Empirical likelihood inference for missing survey data under unequal probability sampling
Section 6. Conclusion and future perspectives

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A fractional imputation is proposed and an associated EL is developed for making inference on the population mean $\mu$ of a function of a variable of interest for survey data that are missing at random under PPS sampling with negligible sampling fractions or with replacement. Two bootstrap and EL based methods, BELP and BELR, are proposed for constructing CIs on $\mu ,$ and are shown to be asymptotically correct. Simulation studies show that the proposed intervals perform better than their naive bootstrap counterparts under various sample size settings. In addition, the proposed BELR intervals are seen to have more accurate coverage probabilities than those of the proposed BELP intervals, particularly in two situations: (i) when the inclusion probabilities differ in size substantially, or (ii) when the sample size is not too large. Moreover, the proposed BELR intervals exhibit notably better coverage probabilities than the EL-ratio intervals with estimated scaling constant (SELR intervals) in the above case (i).

For parameters defined by smooth estimating equations under an IID setting, Tang and Qin (2012), using an inverse-probability-weighted fractional imputation, achieved two most desirable properties of EL inference with data missing at random: (1) the associated MELE attains the semi-parametric efficiency bound, and (2) the corresponding EL ratio follows a simple chi-square limiting distribution. However, their method requires both the observed and the missing data to be imputed, which is unlikely to be accepted in practice in survey studies. We are working on extending Tang and Qin (2012) to a survey setup while avoiding imputing observed data points.

Our future work also includes extending the current methods to tackling missing data from stratified sampling and multistage sampling, as well as to the case where the sampling fraction is non-negligible.

Appendix A

Define, for all $k\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}K,$

$$\begin{array}{ll}{Q}_k\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in U}1\left(z_i\mathrm{<mo>=</mo>}k\right){\left(n{d}_i\right)}^{-1}}\mathrm{,}\hfill \\ {\overline{H}}_k\hfill & \mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle \sum _{i\in U}1\left(z_i\mathrm{<mo>=</mo>}k\right)\left\{h\left(y_i\right)-{\mu }_N\right\}}\mathrm{,}\hfill \end{array}$$

and

$$S_{H_k}^2\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle \sum _{i\in U}1\left(z_i\mathrm{<mo>=</mo>}k\right){\left\{h\left(y_i\right)-{\mu }_N\right\}}^2\left(n{d}_i/N\right)-{\overline{H}}_k^2/Q_k}\mathrm{.}$$

Theorems 1-3 are established under the following regularity conditions.

Conditions (R.1) and (R.2) ensure that the population has regular behaviour when embedded in a asymptotic sequence. Condition (R.3) is a standard condition in EL inference as used by Chen and Sitter (1999).

The constant ${\sigma }_N$ in Theorem 1 is given by the positive square root of

$${\sigma }_N^2\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K\left[\left\{P_k^{-1}+\left(1-P_k\right)J^{-1}\right\}{S}_{H_k}^2+{\overline{H}}_k^2/Q_k\right]}.$$

The constant $c_N$ in Theorem 2 is given by

$$c_N\mathrm{<mo>=</mo>}\frac{{\sigma }_N^2}{{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K\left[\left\{P_k+\left(1-P_k\right)J^{-1}\right\}{S}_{H_k}^2+{\overline{H}}_k^2/Q_k\right]}}\mathrm{.}$$

Design-consistent estimators of ${\sigma }_N^2$ and $c_N$ can be obtained by plugging in the following design-consistent estimators of $P_k,$ $Q_k,$ ${\overline{H}}_K$ and $S_{H_k}^2$ for $k\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}K:$

$$\begin{array}{ll}{\widehat{P}}_k\hfill & \mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\in s}{\delta }_{i}1\left(z_i=k\right)d_i}}{{\displaystyle {\sum }_{i\in s}1\left(z_i=k\right)d_i}}\mathrm{,}{\widehat{Q}}_k\mathrm{<mo>=</mo>}{n}^{-1}{\displaystyle \sum _{i\in s}1\left(z_i\mathrm{<mo>=</mo>}k\right)}\mathrm{,}\hfill \\ {\widehat{\overline{H}}}_k\hfill & \mathrm{<mo>=</mo>}{N}^{-1}{P}_k^{-1}{\displaystyle \sum _{i\in s}{\delta }_{i}1\left(z_i\mathrm{<mo>=</mo>}k\right)d_i\left(h\left(y_i\right)-\widehat{\mu }\right)}\mathrm{,}\hfill \\ {\widehat{S}}_{H_k}^2\hfill & \mathrm{<mo>=</mo>}{N}^{-1}{P}_k^{-1}{\displaystyle \sum _{i\in s}{\delta }_{i}1\left(z_i\mathrm{<mo>=</mo>}k\right){\left(h\left(y_i\right)-\widehat{\mu }\right)}^2\left(n{d}_i^2/N\right)-{\widehat{Q}}_k^{-1}{\widehat{\overline{H}}}_k^2}\mathrm{.}\hfill \end{array}$$

Appendix B

We now give proofs for Theorems 1-3. Let $E(\cdot )$ and $\text{Var}(\cdot )$ denote the expectation and variance operators with respect to the sampling design, respectively. We consider PPS sampling with replacement in the proofs. All results also apply to PPS sampling without replacement with negligible sampling fractions.

Lemma 1. Under condition (R.1), $n_k/r_k\mathrm{<mo>=</mo>}{P}_k^{-1}\mathrm{<mo>=</mo>}{o}_p\left(1\right).$

Proof of Lemma 1. Recall that $n_k\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}}1\left(z_i\mathrm{<mo>=</mo>}k\right),$ so we have $\text{E}\left(n_k/n\right)\mathrm{<mo>=</mo>}{{\displaystyle {\sum }_{i\in U}1\left(z_i\mathrm{<mo>=</mo>}k\right)\left(n{d}_i\right)}}^{-1}\mathrm{<mo>=</mo>}{Q}_k\mathrm{<mo>=</mo>}O\left(1\right)$ by (R.1). Under PPS sampling with replacement, we have

$$\text{Var}\left(n_k/n\right)\mathrm{<mo>=</mo>}{n}^{-1}\left\{{\displaystyle \sum _{i\in U}1\left(z_i\mathrm{<mtext>\hspace{0.17em}</mtext><mo>=</mo><mtext>\hspace{0.17em}</mtext>}k\right)}{\left(n{d}_i\right)}^{-1}-{\text{E}}^2\left(n_k/n\right)\right\}\mathrm{<mo>=</mo>}{n}^{-1}\left\{Q_k-Q_k^2\right\}\mathrm{<mo>=</mo>}o\left(1\right)\mathrm{.}$$

Therefore, by Markov’s inequality, we easily obtain $n_k/n\mathrm{<mo>=</mo>}\text{E}\left(n_k/n\right)+o_p\left(1\right)\mathrm{<mo>=</mo>}{Q}_k+o_p\left(1\right).$

Moreover, since $r_k\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}{\delta }_{i}1\left(z_i\mathrm{<mo>=</mo>}k\right)},$ by the MAR assumption, we have $\text{E}\left(r_k/n\right)\mathrm{<mo>=</mo>}\text{E}\left({\delta }_i|z_i\mathrm{<mo>=</mo>}k\right)\text{E}\left(n_k/n\right)\mathrm{<mo>=</mo>}{P}_k\text{E}\left(n_k/n\right)\mathrm{<mo>=</mo>}{P}_k{Q}_k.$ Similar to the case of $n_k/n,$ we can show that $\text{Var}\left(r_k/n\right)\mathrm{<mo>=</mo>}o\left(1\right),$ so we have $r_k/n\mathrm{<mo>=</mo>}{P}_k{Q}_k+o_p\left(1\right).$

We hence conclude that $n_k/r_k\mathrm{<mo>=</mo>}\left(n_k/n\right)/\left(r_k/n\right)\mathrm{<mo>=</mo>}{P}_k^{-1}+o_p\left(1\right).$

Lemma 2. Under conditions (R.1) and (R.2),

$${\sigma }_N^{-1}\sqrt{n}{N}^{-1}{\displaystyle \sum _{i\in s}{\tilde{h}}_i\left({\mu }_N\right)}\stackrel{d}{\to }N\left(\mathrm{0,}1\right)\mathrm{.}$$

Proof of Lemma 2. The following decomposition holds:

$$\sqrt{n}{N}^{-1}{\displaystyle \sum _{i\in s}{\tilde{h}}_i\left({\mu }_N\right)}\mathrm{<mo>=</mo>}{U}_n+V_n\mathrm{,}$$

where $U_n=\sqrt{n}{N}^{-1}{\displaystyle {\sum }_{i\in s}}\left[{\tilde{h}}_i\left({\mu }_N\right)-\text{E}\left\{{\tilde{h}}_i\left({\mu }_N\right)|{\mathcal{F}}_n\right\}\right]$ and $V_n\mathrm{<mo>=</mo>}\sqrt{n}{N}^{-1}{\displaystyle {\sum }_{i\in s}\text{E}\left\{{\tilde{h}}_i\left({\mu }_N\right)|{\mathcal{F}}_n\right\}}.$ Noting that $\text{E}\left\{{\tilde{h}}_i\left({\mu }_N\right)|{\mathcal{F}}_n\right\}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^{K}1\left(z_i\mathrm{<mo>=</mo>}k\right)\left[{\delta }_i{d}_i\left\{h\left(y_i\right)-{\mu }_N\right\}+\left(1-{\delta }_i\right)r_k^{-1}{\widehat{H}}_{kr}\right]},$ where ${\widehat{H}}_{kr}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}1\left(z_i\mathrm{<mo>=</mo>}k\right)}{\delta }_i{d}_i\left\{h\left(y_i\right)-{\mu }_N\right\},$ we get

$$U_n\mathrm{<mo>=</mo>}\frac{\sqrt{n}}{N}{\displaystyle \sum _{i\in s}{\eta }_i}\text{and}{V}_n\mathrm{<mo>=</mo>}\frac{\sqrt{n}}{N}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\frac{n_k}{r_k}{\widehat{H}}_{kr}(\text{B}\text{.1})$$

with

$${\eta }_i\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K\left(1-{\delta }_i\right)1\left(z_i\mathrm{<mo>=</mo>}k\right)}{J}^{-1}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^J\left[d_{ij}^{*}\left\{h\left(y_{ij}^{*}\right)-{\mu }_N\right\}-r_k^{-1}{\widehat{H}}_{kr}\right]}\mathrm{.}$$

We now work out the conditional limiting distributions of $U_n$ given the sample data ${\mathcal{F}}_n.$ Since the imputation is carried out independently across $i\in s,$ ${\eta }_i$ are conditionally independent random variables given ${\mathcal{F}}_n.$ Note that ${\eta }_i$ have zero conditional mean, $E\left({\eta }_i|{\mathcal{F}}_n\right)\mathrm{<mo>=</mo>\; 0}$ and common conditional variance for all $i\in s.$ Moreover, we have

$$\text{Var}\left(U_n|{\mathcal{F}}_n\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left(\frac{n_k}{r_k}-1\right)\frac{1}{J}\left[\frac{n}{N^2}{\displaystyle \sum _{i\in s}1\left(z_i\mathrm{<mo>=</mo>}k\right)}{\delta }_i{d}_i^2{\left\{h\left(y_i\right)-{\mu }_N\right\}}^2-\frac{n}{r_k}{\left(\frac{{\widehat{H}}_{kr}}{N}\right)}^2\right]\mathrm{.}$$

By Lemma 1, $n_k/r_k\mathrm{<mo>=</mo>}{P}_k^{-1}+o_p\left(1\right),$ and in the proof of Lemma 1, we have shown $n/r_k\mathrm{<mo>=</mo>}{P}_k^{-1}{Q}_k^{-1}+o_p\left(1\right).$ Under condition (R.2)(a), we can show that

$$\frac{n}{N^2}{\displaystyle \sum _{i\in s}1\left(z_i\mathrm{<mo>=</mo>}k\right)}{\delta }_i{d}_i^2{\left\{h\left(y_i\right)-{\mu }_N\right\}}^2\mathrm{<mo>=</mo>}\frac{1}{N}{P}_k{{\displaystyle \sum _{i\in s}1\left(z_i\mathrm{<mo>=</mo>}k\right)\left\{h\left(y_i\right)-{\mu }_N\right\}}}^2\left(n{d}_i/N\right)+o_p\left(1\right)$$

and ${\widehat{H}}_{kr}/N\mathrm{<mo>=</mo>}{P}_k{\overline{H}}_k+o_p\left(1\right).$ Therefore,

$$s_u^2\text{:=}\text{Var}\left(U_n|{\mathcal{F}}_n\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K\left(1-P_k\right)}{J}^{-1}{S}_{H_k}^2+o_p\left(1\right)\mathrm{<mo>=</mo>}{O}_p\left(1\right)\mathrm{.}(\text{B}\text{.2})$$

By the Berry-Essen Theorem (Chow and Teicher, 1997, Section 9.1), we have

$$\underset{t}{{\displaystyle \sup }}\left|P\left(U_n\le t{s}_u|{\mathcal{F}}_n\right)-\Phi \left(t\right)\right|\le c_{{ϵ}^{{}^{\prime }}}{\left({\iota }_u/s_u\right)}^{2+{ϵ}^{{}^{\prime }}}\text{}\mathrm{,}$$

where ${ϵ}^{{}^{\prime }}\in \left(\mathrm{0,}ϵ\right)$ and $c_{{ϵ}^{{}^{\prime }}}$ are some constants that do not rely on $n,$ and

$${\iota }_u^{2+{ϵ}^{{}^{\prime }}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in s}\text{E}\left\{{\left(\sqrt{n}{N}^{-1}{\eta }_i\right)}^{2+{ϵ}^{{}^{\prime }}}|{\mathcal{F}}_n\right\}}\mathrm{<mo>=</mo>}\frac{n^{2+{ϵ}^{{}^{\prime }}/2}}{N^{2+{ϵ}^{{}^{\prime }}}}\left\{\frac{1}{n}{\displaystyle \sum _{i\in s}\text{E}\left({\eta }_i^{2+{ϵ}^{{}^{\prime }}}|{\mathcal{F}}_n\right)}\right\}\mathrm{.}$$

We can further show that, under condition (R.2)(a), $n^{-1}{\displaystyle {\sum }_{i\in s}}\text{E}\left({\eta }_i^{2+{ϵ}^{{}^{\prime }}}|{\mathcal{F}}_n\right)\mathrm{<mo>=</mo>}{O}_p\left(1\right),$ which implies that ${\iota }_u^{2+{ϵ}^{{}^{\prime }}}\mathrm{<mo>=</mo>}{o}_p\left(1\right).$ Therefore,

$$\underset{t}{{\displaystyle \sup }}\left|P\left(U_n\le t{s}_u|{\mathcal{F}}_n\right)-\Phi \left(t\right)\right|\mathrm{<mo>=</mo>}{o}_p\left(1\right)\mathrm{.}(\text{B}\text{.3})$$

We next find the limiting distribution of $V_n.$ It can be shown that, for each $k\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}K,$

$$\frac{1}{N}\frac{n_k}{r_k}{\widehat{H}}_{kr}\mathrm{<mo>=</mo>}\frac{1}{n}{\displaystyle \sum _{i\in s}1\left(z_i\mathrm{<mo>=</mo>}k\right)}\left[\frac{{\delta }_i}{P_k}\left\{\frac{n}{N}{d}_i\left(h\left(y_i\right)-{\mu }_N\right)-\frac{{\overline{H}}_k}{Q_k}\right\}+\frac{{\overline{H}}_k}{Q_k}\right]+o_p\left(n^{-1/2}\right)\mathrm{.}$$

Hence, by (B.1), we have $V_n\mathrm{<mo>=</mo>}\frac{\sqrt{n}}{n}{\displaystyle {\sum }_{i\in s}{\zeta }_i}+o_p\left(1\right),$ where

$${\zeta }_i\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K}1\left(z_i\mathrm{<mo>=</mo>}k\right)}\left[\frac{{\delta }_i}{P_k}\left\{\frac{n}{N}{d}_i\left(h\left(y_i\right)-{\mu }_N\right)-\frac{{\overline{H}}_k}{Q_k}\right\}+\frac{{\overline{H}}_k}{Q_k}\right]\mathrm{.}$$

Under sampling with replacement, ${\zeta }_i,$ $i\in s,$ are independent random variables. Moreover, we get $\text{E}\left({\displaystyle {\sum }_{i\in s}{\zeta }_i}\right)\mathrm{<mo>=</mo>\; 0}$ and

$$s_v^2\text{:=}\text{Var}\left(V_n\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left\{P_k^{-1}{S}_{H_k}^2+{\overline{H}}_k^2/Q_k\right\}\mathrm{.}(\text{B}\text{.4})$$

Under condition (R.2)(a), the conditions of the Lyanunov central limit theorem are satisfied, so we have

$$s_v^{-1}{V}_n\stackrel{d}{\to }N\left(\mathrm{0,}1\right)\mathrm{.}(\text{B}\text{.5})$$

By (B.3) and (B.5), and observing that $s_v/s_u$ converges to a constant limit under condition (R.1), all conditions of Theorem 2 of Chen and Rao (2007) are verified. Accordingly, we have

$${\left(s_u^2+s_v^2\right)}^{-1/2}\left(U_n+V_n\right)\stackrel{d}{\to }N\left(\mathrm{0,}1\right)\mathrm{.}$$

Noticing that $s_u^2+s_v^2\mathrm{<mo>=</mo>}{\sigma }_N^2+o_p\left(1\right)$ and $U_n+V_n\mathrm{<mo>=</mo>}\sqrt{n}{N}^{-1}{\displaystyle {\sum }_{i\in s}{\tilde{h}}_i\left({\mu }_N\right)},$ the claimed result is proved.

Proof of Theorem 1. By (3.3) and (2.1), we have

$$\sqrt{n}\left(\widehat{\mu }-{\mu }_N\right)\mathrm{<mo>=</mo>}\sqrt{n}{\tilde{d}}^{-1}{\displaystyle \sum _{i\in s}{\tilde{h}}_i\left({\mu }_N\right)}\mathrm{,}(\text{B}\text{.6})$$

where $\tilde{d}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K}1\left(z_i\mathrm{<mo>=</mo>}k\right)\left\{{\delta }_i{d}_i+\left(1-{\delta }_i\right)J^{-1}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^J{d}_{ij}^{*}}\right\}.$

Note that, given the sample data ${\mathcal{F}}_n,$ $\text{E}\left(N^{-1}\tilde{d}|{\mathcal{F}}_n\right)\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K\left(n_k/r_k\right)}{\displaystyle {\sum }_{i\in s}}1\left(z_i\mathrm{<mo>=</mo>}k\right){\delta }_i{d}_i.$ By Lemma 1, we have $n_k/r_k\mathrm{<mo>=</mo>}{P}_k^{-1}+o_p\left(1\right).$ Moreover, we can show that, under condition (R.2)(b),

$$\begin{array}{ll}{N}^{-1}{\displaystyle \sum _{i\in s}}1\left(z_i\mathrm{<mo>=</mo>}k\right){\delta }_i{d}_i\hfill & \mathrm{<mo>=</mo>}\text{E}\left\{N^{-1}{\displaystyle \sum _{i\in s}1\left(z_i\mathrm{<mo>=</mo>}k\right){\delta }_i{d}_i}\right\}+o_p\left(1\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}^{-1}{P}_k{\displaystyle \sum _{i\in U}1\left(z_i\mathrm{<mo>=</mo>}k\right)}+o_p\left(1\right)\mathrm{.}\hfill \end{array}$$

Therefore,

$$\text{E}\left(N^{-1}\tilde{d}|{\mathcal{F}}_n\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K{P}_k^{-1}\left\{N^{-1}{P}_k{\displaystyle \sum _{i\in U}1\left(z_i\mathrm{<mo>=</mo>}k\right)}\right\}}+o_p\left(1\right)\mathrm{<mo>=</mo>\; 1}+o_p\left(1\right)\mathrm{.}$$

In addition, it can be shown that given ${\mathcal{F}}_n,$ $N^{-1}\tilde{d}\mathrm{<mo>=</mo>}\text{E}\left(N^{-1}\tilde{d}|{\mathcal{F}}_n\right)+o_p\left(1\right).$ The above results imply that

$$N^{-1}\tilde{d}\mathrm{<mo>=</mo>}1+o_p\left(1\right)\mathrm{.}(\text{B}\text{.7})$$

Combining (B.6), (B.7) and Lemma 2, we obtain the desired result.

Proof of Theorem 2. By (3.2) and (3.4), we have

$$R\left({\mu }_N\right)\mathrm{<mo>=</mo>}-2l_n\left({\mu }_N\right)\mathrm{<mo>=</mo>}2{\displaystyle \sum _{i\in s}}\log \left\{1+{\lambda }_N{\tilde{h}}_i\left({\mu }_N\right)\right\}\mathrm{.}(\text{B}\text{.8})$$

where ${\lambda }_N$ satisfies $n^{-1}{\displaystyle {\sum }_{i\in s}{\tilde{h}}_i\left({\mu }_N\right)}/\left\{1+{\lambda }_N{\tilde{h}}_i\left({\mu }_N\right)\right\}\mathrm{<mo>=</mo>\; 0.}$ Using the same argument as used by Owen (2001, Section 11.2, Proof of Theorem 3.2), we can show that, under condition (R.3), ${\lambda }_N\mathrm{<mo>=</mo>}{O}_p\left(n^{-1/2}\right)$ and

$${\lambda }_N\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\in s}{\tilde{h}}_i\left({\mu }_N\right)}}{{\displaystyle {\sum }_{i\in s}{\tilde{h}}_i^2\left({\mu }_N\right)}}+o_p\left(n^{-1/2}\right)\mathrm{.}$$

By this expression of ${\lambda }_N,$ (B.8) and a Taylor’s expansion, we obtain

$$\begin{array}{ll}R\left({\mu }_N\right)\hfill & \mathrm{<mo>=</mo>\; 2}{\lambda }_N{\displaystyle \sum _{i\in s}{\tilde{h}}_i\left({\mu }_N\right)}-{\lambda }_N^2{\displaystyle \sum _{i\in s}{\tilde{h}}_i^2\left({\mu }_N\right)}+o_p\left(1\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{{\left\{\sqrt{n}{N}^{-1}{\sigma }_N^{-1}{\displaystyle {\sum }_{i\in s}{\tilde{h}}_i\left({\mu }_N\right)}\right\}}^2}{n{N}^{-2}{\displaystyle {\sum }_{i\in s}{\tilde{h}}_i^2\left({\mu }_N\right)}/{\sigma }_N^2}+o_p\left(1\right)\mathrm{.}(\text{B}\text{.9})\hfill \end{array}$$

Note that $n{N}^{-2}{\displaystyle {\sum }_{i\in s}{\tilde{h}}_i^2\left({\mu }_N\right)}\mathrm{<mo>=</mo>}{A}_1+A_2,$ where

$$A_1\mathrm{<mo>=</mo>}{n}^{-1}{\displaystyle \sum _{i\in s}}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K{\delta }_{i}1\left(z_i\mathrm{<mo>=</mo>}k\right)}{\left\{h\left(y_i\right)-{\mu }_N\right\}}^2{\left(n{d}_i/N\right)}^2\mathrm{,}$$

and

$$A_2\mathrm{<mo>=</mo>}{n}^{-1}{\displaystyle \sum _{i\in s}}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K\left(1-{\delta }_i\right)}1\left(z_i\mathrm{<mo>=</mo>}k\right)J^{-2}{\left[{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^J}\left\{h\left(y_{ij}^{*}\right)-{\mu }_N\right\}\left(n{d}_{ij}^{*}/N\right)\right]}^2\mathrm{.}$$

Under condition (R.2)(a), we can show that $A_1\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{P}_k\left(S_{H_k}^2+{\overline{H}}_k^2/Q_k\right)}+o_p\left(1\right)$ and $A_2\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K}\left(1-P_k\right)\left(J^{-1}{S}_{H_k}^2+{\overline{H}}_k^2/Q_k\right)+o_p\left(1\right).$ Hence,

$$n{N}^{-2}{\displaystyle \sum _{i\in s}{\tilde{h}}_i^2\left({\mu }_N\right)}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K\left[\left\{P_k+\left(1-P_k\right)J^{-1}\right\}{S}_{H_k}^2+{\overline{H}}_k^2/Q_k\right]}+o_p\left(1\right)\mathrm{.}$$

Theorem 2 is then proved by substituting the above expression into (B.9) and applying Theorem 1.

For the proof of Theorem 3, we introduce the following Lemma.

Lemma 3. Under conditions (R.1) and (R.2),

$$\underset{t}{{\displaystyle \sup }}\left|P\left({\sigma }_N^{-1}\sqrt{n}{N}^{-1}\left\{{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n{\tilde{h}}_{b\text{}\mathrm{,}i}\left({\mu }_N\right)}-{\stackrel{⌣}{h}}_{+}\left({\mu }_N\right)\right\}\le t|{\mathcal{F}}_n\right)-\Phi \left(t\right)\right|\mathrm{<mo>=</mo>}{o}_p\left(1\right)\mathrm{.}$$

where ${\stackrel{⌣}{h}}_{+}\left({\mu }_N\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}{\stackrel{⌣}{h}}_i\left({\mu }_N\right)}.$

Proof of Lemma 3. Let ${\mathcal{G}}_{b\text{}\mathrm{,}n}$ denote the bootstrap sample $\left\{({{\displaystyle y}}_{b,i},{{\displaystyle d}}_{b,i},{{\displaystyle z}}_{b,i},{{\displaystyle \delta }}_{b,i}):i=1,\mathrm{...},n\right\}.$ We have the following decomposition:

$$\sqrt{n}{N}^{-1}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n{\tilde{h}}_{b\text{}\mathrm{,}i}\left({\mu }_N\right)}\mathrm{<mo>=</mo>}{\mathbb{U}}_n+{\mathbb{V}}_n\mathrm{,}$$

where ${\mathbb{U}}_n\mathrm{<mo>=</mo>}\sqrt{n}{N}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^n\left[{\tilde{h}}_{b\text{}\mathrm{,}i}\left({\mu }_N\right)-\text{E}\left\{{\tilde{h}}_{b\text{}\mathrm{,}i}\left({\mu }_N\right)|{\mathcal{G}}_{b\text{}\mathrm{,}n}\right\}\right]}$ and

$${\mathbb{V}}_n\mathrm{<mo>=</mo>}\sqrt{n}{N}^{-1}\left[{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n\text{E}\left\{{\tilde{h}}_{b\text{}\mathrm{,}i}\left({\mu }_N\right)|{\mathcal{G}}_{b\text{}\mathrm{,}n}\right\}}-{\stackrel{⌣}{h}}_{+}\left({\mu }_N\right)\right]\mathrm{.}$$

Similar to the proof of Lemma 2, we can show that

$$\underset{t}{{\displaystyle \sup }}\left|P\left({\mathbb{U}}_n\le t{s}_u|{\mathcal{G}}_{b\mathrm{,}n}\right)-\Phi \left(t\right)\right|\mathrm{<mo>=</mo>}{o}_p\left(1\right)\mathrm{,}(\text{B}\text{.10})$$

where $s_u$ is defined in (B.2).

We next give the conditional limiting distribution of ${\mathbb{V}}_n$ given ${\mathcal{F}}_n.$ It can be shown that ${\mathbb{V}}_n\mathrm{<mo>=</mo>}\sqrt{n}{N}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^n{\xi }_{b\text{}\mathrm{,}i}}+o_p\left(1\right),$ where

$${\xi }_{b\text{}\mathrm{,}i}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K\left[P_k^{-1}{\delta }_{b\text{}\mathrm{,}i}1\left(z_{b\text{}\mathrm{,}i}\mathrm{<mo>=</mo>}k\right)\left\{d_{b\text{}\mathrm{,}i}\left(h\left(y_{b\text{}\mathrm{,}i}\right)-{\mu }_N\right)-r_k^{-1}{\widehat{H}}_{kr}\right\}+1\left(z_{b\text{}\mathrm{,}i}\mathrm{<mo>=</mo>}k\right)r_k^{-1}{\widehat{H}}_{kr}\right]}-n^{-1}{\stackrel{⌣}{h}}_{+}\left({\mu }_N\right)\mathrm{.}$$

Conditioned on ${\mathcal{F}}_n,$ ${\xi }_{b\text{}\mathrm{,}i}$ are IID across $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n,$ and we can show that $\text{E}\left({\xi }_{b\text{}\mathrm{,}i}|{\mathcal{F}}_n\right)\mathrm{<mo>=</mo>\; 0}$ and

$$\begin{array}{ll}\text{Var}\left(\frac{\sqrt{n}}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n{\xi }_{b\text{}\mathrm{,}i}|{\mathcal{F}}_n}\right)\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K{P}_k^{-2}\frac{n}{N^2}}{\displaystyle \sum _{i\in s}{\delta }_{i}1\left(z_i\mathrm{<mo>=</mo>}k\right)}{\left\{d_i\left(h\left(y_i\right)-{\mu }_N\right)-\frac{1}{r_k}{\widehat{H}}_{kr}\right\}}^2\hfill \\ \hfill & +{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\frac{n}{N^2}{n}_k{\left\{\frac{1}{r_k}{\widehat{H}}_{kr}\right\}}^2\mathrm{.}\hfill \end{array}$$

We can show that the first term on the right hand side (RHS) equals ${\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{P}_k^{-1}{S}_{H_k}^2}+o_p\left(1\right)$ and the second term on the RHS equals ${\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K{\overline{H}}_k^2/Q_k}+o_p\left(1\right).$ Therefore

$$\text{Var}\left(\sqrt{n}{N}^{-1}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n{\xi }_{b\text{}\mathrm{,}i}|{\mathcal{F}}_n}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K\left\{P_k^{-1}{S}_{H_k}^2+{\overline{H}}_k^2/Q_k\right\}}+o_p\left(1\right)\mathrm{<mo>=</mo>}{s}_v^2+o_p\left(1\right),$$

where $s_v^2$ is defined in (B.4). By verifying the conditions of the Berry-Essen Theorem as in the proof of Lemma 2, we get

$$\underset{t}{{\displaystyle \sup }}\left|P\left({\mathbb{V}}_n\le t{s}_v|{\mathcal{F}}_n\right)-\Phi \left(t\right)\right|\mathrm{<mo>=</mo>}{o}_p\left(1\right)\mathrm{.}(\text{B}\text{.11})$$

By (B.10) and (B.11), and applying Theorem 2 of Chen and Rao (2007), we obtain the claimed results.

Proof of Theorem 3. We first prove (4.2). By (3.3) and (2.1), we have

$$\sqrt{n}\left({\widehat{\mu }}_b-{\mu }_N\right)\mathrm{<mo>=</mo>}\sqrt{n}{\tilde{d}}_b^{-1}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n{\tilde{h}}_{b\text{}\mathrm{,}i}\left({\mu }_N\right)}\mathrm{,}$$

where ${\tilde{d}}_b\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^n}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^{K}1\left(z_{b\text{}\mathrm{,}i}\mathrm{<mo>=</mo>}k\right)}\left\{{\delta }_{b\text{}\mathrm{,}i}{d}_{b\text{}\mathrm{,}i}+\left(1-{\delta }_{b\text{}\mathrm{,}i}\right)J^{-1}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^J{d}_{b\text{}\mathrm{,}ij}^{*}}\right\}.$ By (4.1), we have

$$\sqrt{n}\left(\stackrel{⌣}{\mu }-{\mu }_N\right)\mathrm{<mo>=</mo>}\sqrt{n}{\stackrel{⌣}{d}}^{-1}{\displaystyle \sum _{i\in s}{\stackrel{⌣}{h}}_i\left({\mu }_N\right)}\mathrm{<mo>=</mo>}\sqrt{n}{\stackrel{⌣}{d}}^{-1}{\stackrel{⌣}{h}}_{+}\left({\mu }_N\right)\mathrm{,}$$

where $\stackrel{⌣}{d}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K}\left(n_k/r_k\right)1\left(z_i\mathrm{<mo>=</mo>}k\right){\delta }_i{d}_i.$

It is straightforward to show that ${\tilde{d}}_b/N\mathrm{<mo>=</mo>}\stackrel{⌣}{d}/N+o_p\left(1\right)$ and $\stackrel{⌣}{d}/N\mathrm{<mo>=</mo>\; 1}+o_p\left(1\right)$ under condition (R.2)(b). Therefore,

$$\sqrt{n}{N}^{-1}\left({\widehat{\mu }}_b-\stackrel{⌣}{\mu }\right)\mathrm{<mo>=</mo>}\sqrt{n}{N}^{-1}\left\{{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n{\tilde{h}}_{b\text{}\mathrm{,}i}\left({\mu }_N\right)}-{\stackrel{⌣}{h}}_{+}\left({\mu }_N\right)\right\}+o_p\left(1\right)\mathrm{.}$$

Then, by Lemma 3, we have

$$\underset{t}{{\displaystyle \sup }}\left|P\left(\sqrt{n}{N}^{-1}\left({\widehat{\mu }}_b-\stackrel{⌣}{\mu }\right)\le t{\sigma }_N|{\mathcal{F}}_n\right)-\Phi \left(t\right)\right|\mathrm{<mo>=</mo>}{o}_p\left(1\right)\mathrm{.}$$

This, combined with Theorem 1 and Polya’s Theorem, completes the proof of (4.2).

The result (4.3) can be proved based on (4.2) and by following the same arguments as used in the proof of Theorem 2.

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