Model checking for generalized partially linear models

Abstract

We propose a residual-marked empirical process test to check goodness of fit for generalized partially linear models. The proposed test can gain dimension reduction, is shown to be consistent, and can detect root-n local alternatives. We further establish asymptotic distributions of the proposed test under the null hypothesis and analyze asymptotic properties under the local and global alternatives, and suggest a bootstrap procedure for calculating the critical value. We investigate its numerical performance by simulation experiments and illustrate its utilization in two real data examples.

Appendix: Technical details

We need the representations of $\widehat{\varvec{\beta }}_n-\varvec{\beta }_0$ and $\widehat{m}_n(Z_i) - m_0(Z_i)$, which play a critical role in the proofs of the main results and need the following assumptions modified from Wang et al. (2011).

Let $\nu$ be a positive integer and $\alpha \in (0,1]$ such that $\zeta =\nu +\alpha >2$. Let ${\mathcal {H}}(\zeta )$ be the collection of functions G on [0, 1] whose $\nu$th derivative, $G^{(\nu )}$, exists and satisfies a Lipschitz condition of order $\alpha$, $|G^{(\nu ) }(s^{*})-G^{(\nu )}(s) | \le C| {s^{*}}-s| ^{\alpha }$, for $0\le {s^{*}}, s\le 1$, where C is a positive constant. Let $\rho _{\ell }(s)=\{{dG(s)/ds}\}^{\ell }/ \sigma \{G(s)\}$ and $q_{\ell }(s, y) =\partial ^{\ell }/\partial s^{\ell }Q\{G(s),y\}$, so that

$$\begin{aligned} q_{1}(s,y)= & {} \partial /\partial s Q\{G(s),y\} =\{y-G(s)\} \rho _{1}(s),\\ q_{2}(s,y)= & {} \partial ^{2}/\partial s^{2}Q\{G(s),y\} =\{y-G(s)\} \rho _{1}^{\prime }(s) -\rho _{2}(s). \end{aligned}$$

Write $\textbf{A}^{\otimes 2}=\textbf{A} \textbf{A}^{\top }$ for any matrix or vector $\textbf{A}$. We make the following assumptions.

• (C1) The function $m^{(2)}(\cdot )$ is continuous and $m(\cdot )\in {\mathcal {H}}(\zeta )$.

• (C2) The function $q_{2}(s, y) <0$ and $c_{q}<| q_{2}^{k}(s, y) | <C_{q}$ ($k=0,1$) for $s\in R$ and y in the range of the response variable.

• (C3) The distribution of Z is absolutely continuous and its density f is bounded away from zero and infinity on [0, 1].

• (C4) The random vector X satisfies that

  $$\begin{aligned} c\le E(X^{\otimes 2}|Z=z)\le C. \end{aligned}$$

• (C5) The number of knots $N_n$ satisfies $n^{1/(2\zeta ) }\ll N_{n}\ll n^{1/4}$.

• (C6) For $\rho _{\ell }$, we have

  $$\begin{aligned} |\rho _{\ell }(s_0) | \le C_{\rho }\text { and }| \rho _{\ell }(s) -\rho _{\ell }( s_0) | \le C_{\rho }^{*}|s-s_0| \text { for all }|s-s_0| \le {C_{s},\ \ell =1,2}, \end{aligned}$$

  where and below s and $s_0$ appearing in $\rho _{\ell }(\cdot )$ correspond to $X^{\top }\varvec{\beta }+m(Z)$ and $X^{\top }\varvec{\beta }_0+m_0(Z)$, respectively.

• (C7) There exists a positive constant $C_{0}$, such that $E[\{Y-G(X^{\top }\varvec{\beta }+m(Z))\}^{2}|V] \le C_{0}$, almost surely.

Let $\alpha _n=n^{-1/4}\log n$.

$$\begin{aligned} \widehat{\varvec{\beta }}_n-\varvec{\beta }_0 = \{E(S_{1,2} \widetilde{X}_1 \widetilde{X}_1^{\top })\}^{-1} \frac{1}{n} \sum \limits ^n_{i=1} S_{i,1} \widetilde{X}_i+o_p (n^{-1/2}), \end{aligned}$$

(A.1)

and

$$\begin{aligned} \widehat{m}_n(Z_i) - m_0(Z_i) = \{E(S_{i,2} |Z_i)\}^{-1} E(S_{i,2} X_i^{\top } |Z_i) (\widehat{\varvec{\beta }}_n -\varvec{\beta }_0)+o_p(\alpha _n). \end{aligned}$$

(A.2)

The proof of (A.1) is referred to the proof of Theorem 1 of Wang et al. (2011), and the proof of (A.2) is referred to the proof of the last line on page 1847 of Wang et al. (2011).

Proof of Theorem 1

By the definition of $M_{n}(u, W)$, we have

$$\begin{aligned} M_{n}(u, W)= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n [ Y_i-G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}] I(V_i^{\top } W\le u)\nonumber \\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n [Y_i-G\{X_i^{\top }\varvec{\beta }_0+m_0(Z_i)\}] I(V_i^{\top } W\le u)\nonumber \\{} & {} -\frac{1}{\sqrt{n}}\sum _{i=1}^n [G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}-G\{X_i^{\top }\varvec{\beta }_0+m_0(Z_i)\}]I(V_i^{\top } W\le u)\nonumber \\:= & {} B_{n1}(u, W)-B_{n2}(u, W). \end{aligned}$$

(A.3)

It follows from model (2) that

$$\begin{aligned} B_{n1}(u, W)=\frac{1}{\sqrt{n}}\sum _{i=1}^n \varepsilon _i I(V_i^{\top } W\le u). \end{aligned}$$

(A.4)

Let us examine $X_i^{\top } (\widehat{\varvec{\beta }}_n-\varvec{\beta }_0)+\widehat{m}_n(Z_i)-m_0(Z_i)$. This expression can be simplified as follows using (A.1) and (A.2).

$$\begin{aligned}{} & {} X_i^{\top } (\widehat{\varvec{\beta }}_n-\varvec{\beta }_0)+\widehat{m}_n(Z_i)-m_0(Z_i)\\{} & {} \quad =X_i^{\top } (\widehat{\varvec{\beta }}_n-\varvec{\beta }_0)- \{E(S_{i,2} |Z_i)\}^{-1} E(S_{i,2} X_i^{\top } |Z_i) (\widehat{\varvec{\beta }}_n -\varvec{\beta }_0)+o_p(\alpha _n)\\{} & {} \quad =\widetilde{X}_i^{\top } (\widehat{\varvec{\beta }}_n-\varvec{\beta }_0)+o_p(n^{-1/2})\\{} & {} \quad =\widetilde{X}_i^{\top } \{E(S_{1,2} \widetilde{X}_1 \widetilde{X}_1^{\top })\}^{-1} \sum _{j=1}^n S_{j,1}\widetilde{X}_j+o_p(\alpha _n) \end{aligned}$$

The second term $B_{n2}(u, W)$ in (A.3) can be simplified as follows:

$$\begin{aligned}{} & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n \left[ \frac{1}{n}\widetilde{X}_i^{\top } \{E(S_{1,2} \widetilde{X}_1 \widetilde{X}_1^{\top })\}^{-1} \sum _{j=1}^n S_{j,1}\widetilde{X}_j\right] I(V_i^{\top } W\le u)\\{} & {} \quad \qquad G'\{X_i^{\top }\varvec{\beta }_0 +m_0(Z_i)\}+o_p(\alpha _n)\\{} & {} \quad = \frac{1}{\sqrt{n}}\sum _{j=1}^n \left[ \frac{1}{n}\sum _{i=1}^n\widetilde{X}_i^{\top } \{E(S_{1,2} \widetilde{X}_1 \widetilde{X}_1^{\top })\}^{-1} I(V_i^{\top } W\le u)\right. \\{} & {} \quad \qquad \left. G'\{X_i^{\top }\varvec{\beta }_0+m_0(Z_i)\}\right] S_{j,1}\widetilde{X}_j +o_p(\alpha _n)\\{} & {} \quad = \frac{1}{\sqrt{n}}\sum _{j=1}^n E\left[ \widetilde{X}_1^{\top } \{E(S_{1,2} \widetilde{X}_1 \widetilde{X}_1^{\top })\}^{-1} I(V_1^{\top } W\le u) G'\{X_1^{\top }\varvec{\beta }_0+m_0(Z_1)\}\right] \\{} & {} \quad \qquad \times \frac{G'\{X_j^{\top }\varvec{\beta }_0+m_0(Z_j)\}}{\sigma (G\{X_j^{\top }\varvec{\beta }_0 +m_0(Z_j)\})}\varepsilon _j \widetilde{X}_j+o_p(\alpha _n). \end{aligned}$$

Recall $\Gamma (u)=E\left[ \widetilde{X}_1^{\top } \{E(S_{1,2} \widetilde{X}_1 \widetilde{X}_1^{\top })\}^{-1} I(V_1^{\top } W\le u) G'\{X_1^{\top }\varvec{\beta }_0+m_0(Z_1)\}\right]$. As a result, we have

$$\begin{aligned} B_{n2}(u, W)=\frac{1}{\sqrt{n}}\Gamma (u)\sum _{i=1}^{n} \frac{G'\{X_i^{\top }\varvec{\beta }_0+m_0(Z_i)\}}{\sigma (G\{X_i^{\top }\varvec{\beta }_0+m_0(Z_i)\})}\widetilde{X}_i \varepsilon _i+o_p(1). \end{aligned}$$

(A.5)

So, we have the following expression for $M_{n}(u, W)$.

$$\begin{aligned} M_{n}(u, W)\!=\!\frac{1}{\sqrt{n}}\sum _{i=1}^n \left[ I(V_i^{\top } W\!\le \! u)-\Gamma (u)\frac{G'\{X_i^{\top }\varvec{\beta }_0+m_0(Z_i)\}}{\sigma (G\{X_i^{\top }\varvec{\beta }_0+m_0(Z_i)\})}\widetilde{X}_i\right] \varepsilon _i+o_p(1).\nonumber \\ \end{aligned}$$

(A.6)

It is easy to see that $I(V^{\top } W\le u)$ is monotone with respect to u. By Lemma 9.10 of Kosorok (2008), the function class $\{I(V^{\top } W\le u): u\in {\mathbb {R}}^1\}$ is a VC-class. Similarly the function class $\{\Gamma (u): u\in {\mathbb {R}}^1\}$ is a VC-class as well. By Theorem 2.6.8 of van der Vaart and Wellner (1996), the function classes $\{\varepsilon I(V^{\top } W\le u): u\in {\mathbb {R}}^1\}$ and the class $\{\Gamma (u)\frac{G'\{X^{\top }\varvec{\beta }_0+m_0(Z)\}}{\sigma (G\{X^{\top }\varvec{\beta }_0 +m_0(Z)\})}\widetilde{X}: u\in {\mathbb {R}}^1\}$ are all VC-class. Then, by Lemma 9.17 of Kosorok (2008), the function class $\{\Psi _{u}({u},{y},\varepsilon ,w): u\in {\mathbb {R}}^1\}$ is a VC-class. By Theorem 2.6.7 and Theorem 2.5.2 of van der Vaart and Wellner (1996), we can prove that the estimated empirical process $M_{n}(u, W)$ converges weakly to M(u) in the Skorokhod space $S[\Pi ]$. By the continuous mapping theorem, we prove the result for $T_{n}$. $\square$

Proof of Theorem 2

Under the local alternatives (8), we have

$$\begin{aligned} \widehat{\varvec{\beta }}_n-\varvec{\beta }_0= & {} \{E(S_{1,2} \widetilde{X}_1 \widetilde{X}_1^{\top })\}^{-1} \frac{1}{n} \sum \limits ^n_{i=1} S_{i,1} \widetilde{X}_i \varepsilon _i\\{} & {} + \{E(S_{1,2} \widetilde{X}_1\widetilde{X}_1^{\top })\}^{-1} E\{S_{1,2}\widetilde{X}_1\widetilde{D}(V_1)\}+o_p (n^{-1/2}). \end{aligned}$$

Along the line to prove Theorem 1, we can validate that, under the alternatives (8),

$$\begin{aligned} M_{n}(u, W)= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n \left[ I(V_i^{\top } W\le u)-\Gamma (u)\frac{G'\{X_i^{\top }\varvec{\beta }_0+_0m(Z_i)\}}{\sigma (G\{X_i^{\top }\varvec{\beta }_0+m_0(Z_i)\})} \widetilde{X}_i\right] \varepsilon _i\\{} & {} +r_n n^{1/2}E\left[ \frac{G'\{X^{\top }\varvec{\beta }_0+m_0(Z)\}}{\sigma (G\{X^{\top }\varvec{\beta }_0+m_0(Z)\})} \widetilde{D}(V_1)I(V_1^{\top } W\le u)\right] +o_p(1). \end{aligned}$$

When $r_nn^{1/2}\rightarrow \infty$, the second term tends to infinity. So the first assertion holds. When $r_nn^{1/2}\rightarrow C_r$, we have

$$\begin{aligned} T_{n}{\longrightarrow }\int \left( M(u)+C_r E\left[ \frac{G'\{X^{\top }\varvec{\beta }_0+m_0(Z)\}}{\sigma (G\{X^{\top }\varvec{\beta }_0+m_0(Z)\})} \widetilde{D}(V_1)I(V_1^{\top } W\le u)\right] \right) ^2F(du). \end{aligned}$$

$\square$

Proof of Theorem 3

Let $\widehat{\varvec{\beta }}_n^*$ and $\widehat{m}_n^*(\cdot )$ be the regression spline-based estimators of $\varvec{\beta }$ and $m(\cdot )$ based on the bootstrap samples $\{(V_i, Y_i^*), i=1, \cdots , n\}$, where $Y^*_i$ has the success probability $p_i$. Analogously to establish (A.1) and (A.2), we can prove that

$$\begin{aligned} \widehat{\varvec{\beta }}_n^*-\widehat{\varvec{\beta }}_n= \sum _{i=1}^n \frac{G'\{X_i^{\top }\widehat{\varvec{\beta }}_n +\widehat{m}_n(Z_i)\}}{\sigma (G\{X_i^{\top }\widehat{\varvec{\beta }}_n +\widehat{m}_n(Z_i)\})} \widetilde{X}_i[Y_i^*-G\{X_i^{\top }\widehat{\varvec{\beta }}_n^*+\widehat{m}_n^*(Z_i)\}. \end{aligned}$$

(A.7)

$$\begin{aligned} \widehat{m}_n^*(Z_i) - \widehat{m}_n(Z_i) = \{E(S_{i,2} |Z_i)\}^{-1} E(S_{i,2} X_i^{\top } |Z_i)(\widehat{\varvec{\beta }}_n^* -\widehat{\varvec{\beta }}_n)+o_p(\alpha _n). \end{aligned}$$

(A.8)

Write the bootstrap version of $M_{n}(u, W)$ as $M^*_{n}(u, W)=1/\sqrt{n}\sum _{i=1}^n [Y^*_i-G\{X_i^{\top }\widehat{\varvec{\beta }}_n^*+\widehat{m}_n^*(Z_i)\}]I(V_i^{\top } W\le u)$. Note that $Y^*_i-G\{X_i^{\top }\widehat{\varvec{\beta }}_n^*+\widehat{m}_n^*(Z_i)\} =[Y_i^*-G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}] -[G\{X_i^{\top }\widehat{\varvec{\beta }}_n^*+\widehat{m}_n^*(Z_i)\}-G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}].$ Then, we have

$$\begin{aligned} M^*_{n}(u, W)= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\Big ([Y_i^*-G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}] \end{aligned}$$

(A.9)

$$\begin{aligned}{} & {} -[G\{X_i^{\top }\widehat{\varvec{\beta }}_n^*+\widehat{m}_n^*(Z_i)\}-G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}]\Big ) I(V_i^{\top } W\le u)\nonumber \\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n [Y_i^*-G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}] I(V_i^{\top } W\le u)\nonumber \\{} & {} -\frac{1}{\sqrt{n}}\sum _{i=1}^n [G\{X_i^{\top }\widehat{\varvec{\beta }}_n^*+\widehat{m}_n^*(Z_i)\}-G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}]I(V_i^{\top } W\le u) \nonumber \\:= & {} M^*_{n1}(u, W)-M^*_{n2}(u, W). \end{aligned}$$

(A.10)

Applying (A.7) along with the similar proof to that for (A.5) yields that

$$\begin{aligned} M^*_{n2}(u, W)= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n[G\{X_i^{\top }\widehat{\varvec{\beta }}_n^*+\widehat{m}_n^*(Z_i)\}-G\{X_i^{\top }\widehat{\varvec{\beta }}_n +\widehat{m}_n(Z_i)\}]I(V_i^{\top } W\le u)\nonumber \\= & {} \frac{1}{\sqrt{n}}\textrm{E}\left\{ \frac{G'\{X_i^{\top }\widehat{\varvec{\beta }}_n +\widehat{m}_n(Z_i)\}}{\sigma (G\{X_i^{\top }\widehat{\varvec{\beta }}_n +\widehat{m}_n(Z_i)\})}\widetilde{X}_i I(V^{\top } W\le u)\right\} \nonumber \\{} & {} \sum _{i=1}^{n} [Y_i^*-G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}]+o_p(1). \end{aligned}$$

(A.11)

It follows from (A.9)–(A.11) that

$$\begin{aligned} M^*_{n}(u, W)= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n [Y_i^*-G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}] [I(V_i^{\top } W\le u)\nonumber \\{} & {} -\frac{1}{\sqrt{n}}\Gamma (u) \sum _{i=1}^{n}\frac{G'\{X_i^{\top }\widehat{\varvec{\beta }}_n +\widehat{m}_n(Z_i)\}}{\sigma (G\{X_i^{\top }\widehat{\varvec{\beta }}_n +\widehat{m}_n(Z_i)\})}\nonumber \\{} & {} \quad \widetilde{X}_i[Y_i^*-G\{X_i^{\top }\widehat{\varvec{\beta }}_n+\widehat{m}_n(Z_i)\}]+o_p(1). \end{aligned}$$

(A.12)

Note that $E(Y_i^*|\textrm{data})=G\{X_i^{\top }\widehat{\varvec{\beta }}_n +\widehat{m}_n(Z_i)\}.$ The similar arguments to the proof of Theorem 1 along the line with the proof of Theorem 2 in Dikta et al. (2006) can prove that the conditional distribution of $T^*_{n}$ converges in distribution to the limiting null distribution of $T_{n}$.

Note that the validity of (A.7) is independent of $D(V)=0$. We can similarly prove that the conditional distribution of $T^*_{n}$ converges in distribution to the limiting alternative distribution of $T_{n}$. Theorem 3 follows. $\square$