Semiparametric quantile regression imputation for a complex survey with application to the Conservation Effects Assessment Project
Section 3. Asymptotic distributions and variance estimation

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We derive an asymptotic normal distribution for the QRI estimator $\widehat{\theta }$ defined in (2.16), although the estimator ${\widehat{\theta }}_J,$ defined in (2.19), with a finite number of $\left(J\right)$ imputations is necessary in practice. This approach of developing theory under an assumption of an infinite number of imputed values has been used previously. See, for example, Clayton, Spiegelhalter, Dunn and Pickles (1998) and Robins and Wang (2000). The simulations in Section 5 demonstrate that the asymptotic normal distribution derived for $J\mathrm{<mo>=</mo>}\infty$ is a reasonable approximation for the distribution of the estimator constructed with finite $J.$ We outline the main concepts underlying the proofs of lemma 1, lemma 2, and Theorem 1, deferring details to Section B of the online supplement https://github.com/emilyjb/Semiparametric-QRI-Supplement/blob/master/SupplementToQRI.pdf, (Berg and Yu, 2016).

The derivation of the asymptotic distribution of $\widehat{\theta }$ proceeds in three main steps. Lemma 1 gives the asymptotic distribution of the estimators of the quantile regression coefficients. Lemma 2 presents the asymptotic distribution of the estimating equation (2.17). These two lemmas are analogous to lemma 1 and lemma 2 of Chen and Yu (2016). Theorem 1 then provides the asymptotic distribution of $\widehat{\theta }.$

3.1  Asymptotic normality of $\widehat{\theta }$

We consider a sequence of samples and finite populations indexed by $N,$ where the sample size $n\to \infty$ as $N\to \infty .$ To define the regularity conditions, we introduce the notation ${\mathcal{F}}_N$ to represent an element of the sequence of finite populations with size $N$ and use the notation “ $|{\mathcal{F}}_N$ ” to indicate that the reference distribution is the distribution based on repeated sampling conditional on the finite population of size $N$ . For example, $E\left[\widehat{Y}|{\mathcal{F}}_N\right]$ and $V\left\{\widehat{Y}|{\mathcal{F}}_N\right\},$ respectively, denote the conditional expectation and variance of the outcome $\widehat{Y}$ with respect to the randomization distribution generated from repeated sampling from ${\mathcal{F}}_N.$ Similarly, $\widehat{Y}\stackrel{d}{{\displaystyle \to }}Y|{\mathcal{F}}_N$ a.s., means that $\widehat{Y}$ converges in distribution to $Y$ almost surely with respect to the process of repeated sampling from the sequence of finite populations as $N\to \infty .$ The convergence is with probability 1 because ${\mathcal{F}}_N$ is a random realization from the superpopulation model (2.1).

The regularity conditions on the sample design and tuning parameters for the estimator of the B-spline model are as follows:

1. Any variable $v_i$ such that $E\left[{\left|v_i\right|}^{2+\delta }\right]\mathrm{<mo><</mo>}\infty ,$ where $\delta \mathrm{<mo>></mo>\; 0},$ satisfies,

$$\sqrt{n}\left({\overline{v}}_{\text{HT}}-{\overline{v}}_N\right)|{\mathcal{F}}_N\stackrel{d}{{\displaystyle \to }}N\left(\mathrm{0,}{V}_{\infty }\right)\text{a}\text{.s}\mathrm{.,}(3.1)$$

• where $\left({\overline{v}}_{\text{HT}}\mathrm{,}{\overline{v}}_N\right)\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N}\left({\pi }_i^{-1}{v}_i{I}_i\mathrm{,}{v}_i\right),$ $V_{\infty }\mathrm{<mo>=</mo>}{{\displaystyle \lim }}_{N\to \infty }{V}_N,$ and $V_N\mathrm{<mo>=</mo>}nV\left\{{\overline{v}}_{\text{HT}}|{\mathcal{F}}_N\right\}$ is the conditional variance of the Horvitz-Thompson mean, ${\overline{v}}_{\text{HT}},$ given ${\mathcal{F}}_N.$

2. $n{n}_B^{-1}\to 1$ and $n_B{N}^{-1}\to f_{\infty }\in \left[\mathrm{0,}1\right]\mathrm{,}$ where $n_B$ is the expected sample size.
3. There exist constants $C_1,$ $C_2,$ and $C_3$ such that $\mathrm{0\; <mo><</mo>}{C}_1\le n_B{N}^{-1}{\pi }_i^{-1}\le C_2\mathrm{<mo><</mo>}\infty ,$ and

$$\left|n_B\left({\pi }_{ij}-{\pi }_i{\pi }_j\right){\pi }_i^{-1}{\pi }_j^{-1}\right|\le C_3\mathrm{<mo><</mo>}\infty \text{a}\text{.s}\text{.}(3.2)$$

4. The value determining the number of interior knots $K_n=O\left(n_B^{\frac{1}{2p+3}}\right).$
5. ${\lambda }_n\mathrm{<mo>=</mo>}O\left(n_B^{\nu }\right)$ for $\nu \le {\left(2p+3\right)}^{-1}\left(p+m+1\right).$

Condition 3 is also used in Fuller (2009a). Condition 3 holds for simple random sampling, where $\left({\pi }_{ij}-{\pi }_i{\pi }_j\right){\pi }_i^{-1}{\pi }_j^{-1}\mathrm{<mo>=</mo>}{n}^{-1}\left(n-1\right){\left(N-1\right)}^{-1}N-1,$ and for Poisson sampling, where $\left({\pi }_{ij}-{\pi }_i{\pi }_j\right){\pi }_i^{-1}{\pi }_j^{-1}\mathrm{<mo>=</mo>\; 0.}$ Fuller (2009a) explains that condition 3 holds for many stratified designs and that the designer has the control to ensure condition 3.

Under assumptions 4-5, Barrow and Smith (1978) show that a ${\beta }_{\tau }^{*}$ exists that satisfies,

$${\text{sup}}_{x\in \left[M_1\text{}\mathrm{,}{M}_2\right]}\left|q_{\tau }\left(x\right)-b_{\tau }^a\left(x\right)-B{\left(x\right)}^{\prime }{\beta }_{\tau }^{*}\right|\mathrm{<mo>=</mo>}o\left(K_n^{-\left(p+1\right)}\right)\mathrm{,}(3.3)$$

where $B{\left(x\right)}^{\prime }{\beta }_{\tau }^{*}$ is the best $L_{\infty }$ approximation for $q_{\tau }\left(x\right),$ and $b_{\tau }^{\left(a\right)}\left(x\right)$ is a bias of the B-spline approximation for the true quantile function, satisfying, $b_{\tau }^a\left(x\right)\mathrm{<mo>=</mo>}O\left(K_n^{-\left(p+1\right)}\right).$ For details of the form of the bias term, see Chen and Yu (2016) and Yoshida (2013). The property (3.3) is used extensively in the derivation of lemma 1.

The proofs of both lemma 1 and lemma 2 use a result given in Theorem 1.3.6 of Fuller (2009b). Because of the importance of this theorem to the results of this section, we state this theorem as Fact 1:

Fact 1. (Theorem 1.3.6 of Fuller (2009b)): Suppose

$$\left(\widehat{\theta }-{\theta }_N\right)|{\mathcal{F}}_N\stackrel{d}{{\displaystyle \to }}N\left(\mathrm{0,}{V}_{11}\right)\text{a}\text{.s}\mathrm{.,}\text{and}{\theta }_N-{\theta }_o\stackrel{d}{{\displaystyle \to }}N\left(\mathrm{0,}{V}_{22}\right)\mathrm{.}(3.4)$$

Then, $\left(\widehat{\theta }-{\theta }_o\right)\stackrel{d}{{\displaystyle \to }}N\left(\mathrm{0,}{V}_{11}+V_{22}\right)\mathrm{.}$

Note that $V_{11}$ in Fact 1 is a fixed limit and not a design variance because the design variance is a random function of the finite population in this framework. The condition $\left(\widehat{\theta }-{\theta }_N\right)|{\mathcal{F}}_N\stackrel{d}{{\displaystyle \to }}N\left(\mathrm{0,}{V}_{11}\right)\text{a}\text{.s}\text{.,}$ holds for a broad class of designs, such as those discussed in Isaki and Fuller (1982).

Lemma 1. Under assumptions 1-5 and for fixed $x_i\in \left[M_1\mathrm{,}{M}_2\right]$ and $\tau \in \left(\mathrm{0,}1\right),$

$$\sqrt{\frac{n}{K_n}}\left({\widehat{q}}_{\tau }\left(x_i\right)-B{\left(x_i\right)}^{\prime }{\beta }_{\tau }^{*}+b_{\tau }^{\lambda }\left(x_i\right)\right)\stackrel{d}{{\displaystyle \to }}N\left(\mathrm{0,}B{\left(x_i\right)}^{\prime }{\Sigma }_{\infty }\left(\tau \right)B\left(x_i\right)\right)\mathrm{,}(3.5)$$

and

$$\sqrt{\frac{n}{K_n}}\left({\widehat{q}}_{\tau }\left(x_i\right)-q_{\tau }\left(x_i\right)+b_{\tau }^a\left(x_i\right)+b_{\tau }^{\lambda }\left(x_i\right)\right)\stackrel{d}{{\displaystyle \to }}N\left(\mathrm{0,}B{\left(x_i\right)}^{\prime }{\Sigma }_{\infty }\left(\tau \right)B\left(x_i\right)\right)\mathrm{,}(3.6)$$

where

$$\begin{array}{ll}{b}_{\tau }^{\lambda }\left(x_i\right)\hfill & \mathrm{<mo>=</mo>}\underset{N\to \infty }{{\displaystyle \lim }}\frac{{\tilde{\lambda }}_n}{n}B{\left(x_i\right)}^{\prime }{\Omega }_n{\left(\tau \right)}^{-1}{D}_m^{\prime }{D}_m{\beta }_{\tau }^{*}\mathrm{,}(3.7)\hfill \\ {\Omega }_n\left(\tau \right)\hfill & \mathrm{<mo>=</mo>}H\left(\tau \right)+\frac{{\tilde{\lambda }}_n}{n}{D}_m^{\prime }{D}_m\mathrm{,}\hfill \\ {\Sigma }_{\infty }\left(\tau \right)\hfill & \mathrm{<mo>=</mo>}\underset{N\to \infty }{{\displaystyle \lim }}\frac{1}{K_n}{\Omega }_n{\left(\tau \right)}^{-1}\left(V_{\mathrm{1,}\infty }\left(\tau \right)+f_{\infty }\tau \left(1-\tau \right)\Phi \right){\Omega }_n{\left(\tau \right)}^{-1}\mathrm{,}\hfill \\ H\left(\tau \right)\hfill & \mathrm{<mo>=</mo>}E\left[p_{i}B\left(x_i\right)f_{y|x\text{},i}\left(q_{\tau i}\right)B{\left(x_i\right)}^{\prime }\right]\mathrm{,}\hfill \\ \Phi \hfill & \mathrm{<mo>=</mo>}E\left[p_{i}B\left(x_i\right)B{\left(x_i\right)}^{\prime }\right]\mathrm{,}\hfill \\ V_{\mathrm{1,}\infty }\left(\tau \right)\hfill & \mathrm{<mo>=</mo>}\underset{N\to \infty }{{\displaystyle \lim }}\frac{n}{N^2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^N}\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}{\delta }_i{\delta }_{j}B\left(x_i\right){\psi }_{\tau }\left(u_{i\tau }\right)B{\left(x_j\right)}^{\prime }{\psi }_{\tau }\left(u_{j\tau }\right)\mathrm{,}\hfill \end{array}$$

$u_{i\tau }=y_i-B{\left(x_i\right)}^{\prime }{{\displaystyle \beta }}_{\mathrm{<mtext>\hspace{0.17em}</mtext>\tau }}^{<mtext>\hspace{0.17em}</mtext>*}<mtext>\; </mtext>,$ ${\psi }_{\tau }\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{<mo\; stretchy="false">)</mo>\; <mo>=</mo>}\tau -I\left[u\mathrm{<mo><</mo>\; 0}\right],$ ${\tilde{\lambda }}_n\mathrm{<mo>=</mo>}n\widehat{N}{N}^{-1}{\lambda }_n,$ $\widehat{N}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^n}{\pi }_i^{-1}\text{},$ and $f_{y|x\text{},i}\left(q\right)$ is the pdf of $y_i$ given $x_i$ evaluated at $q.$

The main idea of the proof of lemma 1 is to show that the estimator of the quantile regression coefficient has a Bahadur representation given in corollary 1 below:

Corollary 1: By the proof of lemma 1, the estimator of the quantile regression coefficient has the following Bahadur representation:

$$\sqrt{\frac{n}{K_n}}\left({\widehat{\beta }}_{\tau }-{\beta }_{\tau }^{*}+\frac{{\tilde{\lambda }}_n}{n}{\Omega }_n{\left(\tau \right)}^{-1}{D}_m^{\prime }{D}_m{\beta }_{\tau }^{*}\right)\mathrm{<mo>=</mo>}\sqrt{\frac{n}{K_n}}{\Omega }_n{\left(\tau \right)}^{-1}\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n}{\pi }_i^{-1}{\delta }_{i}B\left(x_i\right){\psi }_{\tau }\left(u_{i\tau }\right)+o_p\left(1\right)\mathrm{.}(3.8)$$

The derivation of the Bahadur representation follows the basic approach of Koenker (2005) and Yoshida (2013). To account for the complex sample design, condition (3.2) is used to bound sums of covariances induced by nontrivial second order inclusion probabilities. For independent random variables from an infinite population (as in Chen and Yu (2016), Yoshida (2013) and Koenker (2005)), the corresponding covariances are zero. Given the Bahadur representation (3.8), lemma 1 follows from an application of the regularity condition in (3.1) and Fact 1 to the elements of the Horvitz-Thompson mean in (3.8). The $V_{\mathrm{1,}\infty }$ in ${\Sigma }_{\infty }\left(\tau \right)$ essentially plays the role of $V_{11}$ in Fact 1 and is the limit of the design variance of the Horvitz-Thompson mean. The second term in ${\Sigma }_{\infty }\left(\tau \right)$ is the asymptotic variance of the design-expectation of the Horvitz-Thompson mean and plays the role of $V_{22}$ in Fact 1.

Lemma 2 and Theorem 1 require additional regularity conditions about the estimating equation. The regularity conditions on the estimation are similar to those in Chen and Yu (2016) and are therefore deferred to Section A of the online supplement https://github.com/emilyjb/Semiparametric-QRI-Supplement/blob/master/SupplementToQRI.pdf, (Berg and Yu, 2016).

Lemma 2. Under the assumptions of lemma 1 and the regularity conditions provided in Section A of the online supplement https://github.com/emilyjb/Semiparametric-QRI-Supplement/blob/master/SupplementToQRI.pdf, (Berg and Yu, 2016),

$$\sqrt{n}{G}_n\left({\theta }_o\right)\stackrel{d}{{\displaystyle \to }}N\left(0\mathrm{,}{V}_G\left({\theta }_o\right)\right)\mathrm{,}(3.9)$$

where

$$\begin{array}{ll}{V}_G\left({\theta }_o\right)\hfill & =f_{\infty }V\left\{{\xi }_i\left({\theta }_o\right)\right\}+\underset{N\to \infty }{{\displaystyle \lim }}{V}_{\xi \mathrm{,}N}\left({\theta }_o\right)\mathrm{,}(3.10)\hfill \\ V_{\xi \mathrm{,}N}\hfill & =n{N}^{-2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^N}\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}{\xi }_i\left({\theta }_o\right){\xi }_j{\left({\theta }_o\right)}^{\prime }\text{}\mathrm{,}\hfill \\ {\xi }_i\left({\theta }_o\right)\hfill & ={\delta }_i{g}_i\left(y_i\mathrm{<mo>;</mo>}{\theta }_o\right)+\left(1-{\delta }_i\right){\displaystyle {\int }_0^1}{g}_i\left(q_{\tau }\left(x_i\right)\mathrm{<mo>;</mo>}{\theta }_o\right)d\tau +{\delta }_i{h}_{ni}\left({\theta }_o\right)\mathrm{,}\hfill \\ h_{n_i}\left({\theta }_o\right)\hfill & ={\displaystyle {\int }_0^1}E\left[\left(1-p_j\right){\dot{g}}_{j\text{}\mathrm{,}y}\left(q_{\tau }\left(x_j\right)\mathrm{<mo>;</mo>}{\theta }_o\right)B{\left(x_j\right)}^{\prime }\right]{\Omega }_n{\left(\tau \right)}^{-1}B\left(x_i\right){\psi }_{\tau }\left(u_{i\tau }\right)d\tau \mathrm{,}\hfill \end{array}$$

and ${\dot{g}}_{i\text{}\mathrm{,}y}\left(y_i\mathrm{<mo>;</mo>}{\theta }_o\right)$ is the partial derivative of $g_i\left(a\mathrm{<mo>;</mo>}\theta \right)$ with respect to $a$ evaluated at $y_i.$

The proof of lemma 2 centers on the Taylor expansion given by

$$\begin{array}{ll}{g}_i\left({\widehat{q}}_{\tau }\left(x_i\right)\mathrm{<mo>;</mo>}{\theta }_o\right)\hfill & =g_i\left(q_{\tau }\left(x_i\right)\mathrm{<mo>;</mo>}{\theta }_o\right)+{\dot{g}}_{i\text{}\mathrm{,}y}\left(q_{\tau }\left(x_i\right)\mathrm{<mo>;</mo>}{\theta }_o\right)\left({\widehat{q}}_{\tau }\left(x_i\right)-q_{\tau }\left(x_i\right)\right)\hfill \\ \hfill & +{\ddot{g}}_{i\text{}\mathrm{,}y}\left(q_{\tau }\left(x_i\right)\mathrm{<mo>;</mo>}{\theta }_o\right){\left({\tilde{q}}_{\tau }\left(x_i\right)-q_{\tau }\left(x_i\right)\right)}^2\text{}\mathrm{,}(3.11)\hfill \end{array}$$

where ${\tilde{q}}_{\tau }\left(x_i\right)$ is between ${\widehat{q}}_{\tau }\left(x_i\right)$ and $q_{\tau }\left(x_i\right),$ and ${\ddot{g}}_{i\text{}\mathrm{,}y}\left(q_{\tau }\left(x_i\right)\mathrm{<mo>;</mo>}{\theta }_o\right)$ denotes the vector of partial derivatives of the elements of ${\dot{g}}_{i\text{}\mathrm{,}y}\left(a\mathrm{,}{\theta }_o\right)$ with respect to $a$ evaluated at $q_{\tau }\left(x_i\right).$ By arguments similar to those of Chen and Yu (2016), $n\Vert {\ddot{g}}_{i\text{}\mathrm{,}y}\left(q_{\tau }\left(x_i\right)\mathrm{<mo>;</mo>}{\theta }_o\right){\left({\tilde{q}}_{\tau }\left(x_i\right)-q_{\tau }\left(x_i\right)\right)}^2\Vert \mathrm{<mo>=</mo>}O\left(1\right).$ Lemma 2 then follows from the linear approximation for ${\widehat{q}}_{\tau }\left(x_i\right)-q_{\tau }\left(x_i\right)$ in lemma 1.

Theorem 1. Under the assumptions of lemmas 1 and 2, the QRI estimator $\widehat{\theta }$ defined in (2.16), constructed with $J\mathrm{<mo>=</mo>}\infty ,$ satisfies, $\sqrt{n}\left(\widehat{\theta }-{\theta }_o\right)\stackrel{d}{{\displaystyle \to }}N\left(0\mathrm{,}{\Sigma }_{\theta }\right)\mathrm{,}$ where

$${\Sigma }_{\theta }\mathrm{<mo>=</mo>}{\left[\Gamma {\left({\theta }_o\right)}^{\prime }\Gamma \left({\theta }_o\right)\right]}^{-1}\Gamma {\left({\theta }_o\right)}^{\prime }{V}_G\left({\theta }_o\right)\Gamma \left({\theta }_o\right){\left[\Gamma {\left({\theta }_o\right)}^{\prime }\Gamma \left({\theta }_o\right)\right]}^{-1}\text{}\mathrm{,}(3.12)$$

$G\left(\theta \right)\mathrm{<mo>=</mo>}E\left[G_N\left(\theta \mathrm{,}y\right)\right],$ $G_N\left(\theta \mathrm{,}y\right)\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N}{\delta }_{i}g\left(y_i\mathrm{,}{x}_i\right)\mathrm{,}$ and $\Gamma \left({\theta }_o\right)\mathrm{<mo>=</mo>}E\left[\partial /\partial \theta G_N\left(\theta \right)\right]\mathrm{.}$

By Pakes and Pollard (1989), Theorem 1 is satisfied if the following hold:

1. ${\text{sup}}_{\theta }\left|G_n\left(\theta \right)-G\left(\theta \right)\right|\mathrm{<mo>=</mo>}{o}_p\left(1\right),$
2. For ${\zeta }_n\to 0,$ ${\text{sup}}_{\left|\theta -{\theta }_o\right|\mathrm{<mo><</mo>}{\zeta }_n}\left|G_n\left(\theta \right)-G\left(\theta \right)-G_n\left({\theta }_o\right)\right|\mathrm{<mo>=</mo>}{o}_p\left(n_B^{-0.5}\right),$

where ${\zeta }_n$ is arbitrarily small. Because of the complex sample design, the proof that these conditions hold proceeds in two steps, considering first the deviation $\left|G_n\left(\theta \right)-G_N\left(\theta \right)\right|$ and then the deviation $\left|G_N\left(\theta \right)-G\left(\theta \right)\right|.$ The result then follows from the triangle inequality.

3.2  Variance estimation

We estimate the variance of ${\widehat{\theta }}_J$ using the linearization method (Fuller, 2009b, page 64). We use the asymptotic covariance matrix in (3.12) to estimate the variance of ${\widehat{\theta }}_J,$ the estimator of ${\theta }_o$ defined in (2.19), constructed with a finite number of imputed values. To estimate $V_G\left({\theta }_o\right),$ a design-consistent variance estimator is applied to an estimator of the mean of an estimator of ${\xi }_i\left({\theta }_o\right)$ defined in (3.10). The estimator of ${\xi }_i\left({\theta }_o\right)$ is obtained by replacing ${\theta }_o$ and ${\beta }_{\tau }^{*}$ with estimators ${\widehat{\theta }}_J$ and ${\widehat{\beta }}_{\tau },$ respectively.

The estimator of variance is defined,

$${\widehat{\Sigma }}_{\theta }\mathrm{<mo>=</mo>}{\left[\widehat{\Gamma }{\left({\widehat{\theta }}_J\right)}^{\prime }\widehat{\Gamma }\left({\widehat{\theta }}_J\right)\right]}^{-1}\widehat{\Gamma }{\left({\widehat{\theta }}_J\right)}^{\prime }\left[{\widehat{V}}_{G\text{}\mathrm{,}\infty }\left({\widehat{\theta }}_J\right)\right]\widehat{\Gamma }\left({\widehat{\theta }}_J\right){\left[\widehat{\Gamma }{\left({\widehat{\theta }}_J\right)}^{\prime }\widehat{\Gamma }\left({\widehat{\theta }}_J\right)\right]}^{-1}\text{}\mathrm{,}(3.13)$$

where ${\widehat{V}}_{G\text{}\mathrm{,}\infty }\left({\widehat{\theta }}_J\right)\mathrm{<mo>=</mo>}{\widehat{f}}_{\infty }\widehat{V}\left\{{\widehat{\xi }}_i\left({\widehat{\theta }}_J\right)\right\}+{\widehat{V}}_{\xi \text{}\mathrm{,}N}\left({\widehat{\theta }}_J\right)\mathrm{,}$

$$\begin{array}{ll}\widehat{V}\left\{{\widehat{\xi }}_i\left({\widehat{\theta }}_J\right)\right\}\hfill & =\frac{1}{\widehat{N}}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n}{\pi }_i^{-1}{\widehat{\xi }}_i\left({\widehat{\theta }}_J\right){\widehat{\xi }}_i{\left({\widehat{\theta }}_J\right)}^{\prime }-\frac{1}{\widehat{N}\left(\widehat{N}-1\right)}\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n}{\pi }_i^{-1}{\widehat{\xi }}_i\left({\widehat{\theta }}_J\right)\right){\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n}{\pi }_i^{-1}{\widehat{\xi }}_i\left({\widehat{\theta }}_J\right)\right)}^{\prime }\mathrm{,}(3.14)\hfill \\ {\widehat{V}}_{\xi \text{}\mathrm{,}N}\left({\widehat{\theta }}_J\right)\hfill & =\frac{n}{{\widehat{N}}^2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^n}\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_{ij}{\pi }_i{\pi }_j}{\widehat{\xi }}_i\left({\widehat{\theta }}_J\right){\widehat{\xi }}_j{\left({\widehat{\theta }}_J\right)}^{\prime }\text{}\mathrm{,}\hfill \\ {\widehat{\xi }}_i\left({\widehat{\theta }}_J\right)\hfill & ={\delta }_i{g}_i\left(y_i\mathrm{<mo>;</mo>}{\widehat{\theta }}_J\right)+\left(1-{\delta }_i\right)J^{-1}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^J}{g}_i\left(B{\left(x_i\right)}^{\prime }{\widehat{\beta }}_{{\tau }_j}\text{}\mathrm{<mo>;</mo>}{\widehat{\theta }}_J\right)+{\delta }_i{\widehat{h}}_{ni}\left({\widehat{\theta }}_J\right)\mathrm{,}\hfill \\ {\widehat{h}}_{ni}\left({\widehat{\theta }}_J\right)\hfill & =J^{-1}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^J}{N}^{-1}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^n}{\pi }_k^{-1}\left(1-{\delta }_k\right){\dot{g}}_{k\mathrm{,}y}\left(B{\left(x_k\right)}^{\prime }{\widehat{\beta }}_{{\tau }_j}\text{}\mathrm{<mo>;</mo>}{\widehat{\theta }}_J\right)B{\left(x_k\right)}^{\prime }{\widehat{\Omega }}_n{\left({\tau }_j\right)}^{-1}B\left(x_i\right){\psi }_{\tau }\left({\widehat{u}}_{i{\tau }_j}\right)\mathrm{,}\hfill \\ {\widehat{\Omega }}_n\left({\tau }_j\right)\hfill & \mathrm{<mo>=</mo>}\widehat{H}\left({\tau }_j\right)+\frac{{\widehat{f}}_{\infty }{\tilde{\lambda }}_n}{n}{D}_m^{\prime }{D}_m\mathrm{,}\hfill \\ \widehat{H}\left(\tau \right)\hfill & \mathrm{<mo>=</mo>}\frac{1}{\widehat{N}}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n}{\pi }_i^{-1}{\delta }_{i}B\left(x_i\right){\widehat{f}}_{y|x\text{}\mathrm{,}i}\left({\widehat{q}}_{\tau }\left(x_i\right)\right)B{\left(x_i\right)}^{\prime }\text{}\mathrm{,}\hfill \end{array}$$

${\widehat{f}}_{\infty }\mathrm{<mo>=</mo>}n{\widehat{N}}^{-1},$ $\widehat{N}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^n}{\pi }_i^{-1}\text{},$ and ${\widehat{u}}_{i{\tau }_j}\mathrm{<mo>=</mo>}{y}_i-B{\left(x_i\right)}^{\prime }{\widehat{\beta }}_{{\tau }_j}\text{}.$ An estimator of ${\widehat{f}}_{y|x\text{},i}\left({\widehat{q}}_{\tau }\left(x_i\right)\right)$ is the inverse of an estimator of the derivative of the quantile function and is defined by

$${\widehat{f}}_{y|x\text{},i}\left({\widehat{q}}_{\tau }\left(x_i\right)\right)\mathrm{<mo>=</mo>}\text{max}\left\{\frac{2a_{n\mathrm{,}\tau }}{B{\left(x_i\right)}^{\prime }\left({\widehat{\beta }}_{\tau +a_{n\text{},\tau }}-{\widehat{\beta }}_{\tau -a_{n\text{},\tau }}\right)}\mathrm{,}0\right\}\mathrm{,}(3.15)$$

where the bandwidth $a_{n\mathrm{,}\tau }$ is given by

$$a_{n\text{}\mathrm{,}\tau }\mathrm{<mo>=</mo>}{n}^{-0.2}\left[\frac{4.5\varphi {\left({\Phi }^{-1}\left(\tau \right)\right)}^4}{{\left(2{\Phi }^{-1}{\left(\tau \right)}^2+1\right)}^2}\right]\mathrm{,}(3.16)$$

with $\varphi (\cdot )$ and $\Phi (\cdot ),$ respectively, the pdf and cdf of a standard normal distribution. See Wei, Ma and Carroll (2012) and Koenker (2005) for discussions of (3.15) and (3.16), respectively.

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