Kernel-based methods for combining information of several frame surveys

Abstract

A sample selected from a single sampling frame may not represent adequatly the entire population. Multiple frame surveys are becoming increasingly used and popular among statistical agencies and private organizations, in particular in situations where several sampling frames may provide better coverage or can reduce sampling costs for estimating population quantities of interest. Auxiliary information available at the population level is often categorical in nature, so that incorporating categorical and continuous information can improve the efficiency of the method of estimation. Nonparametric regression methods represent a widely used and flexible estimation approach in the survey context. We propose a kernel regression estimator for dual frame surveys that can handle both continuous and categorical data. This methodology is extended to multiple frame surveys. We derive theoretical properties of the proposed methods and numerical experiments indicate that the proposed estimator perform well in practical settings under different scenarios.

Appendix

1.1 Proof of Theorem 1.

We write

$$\begin{aligned} {\hat{y}}_{\mathrm{np}}^{\mathrm{d}}= & {} \sum _{j \in a}{\widehat{m}}_{j}^A + \sum _{j \in s_a} \left( y_j - {\widehat{m}}_{j}^A \right) w_{A_j} + \sum _{j \in b}{\widehat{m}}_{j}^B \nonumber \\&\quad + \sum _{j \in s_b} \left( y_j - {\widehat{m}}_{j}^B \right) w_{B_j} + \eta \left( \sum _{j \in ab}{\widehat{m}}_{j}^A+ \sum _{j \in s_{ab}} \left( y_j - {\widehat{m}}_{j}^A \right) w_{A_j}\right) \nonumber \\&\quad + (1-\eta )\left( \sum _{j \in ba}{\widehat{m}}_{j}^B + \sum _{j \in s_{ba}} \left( y_j - {\widehat{m}}_{j}^B\right) w_{B_j}\right) . \end{aligned}$$

(20)

Let ${\hat{y}}_{\mathrm{np}}^{a}$ denote the terms $\sum _{j \in a}{\widehat{m}}_{j}^A + \sum _{j \in s_a} (y_j - {\widehat{m}}_{j}^A ) w_{A_j}$. Similarly, ${\hat{y}}_{\mathrm{np}}^{b}$, ${\hat{y}}_\mathrm{np}^{ab}$ and ${\hat{y}}_{\mathrm{np}}^{ba}$ denote the corresponding terms on expansion (20).

We write

$$\begin{aligned} \left( {\hat{y}}_{\mathrm{np}}^{\mathrm{d}}-Y\right)= & {} \left( {\hat{y}}_\mathrm{np}^{\mathrm{a}}-Y_{a}\right) +\left( {\hat{y}}_{\mathrm{np}}^\mathrm{b}-Y_{b}\right) \nonumber \\&+ \eta \left( {\hat{y}}_{\mathrm{np}}^\mathrm{ab}-Y_{ab}\right) +(1-\eta ) \left( {\hat{y}}_{\mathrm{np}}^\mathrm{ba}-Y_{ba}\right) . \end{aligned}$$

(21)

Under (A1), (A2) and (A4)–(A7), $\lambda$ satisfying $0 \le \lambda \le 1$ and taking expectations, Theorem 1 in Breidt and Opsomer (2000) holds for $\left( {\hat{y}}_{\mathrm{np}}^{\mathrm{a}} - Y_{a}\right)$ and the same applies to terms $\left( {\hat{y}}_\mathrm{np}^{\mathrm{b}} - Y_{b}\right)$, $\left( {\hat{y}}_{\mathrm{np}}^{\mathrm{ab}} - Y_{ab}\right)$ and $\left( {\hat{y}}_{\mathrm{np}}^{\mathrm{ba}} - Y_{ba}\right)$. Then, the result follows.

1.2 Proof of Theorem 2.

We write

$$\begin{aligned} \left( {\hat{y}}_{\mathrm{np}}^{\mathrm{d}}-Y \right)= & {} \sum _{j \in a} \big (y_j - {\widehat{m}}_{j}^A \big )( w_{A_j}I_{js_A} -1) + \sum _{j \in ab} \big (y_j - {\widehat{m}}_{j}^A \big )(w_{A_j}I_{js_A} -1)\eta \nonumber \\&\quad + \sum _{j \in b} \big (y_j - {\widehat{m}}_{j}^B \big )( w_{B_j}I_{js_B} -1) + \sum _{j \in ba} \big (y_j - {\widehat{m}}_{j}^B \big )(w_{B_j}I_{js_B} -1)(1-\eta ), \end{aligned}$$

where $I_{js_A}=1$ if $j \in s_A$ and $I_{js_A}=0$ otherwise, and $I_{js_B}=1$ if $j \in s_B$ and $I_{js_B}=0$.

The design variance is given by

$$\begin{aligned} V_d({\hat{y}}^{d}_{\mathrm{np}})= & {} E_{dA} \left( \sum _{j \in a} \big (y_j - {\widehat{m}}_{j}^A \big )( w_{A_j}I_{js_A} -1) + \sum _{j \in ab} \big (y_j - {\widehat{m}}_{j}^A \big )(w_{A_j}I_{js_A} -1)\eta \right) ^2 \nonumber \\&\quad + E_{dB} \left( \sum _{j \in b} \big (y_j - {\widehat{m}}_{j}^B \big )( w_{B_j}I_{js_B} -1) + \sum _{j \in ba} \big (y_j - {\widehat{m}}_{j}^B \big )(w_{B_j}I_{js_B} -1)(1-\eta ) \right) ^2 \nonumber \\= & {} E_{dA} \left( \sum _{j \in A} \big (y_j - {\widehat{m}}_{j}^A \big )( w_{A_j}I_{js_A} -1)\eta _{A_j}\right) ^2\\&\quad +E_{dB} \left( \sum _{j \in B} \big (y_j - {\widehat{m}}_{j}^B \big )( w_{B_j}I_{js_B} -1)(\eta _{B_j})\right) ^2 , \end{aligned}$$

because the sampling designs $d_A$ and $d_B$ are independent. Let

$$\begin{aligned}&c_A= \sum _{j \in A} (y_j-m_j)( w_{A_j}I_{js_A}-1)\eta _{A_j},&c_B= \sum _{j \in B} (y_j-m_j)( w_{B_j}I_{js_B}-1)\eta _{B_j}, \nonumber \\&t_A= \sum _{j \in A} \big (m_j- {\widehat{m}}_{j}^A\big )( w_{A_j}I_{js_A}-1)\eta _{A_j}&\text {and } t_B= \sum _{j \in B} \big (m_j- {\widehat{m}}_{j}^B\big )( w_{B_j}I_{js_B}-1)\eta _{B_j}. \end{aligned}$$

Then

$$\begin{aligned}&E_{dA} \left( \sum _{j \in A} \big (y_k - {\widehat{m}}_{k}^A \big )( w_{A_j}I_{js_A} -1)\eta _{A_j}\right) ^2 = E_{dA} (c_A +t_A)^2 \nonumber \\&\quad = E_{dA}(c_A^2) +E_{dA}(t_A^2)+ 2E_{dA}(t_A c_A) = E_{dA}(c_A^2) +o(1), \end{aligned}$$

(22)

because of lemma 5 in Breidt and Opsomer (2000), $E_{dA}(t_A^2)=o(1)$, so that $E_{dA}(t_A c_A) \le$$(E_{dA}(t_A^2) E_{dA}(c_A^2))^{1/2} =o(1)$.

Similarly $E_{dB}(t_B c_B)= E_{dB}(c_B^2) +o(1)$.

We have thus that the asymptotic variance of the estimator is given by

$$\begin{aligned} AV_d({\hat{y}}_{\mathrm{np}}^{\mathrm{d}})= E_{dA}(c_A^2)+ E_{dB}(c_B^2). \end{aligned}$$

By using the properties of the Horvitz–Thompson estimator (Horvitz and Thompson 1952) we can deduce

$$\begin{aligned} E_{dA}(c_A^2)= & {} \sum _{k,j \in A} (y_k-m_k)(y_j-m_j) \left( \sum _{s_A \ni k,j} w_{A_k}w_{A_j} (p_d(s_A) -1) \eta _{A_k}\eta _{A_j} \right) \nonumber \\= & {} \sum _{k,j \in A} (y_k-m_k)(y_j-m_j) (w_{A_k}w_{A_j} \pi _{A_{kj}} -1) \eta _{A_k}\eta _{A_j}. \end{aligned}$$

(23)

Using Theorem 3 of Breidt and Opsomer (2000) for the sampling design $d_A$, we obtain that an unbiased estimator of this variance is given by

$$\begin{aligned} \displaystyle \sum _{k,j \in s_A} \frac{\big (y_k - {\widehat{m}}_{k}^A \big )\big (y_j - {\widehat{m}}_{j}^A \big ) \left( w_{A_k}w_{A_j} \pi _{A_{kj}} -1\right) \eta _{A_k}\eta _{A_j} }{\pi _{A_{kj}}}. \end{aligned}$$

A similar expression can be derived for $E_{dB}(c_B^2)$ and then, the result follows.

1.3 Proof of Theorem 3 and 4.

Proofs of Theorems 3 and 4 are similar to proofs of Theorems 1 and 2; the value of $\eta _{A_j}$ and $\eta _{B_j}$ in frames A and B is now assumed by the factors $\frac{1}{mu_j}$ for each frame.