Nonparametric estimation in generalized varying-coefficient models based on iterative weighted quasi-likelihood method

Abstract

This paper focuses on the estimation of the coefficient functions, which is of primary interest, in generalized varying-coefficient models with non-exponential family error. The local weighted quasi-likelihood method which results from local polynomial regression techniques is presented. The nonparametric estimator based on iterative weighted quasi-likelihood method is obtained to estimate coefficient functions. The asymptotic efficiency of the proposed estimator is given. Furthermore, some simulations are carried out to evaluate the finite sample performance of the proposed method, which show that it possesses some advantages to the previous methods. Finally, a real data example is used to illustrate the proposed methodology.

Appendix

1.1 Proof of the Theorem 1

The expression (17) can be obtained easily according to the expression (16) under some assumptions. For brevity, we outline only the proof of the expression (16). For each given point $u$, the conditions $\mathbf C1$–$\mathbf C6$ proposed in the Sect. 3 are needed. For simplicity, we recall that the vector $\hat{\xi }(u_0)$ maximizes (9). Here, we consider the normalized estimator

$$\begin{aligned} \hat{\xi }^{*}(u_0)= & {} \psi ^{-1}[\hat{\varsigma }_1(u_0)-b_1(u_0), \ldots , \hat{\varsigma }_p(u_0)-b_p(u_0), \\&\ldots , h^m\left\{ \hat{\varsigma }_1(u_0)-b_1(u_0)\right\} , \ldots , h^m\left\{ \hat{\varsigma }_p(u_0)-b_p(u_0)\right\} ]^T \end{aligned}$$

where $\psi =(nh)^{-1/2}$. Let $\bar{\eta }(u_0, u, \mathbf x )=\sum \nolimits _{j=1}^p{\left[ b_j(u_0)+ \cdots +b_j^{(m)}(u_0)(u-u_0)\right] x_{j}}$ and $z_i=\left\{ \mathbf X _i^T, \frac{U_i-u_0}{h}\mathbf X _i^T, \ldots , \frac{(U_i-u_0)^m}{h^m}\mathbf X _i^T \right\}$. It can easily be seen that $\hat{\xi }^{*}(u_0)$ maximizes

$$\begin{aligned} \sum \limits _{i=1}^n{\left( \int _{Y_i}^{\mu _i}\frac{Y_i-t}{V(t)}{\hbox {d}}t\right) K_h(U_i-u_0)}, \end{aligned}$$

where $\mu _i=g^{-1}\left( \bar{\eta }(u_0,u,\mathbf x )+\psi \xi ^{*T}(u_0)Z_i\right)$. Equivalently, $\hat{\xi }^{*}(u_0)$ maximizes

$$\begin{aligned} \ln \left( \hat{\xi }^{*}(u_0)\right) =\sum \limits _{i=1}^n{\left[ \int _{Y_i}^{\mu _i} \frac{Y_i-t}{V(t)}{\hbox {d}}t-\int _{Y_i}^{\bar{\mu }_i}\frac{Y_i-t}{V(t)}{\hbox {d}}t\right] K_h(U_i-u_0)}, \end{aligned}$$

where $\bar{\mu }_i=\bar{\eta }(u_0, u, \mathbf x ))$. Condition $\mathbf C5$ implies that the function $\ln (\cdot )$ is concave in $\hat{\xi }^{*}(u_0)$. We have the following expression via a Taylor’s expansion:

$$\begin{aligned} \ln (\hat{\xi }^{*}(u_0))= & {} \psi \sum \limits _{i=1}^n\left\{ q_1\{\bar{\eta }_i(u_0),Y_i\}\xi ^{*T}(u_0) \Gamma _i(u_0) K\left( (U_i-u_0)/h\right) \right\} \\&+\frac{\psi ^2}{2}\sum \limits _{i=1}^n\left\{ q_2\{\bar{\eta }_i(u_0),Y_i\}\left[ \xi ^{*T}(u_0)\Gamma _i(u_0)\right] ^2K\left( (U_i-u_0)/h\right) \right\} \\&+ \frac{\psi ^3}{6}\psi \sum \limits _{i=1}^n\left\{ q_3\{\eta _i(u_0),Y_i\} \left[ \xi ^{*T}(u_0)\Gamma _i(u_0)\right] ^3 K\left( (U_i-u_0)/h\right) \right\} \\= & {} \Xi _n^T \xi ^{*}(u_0)+ \frac{1}{2}\xi ^{*T}(u_0)\Delta _n \xi ^{*}(u_0) + \Omega _n, \end{aligned}$$

where $\bar{\eta }(u_0)=\bar{\eta }(u_0, u, \mathbf x ),\, \eta _i$ is between $\bar{\eta }(u_0)$ and $\bar{\eta }(u_0)+\psi \xi ^{*T}(u_0)\Gamma _i(u_0)$, and

$$\begin{aligned} \Xi _n= & {} \psi \sum \limits _{i=1}^n\left\{ q_1\{\bar{\eta }_i(u_0),Y_i\}\Gamma _i(u_0) K\left( (U_i-u_0)/h\right) \right\} ,\\ \Delta _n= & {} \frac{\psi ^2}{2}\sum \limits _{i=1}^n\left\{ q_2\{\bar{\eta }_i(u_0),Y_i\}\Gamma _i(u_0) \Gamma _i^T(u_0) K\left( (U_i-u_0)/h\right) \right\} , \end{aligned}$$

and

$$\begin{aligned} \Omega _n=\frac{\psi ^3}{6}\sum \limits _{i=1}^n\left\{ q_3\{\eta _i(u_0),Y_i\}(\xi ^{*T}(u_0)\Gamma _i(u_0))^3 K\left( (U_i-u_0)/h\right) \right\} . \end{aligned}$$

Note the fact that $(\Delta _n)_{ij}=(E \Delta _n)_{ij}+ O_p\left\{ \left[ \text {Var}(\Delta _n)_{ij}\right] ^{\frac{1}{2}}\right\}$, and take a Taylor’s expansion of $\eta (u, \mathbf x )$ with respect to $u$ around $|u-u_0|<h$,

$$\begin{aligned} \eta (u, \mathbf x )=\bar{\eta }(u_0, u, \mathbf x )+ \frac{(u-u_0)^{m+1}}{(m+1)!}\eta ^{(m+1)}(u_0, \mathbf x )+o\left( h^{p+1}\right) , \end{aligned}$$

and

$$\begin{aligned} \eta (u, \mathbf x )=-\rho (u, \mathbf x )+o(1). \end{aligned}$$

Therefore, the expect of $\Delta _n$

$$\begin{aligned} E(\Delta _n)= & {} h^{-1}E\left[ q_2\{\eta (u_0), Y\}\Gamma (u_0)\Gamma ^T(u_0) K \left( \frac{U-u_0}{h}\right) \right] \\= & {} h^{-1}E\left[ \{-\rho (u, \mathbf x )+o(1)\}\Gamma (u_0) \Gamma ^T(u_0) K\left( \frac{U-u_0}{h}\right) \right] \\= & {} h^{-1}E\left[ -\rho (u, x)\mathbf X \mathbf X ^T \otimes \left( \begin{array}{ccc} 1 &{} \cdots &{} \frac{(U-u_0)^m}{h^m} \\ \vdots &{} &{} \vdots \\ \frac{(U-u_0)^m}{h^m} &{} \cdots &{} \frac{(U-u_0)^{2m}}{h^{2m}} \\ \end{array} \right) K\left( \frac{U-u_0}{h}\right) \right] +o(1) \\= & {} -f_U(u_0)\Theta \otimes \Pi (u_0)+ o(1) \\\rightarrow & {} -\Delta \end{aligned}$$

where $\Pi (u_0)=E(\rho (u, \mathbf x )\mathbf X \mathbf X ^T|U=u)$. Besides, the element of the variance term can be calculated that $\text {Var}(\Delta _n)_{ij}=E[(\Delta _{nij}-E\Delta _{nij})(\Delta _{nij} -E\Delta _{nij})^T]=O(\psi ^2)$. Therefore,

$$\begin{aligned} \Delta _n=-\Delta +o(1). \end{aligned}$$

(22)

Next, we will compute the expected value of the absolute of $\Omega _n$,

$$\begin{aligned} E(\Omega _n)= & {} E\left[ |\psi ^3\sum _{i=1}^n q_3\{\eta _i(u_0), Y_i\} (\xi ^{*T}(u_0)\Gamma _i(u_0))^3K\left( \frac{U_i-u_0}{h}\right) |\right] \\= & {} h^{-1}\psi E\left[ |q_3\{\eta _1(u_0), Y_1\}\mathbf X _1^3 K\left( \frac{U_i-u_0}{h}\right) |\right] \\= & {} \psi \theta _1 E\left[ |q_3\{\eta _1(u_0), Y_1\}\mathbf X _1^3|\right] . \end{aligned}$$

since $q_3$ is linear in $Y$ with $E(Y_1|(\mathbf X _1,U_1))<\infty$, we have $E\left[ |q_3\{\eta _1(u_0), Y_1\}\mathbf X _1^3|\right] <\infty$, therefore, $E(\Omega _n)=O(\psi )$. Combining the above equations leads to

$$\begin{aligned} \Xi _n^T \xi ^{*}(u_0)+ \frac{1}{2}\xi ^{*T}(u_0)\Delta \xi ^{*}(u_0) + o(1). \end{aligned}$$

By the quadratic approximation lemma (Fan and Gijbels 1996), we have

$$\begin{aligned} \xi ^{*}(u_0)=\Delta ^{-1} \Xi _n + o(1). \end{aligned}$$

(23)

If $\Xi _n$ is a stochastically bounded sequence of random vectors. The asymptotic normality of $\xi ^{*}(u_0)$ follows from that of $\Xi _n$. So, we need to establish the asymptotic normality of $\Xi _n$. To establish its asymptotic normality, the mean and covariance need to be computed. The mean

$$\begin{aligned} E(\Xi _n)= & {} n\psi E\left[ q_1\{\bar{\eta }(u_0), Y\}\Gamma (u_0) K(\frac{U_i-u_0}{h})\right] \nonumber \\= & {} n \psi E\left[ \rho (u, x)\mathbf X \mathbf X ^T \frac{{\mathbf {b}}^{(m+1)}(u_0)(U-u_0)^{m+1}}{(m+1)!} K_h\left( \frac{U_i-u_0}{h}\right) +o(h^{m+1})\right] \nonumber \\= & {} \psi {f_U(u_0)h^{m+1}}\Theta ^{-1}\varvec{\theta } \otimes \Pi (u_0) \frac{{\mathbf {b}}^{(m+1)}(u_0)}{(m+1)!}\left\{ 1+o(1)\right\} , \end{aligned}$$

(24)

where $U_h=\left( 1, \frac{U-u_0}{h}, \ldots , \left( \frac{U-u_0}{h}\right) ^m\right) ^T,\, \varvec{\theta }=(\theta _{m+1}, \theta _{m+2}, \ldots , \theta _{2m+1})^T$, and ${\mathbf {b}}^{(m+1)}(u_0)=\left[ b^{(m+1)}_1(u_0), \ldots , b^{(m+1)}_p(u_0)\right] ^T$. An application of $E(\Xi _n)$ calculated above and the definition of $q_1$, one obtains that

$$\begin{aligned} \text {Var}(\Xi _n)= & {} n\psi ^2 \text {Var}\left[ q_1\left\{ \bar{\eta }(u_0), Y\right\} \Gamma (u_0) K\left( \frac{U_i-u_0}{h}\right) \right] \nonumber \\= & {} h^{-1} \left[ E q^2_1\left\{ \bar{\eta }(u_0), Y\right\} \Gamma (u_0) \Gamma ^T(u_0) K^2\left( \frac{U_i-u_0}{h}\right) +o(h^{m+1})\right] \nonumber \\= & {} f_U(u_0)\Theta ^{*}\otimes \Lambda (u_0) \nonumber \\\equiv & {} \mathbf B +o(1) \end{aligned}$$

(25)

where $\Lambda (u_0)=\frac{\text {Var}(Y|U=u,\mathbf X =\mathbf x )}{\rho (u, \mathbf x )}$. In order to prove that

$$\begin{aligned} \left\{ \text {Var}(\Xi _n)\right\} ^{-1/2}(\Xi _n-E \Xi _n)\xrightarrow {D} N(0, \mathbf I _{p+1}), \end{aligned}$$

(26)

we now employ Cramér-Wold device to derive the asymptotic normality of $\text {Var}(\Xi _n)$: for any unit vector $\mathbf {e}$,

$$\begin{aligned} \left\{ \mathbf{e}^T \text {Var}(\Xi _n)\mathbf{e}\right\} ^{-1/2}(\mathbf {e}^T \Xi _n-E \mathbf {e}^T\Xi _n)\xrightarrow {D} N(0, 1), \end{aligned}$$

(27)

Combining (23), (24), (25) and (26), one has

$$\begin{aligned} \hat{\xi }^{*}(u_0)-\psi ^{-1}h^{m+1}f_U(u_0)\frac{{\mathbf {b}}^{m+1}(u_0)}{(m+1)!} \Delta ^{-1}\varvec{\theta }\otimes \Pi (u_0)\left\{ 1+o(1)\right\} \xrightarrow {D} N\left( 0, \Delta ^{-1}\mathbf B \Delta ^{-1}\right) . \end{aligned}$$

(28)

Therefore, the Theorem (16) holds true. Besides, it is easy to verify the Lyapounov’s condition for that sequence, that is formula (27) can easily be proved. If $m=1$ and $K(\cdot )$ is symmetric, then $\theta _1=0$, so that (17) holds true. The proof is completed.