Nonparametric estimation with mixed data types in survey sampling

Abstract

We consider the problem of finite population mean estimation with mixed data types. A model-assisted estimator based on nonparametric regression is proposed, which can handle discrete and continuous data and incorporates the sampling design in a natural manner. The proposed method shares the design-based properties of the kernel-based model-assisted estimator in the presence of continuous covariates and performs well under different scenarios in simulation experiments.

Appendix

Proof of Theorem 1

We write

$$\begin{aligned} \overline{y}_\mathrm{np}-\overline{Y}= \frac{1}{N} \sum _{j \in U} (y_{j}-m_{j}) \left( \frac{I_{j}}{\pi _{j}}-1\right) + \frac{1}{N}\sum _{j \in U} (\widehat{m}_{j}-m_{j}) \left( 1-\frac{I_{j}}{\pi _{j}}\right) . \end{aligned}$$

(29)

Then

$$\begin{aligned}&E_{d}\left| \overline{y}_\mathrm{np}-\overline{Y} \right| \le E_{d}\left| \sum _{j \in U}\frac{y_{j}-m_{j}}{N} \left( \frac{I_{j}}{\pi _{j}}-1\right) \right| \nonumber \\&+ \left\{ E_{d}\left[ \sum _{j \in U} \frac{(\widehat{m}_{j}-m_{j})^{2}}{N}\right] E_{d}\left[ \sum _{j \in U} \frac{(1-\pi _{j}^{-1}I_{j})^{2}}{N}\right] \right\} ^{1/2}. \end{aligned}$$

(30)

Under assumptions 1–6, there exists $n_{0}$, such that $n \ge n_{0}$ implies lemma 2(ii) in [3],

$$\begin{aligned} \sum _{i \in U} I_{\{|x^{c}-x^{c}_{i}| \le h\}} \ge 1, \end{aligned}$$

(31)

which holds by using the fact that $\lambda \le l_{\lambda }(x^{d}_{i}-x^{d}_{j})\le 1$.

We write

$$\begin{aligned} \lim _{N \rightarrow \infty } \sup \left| \widehat{t}_{jg}/N \right|&= \lim _{N \rightarrow \infty } \sup \left| \sum _{i \in U} \frac{1}{Nh}K\left( \frac{x_{i}^{c}-x_{j}^{c}}{h}\right) l_{\lambda }(x_{i}^{d}-x_{j}^{d})y_{i}^{p_{1}} \frac{I_{i}}{\pi _{i}}\right| \nonumber \\&\le \lim _{N \rightarrow \infty } \sup \sum _{i \in U} \frac{C}{NhM} I_{\{x^{c}_{j}-h \le x^{c}_{i} \le x^{c}_{j}+h|\}}, \end{aligned}$$

(32)

with $p_{1}=0,1$ and using $\lambda \le l_{\lambda }(x^{d}_{i}-x^{d}_{j})\le 1$. As $M < 1$, the same bound is valid for $t_{jg}/N$.

Therefore the $N^{-1}t_{jg}$ are uniformly bounded in $j$ and the $N^{-1}\widehat{t}_{jg}$ are uniformly bounded in $j$ and $s$. The $m_{j}$ are continuous functions of the $t_{jg}$ with denominators uniformly bounded away from 0 by (31). Analogously, the $\widehat{m}_{j}$ are continuous functions of the uniformly bounded $\widehat{t}_{jg}$, well defined by the $\delta /N^{2}$ adjustment in (12). Hence, the $m_{j}$ are uniformly bounded in $j$ and the $\widehat{m}_{j}$ are uniformly bounded in $j$ and $s$.

Under assumptions 1–6, we also have

$$\begin{aligned} \lim _{N \rightarrow \infty } \sup \frac{1}{N} \sum _{j \in U} (y_{j}-m_{j})^{2} < \infty , \end{aligned}$$

(33)

and the first term in (30) converges to 0 as $N \rightarrow \infty$, following the argument of Theorem 1 in [23]. Under assumption 5,

$$\begin{aligned} E_{d}\left( \sum _{j \in U} \frac{(1-\pi _{j}^{-1}I_{j})^{2}}{N}\right) =\sum _{j \in U} \frac{\pi _{j}(1-\pi _{j})}{N\pi _{j}^{2}}\le \frac{1}{M}. \end{aligned}$$

(34)

Using $\lambda \le l_{\lambda }(x^{d}_{i}-x^{d}_{j})\le 1$, lemmas 3, 4 and 5 in [3] continue to hold. Then, using lemma 4,

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{1}{N} \sum _{j \in U} E_{d}(\widehat{m}_{j}- m_{j})^{2}=0, \end{aligned}$$

(35)

and by (34) the second term in (30) converges to 0 as $N \rightarrow \infty$. By the Markov inequality the theorem follows. $\square$

Proof of Theorem 2:

Let

$$\begin{aligned} a_{N}=n^{1/2}\sum _{j \in U}\frac{y_{j}-m_{j}}{N}\left( \frac{I_{j}}{\pi _{j}}-1\right) , \quad b_{N}=n^{1/2}\sum _{j \in U}\frac{m_{j}-\widehat{m}_{j}}{N}\left( \frac{I_{j}}{\pi _{j}}-1\right) ,\qquad \end{aligned}$$

(36)

so that

$$\begin{aligned} nE_{d}(\overline{y}_\mathrm{np}-\overline{Y})^2=E_{d}(a_{N}^{2})+E_{d}(b_{N}^{2})+2E_{d}(a_{N}b_{N}). \end{aligned}$$

(37)

It is immediate that $E_{d}(b_{N}^{2})=o(1)$ by lemma 5 in [3], that also follows because of $\lambda \le l_{\lambda }(x^{d}_{i}-x^{d}_{j})\le 1$. $E_{d}(a_{N}^{2})$ is bounded because $Var_{d}(a_{N})=O(1)$, as $Var_{d}(n^{-1/2}a_{N})=O(1/n)$ for the Horvitz–Thompson mean estimator. Then,

$$\begin{aligned} E_{d}(a_{N}b_{N}) \le (E_{d}(a_{N}^{2})E_{d}(b_{N}^{2}))^{1/2}=o(1). \end{aligned}$$

(38)

Hence,

$$\begin{aligned} nE_{d}(\overline{y}_\mathrm{np}-\overline{Y})^2=E_{d}(a_{N}^{2})+o(1), \end{aligned}$$

(39)

and the theorem follows. $\square$

Proof of Theorems 3 and 4

Under conditions 1–6 and using again that $\lambda \le l_{\lambda }(x^{d}_{i}-x^{d}_{j})\le 1$, the asymptotic normality of the generalized difference estimator follows and a variance consistent estimator of the asymptotic mean square error can be derived, analogously as Theorems 3 and 4 in [3]. $\square$