parametric and nonparametric components. Based on spline-backfitted kernel estimator, we propose empirical likelihood (EL)-based pointwise confidence intervals and simultaneous confidence bands (SCBs) for the nonparametric component functions to make statistical inference. Simulation study strongly supports the asymptotic theory and shows that EL-based SCBs are much easier for implementation and have better performance than Wald-type SCBs. We apply the proposed method to a university retention study and provide SCBs for the effect of the students information.

Appendix A: Assumptions and proofs

We mean by “$\sim$” both sides having the same order as $n\rightarrow \infty$ in the rest of the paper. To obtain EL SCBs, we need the following assumptions which are similar assumptions to construct Wald-type SCBs. See Liu et al. (2017) for more details.

(A1) The additive component functions $m_{\alpha }\in C^{\left( 1\right) }\left[ 0,1\right] ,1\le \alpha \le d_{2}$ with $m_{1}\in C^{\left( 2\right) }\left[ 0,1\right]$, $m_{\alpha }^{\prime }\in \mathrm{Lip}\left( \left[ 0,1\right] , C_{m}\right) , 2\le \alpha \le d_{2}$ for some constant $C_{m}>0$.

(A2) The inverse link function $b^{\prime }$  satisfies $b^{\prime }\in C^{2}\left( {\mathbb {R}}\right) ,b^{\prime \prime }\left( \theta \right) >0,\theta \in {\mathbb {R}}$  while for a compact interval $\Theta$ whose interior contains $m\left( \left[ 0,1\right] ^{d_{2}}\right)$, $C_{b}>\max _{\theta \in \Theta }b^{\prime \prime }\left( \theta \right) \ge \min _{\theta \in \Theta }b^{\prime \prime }\left( \theta \right) >c_{b}$ for constants $C_{b}>c_{b}>0$.

(A3) The conditional variance function $\sigma ^{2}\left( {\mathbf {x}}\right)$ is measurable and bounded. The errors $\left\{ \varepsilon _{i}\right\} _{i=1}^{n}$ satisfy ${\mathsf {E}}\left( \varepsilon _{i}|{\mathbf {T}}_{i}^{{\top }},{\mathbf {X}}_{i}^{{\top }}\right) =0$ and ${\mathsf {E}}\varepsilon _{i}^{6}<\infty$.

(A4) The density function $f\left( {\mathbf {x}}\right)$  of $\left( X_{1},\ldots ,X_{d_{2}}\right)$ is continuous and

$$\begin{aligned} 0<c_{f}\le \inf \limits _{{\mathbf {x}}\in \varvec{\chi }}f\left( {\mathbf {x}} \right) \le \sup \limits _{{\mathbf {x}}\in \varvec{\chi }}f\left( {\mathbf {x}} \right) \le C_{f}<\infty . \end{aligned}$$

The marginal densities $f_{\alpha }\left( x_{\alpha }\right)$ of $X_{\alpha }$ have continuous derivatives on $\left[ 0,1\right]$ as well as the uniform upper bound $C_{f}$ and lower bound $c_{f}$.

(A5) $\left\{ {\mathbf {Z}}_{i}=\left( {\mathbf {T}}_{i}^{{\top }},{\mathbf {X}} _{i}^{{\top }},\varepsilon _{i}\right) \right\} _{i=1}^{n}$ are independent and identically distributed.

(A6) There exist constants $0<c_{\delta }<C_{\delta }<\infty$ and $0<c_{{\mathbf {Q}}}<C_{{\mathbf {Q}}}<\infty$ such that $c_{\delta }\le {\mathsf {E}}(\left| T_{k}\right| ^{2+\delta }\mid \mathbf {X=x})\le C_{\delta }$ for some $\delta >0,$ and $c_{{\mathbf {Q}}}I_{d_{1}\times d_{1}}\le {\mathsf {E}}\left( \mathbf {TT}^{\top }\mid \mathbf {X=x}\right) \le C_{{\mathbf {Q}} }I_{d_{1}\times d_{1}}$.

(A7) The kernel function K is a symmetric probability density, supported on $\left[ -1,1\right]$ and $K\in \mathrm{Lip}\left( \left[ -1,1\right] ,C_{K}\right)$ for some positive constant $C_{K}>0$. The bandwidth $h=h_{n}$ $\sim n^{-1/5}(\log n)^{-1/4}$.

(A8) The number of interior knots satisfies: $N\thicksim n^{1/4}\log n,$ i.e., $c_{N}n^{1/4}$ $\log n\le N\le C_{N}n^{1/4}\log n$ for some positive constants $c_{N}$,$C_{N}.$

Lemma A.1

Under Assumptions (A1)–(A7), as $n\rightarrow \infty$,

$$\begin{aligned} \sqrt{nh}\left[ n^{-1}\sum \limits _{i=1}^{n}Z_{i}\left\{ m_{1}\left( x_{1}\right) \right\} -\mathrm{bias}_{1}\left( x_{1}\right) h^{2} \right] /v_{1}\left( x_{1}\right) \overset{{\mathcal {L}}}{\rightarrow } N\left( 0,1\right) , \end{aligned}$$

(A.1)

with

$$\begin{aligned} \mathrm{bias}{}_{1}\left( x_{1}\right)= & {} \mu _{2}\left( K\right) \left[ m_{1}^{\prime \prime }\left( x_{1}\right) f_{1}\left( x_{1}\right) {\mathsf {E}}\left[ b^{\prime \prime }\left\{ m\left( \mathbf {T,X}\right) \right\} |X_{1}=x_{1} \right] \right. \\&+m_{1}^{\prime }\left( x_{1}\right) \frac{\partial }{\partial x_{1}} \left\{ f_{1}\left( x_{1}\right) {\mathsf {E}}\left[ b^{\prime \prime }\left\{ m\left( \mathbf {T,X}\right) \right\} |X_{1}=x_{1}\right] \right\} \\&\left. -\left\{ m_{1}^{\prime }\left( x_{1}\right) \right\} ^{2}f_{1}\left( x_{1}\right) {\mathsf {E}}\left[ b^{\prime \prime \prime }\left\{ m\left( \mathbf {T,X}\right) \right\} |X_{1}=x_{1}\right] \right] , \\ v_{1}^{2}\left( x_{1}\right)= & {} f_{1}\left( x_{1}\right) {\mathsf {E}}\left\{ \sigma ^{2}\left( {\mathbf {T}},{\mathbf {X}}\right) |X_{1}=x_{1}\right\} \left\| K\right\| _{2}^{2}. \end{aligned}$$

In addition,

$$\begin{aligned}&n^{-1}h\sum \limits _{i=1}^{n}Z_{i}^{2}\left\{ m_{1}\left( x_{1}\right) \right\} \rightarrow _{p}v_{1}^{2}\left( x_{1}\right) , \end{aligned}$$

(A.2)

$$\begin{aligned}&\sup _{x_{1}\in [h,1-h]}\left| n^{-1}\sum \limits _{i=1}^{n}Z_{i}\left\{ m_{1}\left( x_{1}\right) \right\} \right| ={{\mathcal {O}}}_{a.s.}\left( n^{-1/2}h^{-1/2}\log n+h^{2}\right) . \end{aligned}$$

(A.3)

Proof

See the supplement. $\square$

Lemma A.2

Under Assumptions (A1)–(A8), as $n\rightarrow \infty$,

$$\begin{aligned} \sup _{x_{1}\in \left[ h,1-h\right] }\left| n^{-1}\sum \limits _{i=1}^{n} \left[ {\hat{Z}}_{i}\left\{ m_{1}\left( x_{1}\right) \right\} -Z_{i}\left\{ m_{1}\left( x_{1}\right) \right\} \right] \right| ={{\mathcal {O}}}_{a.s.}\left( n^{-1/2}\log n\right) , \end{aligned}$$

(A.4)

and

$$\begin{aligned} \sup _{x_{1}\in \left[ h,1-h\right] }\left| n^{-1}\sum \limits _{i=1}^{n} \left[ {\hat{Z}}_{i}^{2}\left\{ m_{1}\left( x_{1}\right) \right\} -Z_{i}^{2}\left\{ m_{1}\left( x_{1}\right) \right\} \right] \right| ={\scriptstyle {\mathcal {O}}}_{a.s.}\left( n^{-1/5}\right) . \end{aligned}$$

(A.5)

Proof

See the supplement. $\square$

Proof of Theorem 1

See the supplement. $\square$

Proof of Theorem 2

Denote

$$\begin{aligned} G\left( t\right) =h^{1/2}\int _{0}^{1}K_{h}\left( t-s\right) \mathrm{d}W\left( s\right) , \end{aligned}$$

with $W\left( s\right)$ a Wiener process. According to the equation (27) in Sun et al. (2009), one has that

$$\begin{aligned} \sup \left| \mathrm{P}\left[ \frac{r_{h}}{2d_{h}}\left\{ \sup \limits _{x_{1}\in \left[ h,1-h\right] }\left| G\left( x_{1}\right) \right| ^{2}-d_{h}^{2}\right\} <c\right] -e^{-2e^{-c}}\right| ={{\mathcal {O}}}\left\{ \left( \log n\right) ^{-1}\right\} , \end{aligned}$$

(A.6)

where $r_{h},d_{h}$ are defined in Theorem 2. According to equation (A.11) in the supplement, one has

$$\begin{aligned}&-2\log {\tilde{R}}\left\{ m_{1}\left( x_{1}\right) \right\} \\&\quad =n\left[ n^{-1}\sum \limits _{i=1}^{n}Z_{i}^{2}\left\{ m_{1}\left( x_{1}\right) \right\} \right] ^{-1}\left[ n^{-1}\sum \limits _{i=1}^{n}Z_{i} \left\{ m_{1}\left( x_{1}\right) \right\} \right] ^{2}+{\scriptstyle {\mathcal {O}}}_{p}\left( 1\right) \\&\quad =n\left\{ v_{1}^{2}\left( x_{1}\right) h^{-1}\right\} ^{-1}\left[ n^{-1}\sum \limits _{i=1}^{n}Z_{i}\left\{ m_{1}\left( x_{1}\right) \right\} \right] ^{2}+{\scriptstyle {\mathcal {O}}}_{p}\left( 1\right) \\&\quad =\left[ \left( nh\right) ^{1/2}v_{1}^{-1}\left( x_{1}\right) n^{-1}\sum \limits _{i=1}^{n}Z_{i}\left\{ m_{1}\left( x_{1}\right) \right\} \right] ^{2}+{\scriptstyle {\mathcal {O}}}_{p}\left( 1\right) \end{aligned}$$

due to Eq. (A.2). According to equation (A.7) in the supplement and Theorem 1 in Zheng et al. (2016),

$$\begin{aligned}&\sup \limits _{x_{1}\in \left[ h,1-h\right] }\left| \left( nh\right) ^{1/2}v_{1}^{-1}\left( x_{1}\right) n^{-1}\sum \limits _{i=1}^{n}Z_{i}\left\{ m_{1}\left( x_{1}\right) \right\} -G\left( x_{1}\right) /\left\| K\right\| _{2}^{2}\right| \\&\quad ={\scriptstyle {\mathcal {O}}}_{p}\left\{ \left( \log n\right) ^{-1/2}\right\} . \end{aligned}$$

Therefore,

$$\begin{aligned} \sup \limits _{x_{1}\in \left[ h,1-h\right] }\left| -2\log {\tilde{R}} \left\{ m_{1}\left( x_{1}\right) \right\} -\left| G\left( x_{1}\right) \right| ^{2}/\left\| K\right\| _{2}^{4}\right| ={\scriptstyle {\mathcal {O}}}_{p}\left( 1\right) . \end{aligned}$$

Then, the theorem is proved by replacing $G\left( x_{1}\right)$ by $-2\log {\tilde{R}}\left\{ m_{1}\left( x_{1}\right) \right\} \left\| K\right\| _{2}^{4}$ in Eq. (A.6). $\square$

Proof of Theorem 3

See the supplement. $\square$