Kernel density estimation from complex surveys in the presence of complete auxiliary information

Abstract

Auxiliary information is widely used in survey sampling to enhance the precision of estimators of finite population parameters, such as the finite population mean, percentiles, and distribution function. In the context of complex surveys, we show how auxiliary information can be used effectively in kernel estimation of the superpopulation density function of a given study variable. We propose two classes of “model-assisted” kernel density estimators that make efficient use of auxiliary information. For one class we assume that the functional relationship between the study variable Y and the auxiliary variable X is known, while for the other class the relationship is assumed unknown and is estimated using kernel smoothing techniques. Under the first class, we show that if the functional relationship can be written as a simple linear regression model with constant error variance, the mean of the proposed density estimator will be identical to the well-known regression estimator of the finite population mean. If we drop the intercept from the linear model and allow the error variance to be proportional to the auxiliary variable, the mean of the proposed density estimator matches the ratio estimator of the finite population mean. The properties of the new density estimators are studied under a combined design-model-based inference framework, which accounts for the underlying superpopulation model as well as the randomization distribution induced by the sampling design. Moreover, the asymptotic normality of each estimator is derived under both design-based and combined inference frameworks when the sampling design is simple random sampling without replacement. For the practical implementation of these estimators, we discuss how data-driven bandwidth estimators can be obtained. The finite sample properties of the proposed estimators are addressed via simulations and an example that mimics a real survey. These simulations show that the new estimators perform very well compared to standard kernel estimators which do not utilize the auxiliary information.

Appendix: Proofs for Section 2

In this section, we provide the proofs for the results presented in Sect. 2. Sketch proofs for the results presented in Sect. 3 can be found in the Web-based supplementary material.

Lemma A1

Suppose that Assumption A2 holds. Then, the estimator $\hat{f}_{\text {par}}$, defined in (2.4), is bounded and square integrable, and satisfies the inequality in (2.6).

Proof

First, note that the estimator $\hat{f}_{\text {par}}$ can be written as follows:

$$\begin{aligned} \hat{f}_{\text {par}}(y)= & {} \frac{1}{Nh}\sum _{i \in s} d_iK\left( \frac{y-y_i}{h}\right) -\frac{1}{Nh}\sum _{i \in {s}} d_iK\left( \frac{y-\hat{y}_i}{h}\right) +\frac{1}{Nh}\sum _{i \in U}K\left( \frac{y-\hat{y}_i}{h}\right) .\nonumber \\ \end{aligned}$$

(A.1)

Clearly, the kernel K is bounded and square integrable by Assumption A2. Consider the first term on the right hand side of (A.1) and notice that it is a linear combination of the same kernel function evaluated at different points. This term is square integrable since linear combinations of square integrable functions are also square integrable (see Howell 2001, p. 400). Similarly, each of the other two terms on the right hand side of (A.1) is square integrable. Therefore, the estimator $\hat{f}_{\text {par}}$ is square integrable as a function of y. Additionally, since $\hat{f}_{\text {par}}$ integrates out to one and can be negative, it satisfies the condition in (2.6). $\square$

Lemma A2

Under Assumptions A1(i), A2 and A3, the estimator $\hat{f}_{\mathrm{par}}$ has the same limiting design-based distribution as $\tilde{f}_{\mathrm{par}}$.

Proof

We use similar notation to that in Randles (1982). Let $\gamma$ be a mathematical variable, and denote $\hat{f}_{\text {par}}(y)$ as $T_N(\hat{\beta })$ and $\tilde{f}_{\text {par}}(y)$ as $T_N(\beta _{\text {U}})$. Recall the definition of the sample membership indicators: $I_i=1$ if $i\in s$ and $I_i=0$ otherwise, for all $i\in U$. Therefore, $I_i$ is a Bernoulli random variable with mean $\mathrm {E}_{\mathcal {D}}(I_i)=\pi _i$, the sample inclusion probability of unit i. Note that

$$\begin{aligned} \mathrm {E}_{\mathcal {D}}\left[ T_N(\gamma )\right]= & {} \frac{1}{N}\mathrm {E}_{\mathcal {D}}\left[ \sum _{i \in {s}}d_i\left\{ K_{h}(y-y_i)-K_{h}(y-\gamma x_i)\right\} +\sum _{i \in U}K_{h}(y-\gamma x_i)\right] \\= & {} \frac{1}{N}\mathrm {E}_{\mathcal {D}}\left[ \sum _{i \in {U}} I_i d_i\left\{ K_{h}(y-y_i)-K_{h}(y-\gamma x_i)\right\} +\sum _{i \in U}K_{h}(y-\gamma x_i)\right] \\= & {} \frac{1}{N}\sum _{i \in {U}}K_{h}(y-y_i)=f_{\text {U}}(y;h). \end{aligned}$$

Thus, the limiting mean function is

$$\begin{aligned} \mu (\gamma )=\underset{N \rightarrow \infty }{\lim }\mathrm {E}_{\mathcal {D}}\left[ T_N(\gamma )\right] =f(y), \end{aligned}$$

(A.2)

where the second equality in (A.2) follows from the consistency of $f_{\text {U}}(y)$ as an estimator for f(y) (e.g., Parzen 1962). It is clear that $\mu (\gamma )$ has a zero differential at $\gamma =\beta _{\text {U}}$. Now, it follows from Randles (1982, p. 463) that $T_N(\hat{\beta })$ and $T_N(\beta _{\text {U}})$ have the same limiting distribution in the design space, and the proof is complete. $\square$

Throughout the rest of this Appendix and the supplementary material, we use the notation $A_N\simeq B_N$ if $\lim _{N\rightarrow \infty }A_N/B_N=1$.

Proof of Theorem 2.1

Using Lemma A2 together with Assumption A5, we have [by the Corollary on p. 338 of Billingsley (1995)]

$$\begin{aligned} \mathrm {E}_{\mathcal {D}}\left[ \hat{f}_{\text {par}}(y;h)\right]\simeq & {} \mathrm {E}_{\mathcal {D}}\left[ \tilde{f}_{\text {par}}(y;h)\right] \nonumber \\= & {} \frac{1}{N}\mathrm {E}_{\mathcal {D}}\left[ \sum _{i \in {s}}d_i\left\{ K_{h}(y-y_i)-K_{h}(y-\beta _{\text {U}}x_i)\right\} +\sum _{i \in U}K_{h}(y-\beta _{\text {U}}x_i)\right] \nonumber \\= & {} \frac{1}{N}\sum _{i \in {U}}K_{h}(y-y_i)=f_{\text {U}}(y; h). \end{aligned}$$

(A.3)

From (A.3), $\hat{f}_{\text {par}}(y; h)$ is asymptotically design-unbiased for $f_{\text {U}}(y;h)$. It remains to show that the design-variance of $\hat{f}_{\text {par}}(y; h)$ approaches zero in the limit. For this, note that again using Lemma A2 together with Assumption A5, we have

$$\begin{aligned}&\mathrm {Var}_{\mathcal {D}} \left[ \hat{f}_{\text {par}}(y;h)\right] \simeq \mathrm {Var}_{\mathcal {D}}\left[ \tilde{f}_{\text {par}}(y;h)\right] \nonumber \\&\quad =\frac{1}{N^2}\mathrm {Var}_{\mathcal {D}}\left[ \sum \limits _{i \in {s}} d_i\left\{ K_{h}(y-y_i)-K_{h}(y-\beta _{\text {U}}x_i)\right\} +\sum _{i \in U}K_{h}(y-\beta _{\text {U}}x_i)\right] \nonumber \\&\quad =\frac{1}{N^2}\underset{i<j \in U}{\sum \sum }(\pi _i\pi _j-\pi _{ij})\left\{ \frac{K_{h}(y-y_i)-K_{h}(y-\beta _{\text {U}}x_i)}{\pi _i}\right. \nonumber \\&\qquad \left. -\frac{K_{h}(y-y_j)-K_{h}(y-\beta _{\text {U}}x_j)}{\pi _j}\right\} ^2\nonumber \\&\quad \le \frac{1}{N^2}\underset{i<j \in U}{\sum \sum }(\pi _i\pi _j-\pi _{ij})\left\{ \frac{K_{h}(y-y_i)}{\pi _i}+\frac{K_{h}(y-\beta _{\text {U}}x_j)}{\pi _j}\right\} ^2\nonumber \\&\quad \le \frac{1}{N^2}\underset{i<j\in U}{\max }|\pi _i\pi _j-\pi _{ij}|\underset{i<j \in U}{\sum \sum }\left[ \left\{ \frac{K_{h}(y-y_i)}{\pi _i}\right\} ^2+\left\{ \frac{K_{h}(y-\beta _{\text {U}}x_i)}{\pi _j}\right\} ^2\right. \nonumber \\&\qquad \left. +\,2\left\{ \frac{K_{h}(y-y_i)}{\pi _i}\right\} \left\{ \frac{K_{h}(y-\beta _{\text {U}}x_i)}{\pi _j}\right\} \phantom {\int ^{\int ^{\int }}}\right] \nonumber \\&\quad \le \frac{1}{N^2}\underset{i<j\in U}{\max }|\pi _i\pi _j-\pi _{ij}|\underset{i<j \in U}{\sum \sum }\left[ \left\{ h^{-1}K^*/\lambda \right\} ^2+\left\{ h^{-1}K^*/\lambda \right\} ^2+2\left\{ h^{-1}K^*/\lambda \right\} ^2\right] \nonumber \\&\quad =n\underset{i<j\in U}{\max }|\pi _i\pi _j-\pi _{ij}|\frac{1}{nh^2}\frac{(N-1)}{N}\frac{4(K^*)^2}{\lambda ^2}\longrightarrow 0, \end{aligned}$$

(A.4)

where $K^*$ is such that $|K(u)|\le K^*<\infty$ for all u, by the boundedness of K. To reach (A.4), we used the fact that the second summation in the first equality above is a finite population quantity and, hence, considered fixed with respect to the randomization distribution. The equality in (A.4) follows because $\sum _{i \in {s}}d_i\left\{ K_{h}(y-y_i)-K_{h}(y-\beta _{\text {U}}x_i)\right\}$ is the Horvitz-Thompson estimator of the finite population total $\sum _{i \in {U}}\left\{ K_{h}(y-y_i)-K_{h}(y-\beta _{\text {U}}x_i)\right\}$. The zero limit follows from Assumptions A3–A4. $\square$

Proof of Theorem 2.2

Starting with the bias statement, note that

$$\begin{aligned} \mathrm {E}_C\left[ \hat{f}_{\text {par}}(y;h)\right]= & {} \mathrm {E}_{\xi }\left\{ \mathrm {E}_{\mathcal {D}}\left[ \hat{f}_{\text {par}}(y;h)|\mathbf {w}_{\text {U}}, \mathbf {x}_{\text {U}}, \mathbf {y}_{\text {U}}\right] \right\} . \end{aligned}$$

From the proof of Theorem 2.1, we have

$$\begin{aligned} \mathrm {E}_{\mathcal {D}}\left[ \hat{f}_{\text {par}}(y;h)\right]\simeq & {} \mathrm {E}_{\mathcal {D}}\left[ \tilde{f}_{\text {par}}(y;h)\right] =f_{\text {U}}(y; h). \end{aligned}$$

(A.5)

Therefore,

$$\begin{aligned} \mathrm {E}_C\left[ \hat{f}_{\text {par}}(y;h)\right]\simeq & {} \frac{1}{N} \sum _{i \in U}\mathrm {E}_{\xi }\left[ K_{h}(y-Y_i)\right] \nonumber \\= & {} f(y)+\frac{1}{2}h^2c_Kf''(y)+o(h^2), \end{aligned}$$

(A.6)

where (A.6) is a standard result in KDE (cf. Wand and Jones 1995, p. 20).

Next, consider the variance of $\hat{f}_{\text {par}}(y)$ and note that

$$\begin{aligned} \mathrm {Var}_C\left[ \hat{f}_{\text {par}}(y)\right]= & {} \mathrm {E}_{\xi }\left\{ \mathrm {Var}_{\mathcal {D}}\left[ \hat{f}_{\text {par}}(y)|\mathbf {w}_{\text {U}}, \mathbf {x}_{\text {U}}, \mathbf {y}_{\text {U}}\right] \right\} +\mathrm {Var}_{\xi }\left\{ \mathrm {E}_{\mathcal {D}}\left[ \hat{f}_{\text {par}}(y)|\mathbf {w}_{\text {U}}, \mathbf {x}_{\text {U}}, \mathbf {y}_{\text {U}}\right] \right\} \nonumber \\&~{=:}~&J_1+J_2. \end{aligned}$$

(A.7)

Using (A.5), we have

$$\begin{aligned} J_2= & {} \mathrm {Var}_{\xi }\left[ \frac{1}{N} \sum _{i \in U}K_{h}(y-Y_i)\right] =(Nh)^{-1}d_Kf(y)+o\{(Nh)^{-1}\}, \end{aligned}$$

(A.8)

where (A.8) is a standard result in KDE (cf. Wand and Jones 1995, p. 21).

On the other hand, observe that we can rewrite (A.4) as follows:

$$\begin{aligned} \mathrm {Var}_{\mathcal {D}}\left[ \hat{f}_{\text {par}}(y;h)\right]\simeq & {} \frac{1}{N^2}\underset{i, j \in U}{\sum \sum }\left[ \frac{(\pi _{ij}-\pi _i\pi _j)}{\pi _i\pi _j} \left\{ K_{h}(y-Y_i)-K_{h}(y-\beta _{\text {U}}X_i)\right\} \right. \nonumber \\&\left. \times \left\{ K_{h}(y-Y_j)-K_{h}(y-\beta _{\text {U}}X_j)\right\} \right] . \end{aligned}$$

(A.9)

Taking $\Delta _{ij}=(\pi _{ij}-\pi _i\pi _j)/\pi _i\pi _j$, $\Delta _{i}=(1-\pi _i)/\pi _i$ and $\hat{Y}_{\text {U}i}=\beta _{\text {U}} X_i$, (A.9) can be rewritten as follows:

$$\begin{aligned} \mathrm {Var}_{\mathcal {D}}\left[ \hat{f}_{\text {par}}(y;h)\right]\simeq & {} \frac{1}{N^2}\sum _{i \in U}\Delta _{i}\left\{ K_{h}(y-Y_i)-K_{h}(y-\hat{Y}_{\text {U}i})\right\} ^2 +\frac{1}{N^2}\nonumber \\&\times \underset{i,j\in U, i\not = j}{\sum \sum }\Delta _{ij}\left\{ K_{h}(y-Y_i)-K_{h} (y-\hat{Y}_{\text {U}i})\right\} \nonumber \\&\times \left\{ K_{h}(y-Y_j)-K_{h}(y-\hat{Y}_{\text {U}j})\right\} \nonumber \\= & {} \frac{1}{N^2}\sum _{i \in U}\Delta _{i}V^2_i+\frac{1}{N^2}\underset{i,j\in U, i\not = j}{\sum \sum }\Delta _{ij}V_iV_j, \end{aligned}$$

(A.10)

where $V_l=\left\{ K_{h}(y-Y_l)-K_{h}(y-\hat{Y}_{Ul})\right\}$. Then we have

$$\begin{aligned} J_1\simeq & {} \frac{1}{N^2}\sum _{i \in U}\Delta _{i}\mathrm {E}_{\xi }(V^2_i)+\frac{1}{N^2}\underset{i,j\in U, i\not = j}{\sum \sum }\Delta _{ij}\mathrm {E}_{\xi }(V_iV_j)~{=:}~J_{11}+J_{12}. \end{aligned}$$

(A.11)

We work out each term in (A.11) separately. First, note that

$$\begin{aligned} V^2_i=\left\{ K^2_h(Y_i-y)-2K_{h}(Y_i-y)K_{h}(\hat{Y}_{\text {U}i}-y)+K^2_{h}(\hat{Y}_{\text {U}i}-y)\right\} . \end{aligned}$$

By the conditions on the derivatives of K, we can form the following Taylor expansions

$$\begin{aligned} K_{h}(\hat{Y}_{\text {U}i}-y)= & {} \frac{1}{h}K\left( \frac{Y_i-y}{h}\right) +\frac{(\hat{Y}_{\text {U}i}-Y_i)}{h^2}K'\left( \frac{Y_i-y}{h}\right) \nonumber \\&+ \frac{(\hat{Y}_{\text {U}i}-Y_i)^2}{2h^3}K{''}\left( \frac{Y_i-y}{h}\right) +\frac{(\hat{Y}_{\text {U}i}-Y_i)^3}{6h^4}K^{(3)} \left( \frac{\varrho -y}{h}\right) ,\qquad \qquad \end{aligned}$$

(A.12)

$$\begin{aligned} K^2_{h}(\hat{Y}_{\text {U}i}-y)= & {} \frac{1}{h^2}K^2\left( \frac{Y_i-y}{h}\right) +2\frac{(\hat{Y}_{\text {U}i} -Y_i)}{h^3}K\left( \frac{Y_i-y}{h}\right) K'\left( \frac{Y_i-y}{h}\right) \nonumber \\&+\frac{(\hat{Y}_{\text {U}i}-Y_i)^2}{h^4}\left[ K'\left( \frac{Y_i-y}{h} \right) \right] ^2+\frac{(\hat{Y}_{\text {U}i}-Y_i)^2}{h^4}\nonumber \\&\times K \left( \frac{Y_i{-}y}{h}\right) K{''}\left( \frac{Y_i{-}y}{h}\right) {+}\frac{(\hat{Y}_{\text {U}i}{-}Y_i)^3}{6h^5}K^{2(3)}\left( \frac{\varrho {-}y}{h}\right) , \end{aligned}$$

(A.13)

for some $\varrho$ between $Y_i$ and $\hat{Y}_{\text {U}i}$, and $K^{(3)}$ ($K^{2(3)}$) is the third derivative of K ($K^2$). Using (A.12) and (A.13), we can write

$$\begin{aligned} J_{11}= & {} \frac{1}{N^2}\sum _{i \in U}\Delta _{i}\mathrm {E}_{\xi }\left\{ K^2_h(Y_i-y)-2K_{h}(Y_i-y)K_{h} (\hat{Y}_{\text {U}i}-y)+K^2_{h}(\hat{Y}_{\text {U}i}-y)\right\} \nonumber \\= & {} \frac{1}{N^2}\sum _{i \in U}\Delta _{i}\mathrm {E}_{\xi } \left\{ \frac{1}{h^4}\hat{Y}^2_{\text {U}i}\left[ K'\left( \frac{Y_i-y}{h}\right) \right] ^2-\frac{2}{h^4}\hat{Y}_{\text {U}i}Y_i\left[ K'\left( \frac{Y_i-y}{h}\right) \right] ^2\right. \nonumber \\&\left. +\frac{1}{h^4}Y^2_i\left[ K'\left( \frac{Y_i-y}{h}\right) \right] ^2\right\} +o\left( \frac{1}{Nh^3}\right) \nonumber \\&~{=:}~&\frac{1}{N^2}\sum _{i \in U}\Delta _{i}\mathrm {E}_{\xi } \left\{ W_1-2W_2+W_3\right\} +o\left( \frac{1}{Nh^3}\right) , \end{aligned}$$

(A.14)

where the term $o\left( \{Nh^3\}^{-1}\right)$ is obtained by noting that each of $h^{-1}K^{(3)}\left( \frac{\varrho -y}{h}\right)$ and $h^{-2}K^{2(3)}\left( \frac{\varrho -y}{h}\right)$ is $O_p(1)$ (it is not hard to verify that each of $\mathrm {E}_{\xi }\big |h^{-1}K^{(3)}\left( \frac{Y-y}{h}\right) \big |$ and $\mathrm {E}_{\xi }\big |h^{-2}K^{2(3)}\left( \frac{Y-y}{h}\right) \big |$ is O(1)), and by the assumption about the finite population moments of the errors.

We now evaluate the model expectation of each term on the right-hand side of (A.14).

$$\begin{aligned} \frac{1}{N}\mathrm {E}_{\xi }(W_3)= & {} \frac{1}{Nh^4}\int _{\mathbb {R}}y^2_i \left[ K'\left( \frac{y_i-y}{h}\right) \right] ^2f(y_i)dy_i\nonumber \\= & {} \frac{1}{Nh^3}\int _{\mathbb {R}}(y+hz)^2\left[ K'\left( z\right) \right] ^2 f(y+hz)dz\nonumber \\= & {} \frac{1}{Nh^3}\int _{\mathbb {R}}\left[ K'\left( z\right) \right] ^2(y^2+2yhz +h^2z^2)\left[ f(y)+hzf'(y)+\cdots \right] dz\nonumber \\= & {} \frac{1}{Nh^3}y^2f(y)\int _{\mathbb {R}}\left[ K'\left( z\right) \right] ^2dz +\frac{1}{Nh^2}\left[ y^2f'(y)+2yf(y)\right] \int _{\mathbb {R}}z \left[ K'\left( z\right) \right] ^2dz\nonumber \\&+\frac{1}{Nh}\left[ f(y)+2yf'(y)\right] \int _{\mathbb {R}}z^2 \left[ K' \left( z\right) \right] ^2dz+o\left( \frac{1}{Nh}\right) \nonumber \\= & {} \frac{1}{Nh^3}y^2f(y)d_{K'}+\frac{1}{Nh}\left[ f(y)+2yf'(y)\right] c^{*}_{K'}+o\left( \frac{1}{Nh}\right) . \end{aligned}$$

(A.15)

where (A.15) follows from the fact that $\int _{\mathbb {R}}z \{K'\left( z\right) \}^2dz=0$.

Consider $\beta _{\text {U}}=[\sum _{i \in U}x_iy_i][\sum _{i \in U}x^2_i]^{-1}$ as defined under Eq. (2.9). It can be easily shown that under the assumptions of the theorem [cf. Fuller (2009, p. 107)]

$$\begin{aligned} \beta _{\text {U}}-\beta =O_p\left( N^{-1/2}\right) . \end{aligned}$$

(A.16)

Therefore, we can write

$$\begin{aligned} \frac{1}{N}\mathrm {E}_{\xi }(W_2)= & {} \frac{1}{Nh^4}\mathrm {E}_{\xi } \left\{ \beta _{\text {U}}X_iY_i\left[ K'\left( \frac{Y_i-y}{h}\right) \right] ^2\right\} \nonumber \\= & {} \frac{1}{Nh^4}\mathrm {E}_{\xi }\left\{ \left( \beta +O_p(N^{-1/2}) \right) X_iY_i\left[ K'\left( \frac{Y_i-y}{h}\right) \right] ^2\right\} \nonumber \\= & {} \frac{1}{Nh^4}\mathrm {E}_{\xi }\left\{ \beta X_iY_i\left[ K' \left( \frac{Y_i-y}{h}\right) \right] ^2\right\} +O\left( \frac{1}{N^{\frac{3}{2}}h^4} \right) . \end{aligned}$$

(A.17)

But

$$\begin{aligned}&\frac{1}{Nh^4}\mathrm {E}_{\xi }\left\{ X_iY_i\left[ K'\left( \frac{Y_i-y}{h} \right) \right] ^2\right\} \nonumber \\&\quad =\frac{1}{Nh^4}\iint _{\mathbb {R}}x_iy_i\left[ K'\left( \frac{y_i-y}{h} \right) \right] ^2t(x_i|y_i)f(y_i)dx_idy_i\nonumber \\&\quad =\frac{1}{Nh^3} \iint _{\mathbb {R}}w(y+hz)\left[ K'\left( z\right) \right] ^2t(w|y+hz)f(y+hz)dwdz\nonumber \\&\quad =\frac{1}{Nh^3}yf(y)d_{K'}\int _{\mathbb {R}}wt(w|y)dw\nonumber \\&\qquad +\frac{1}{Nh}c^{*}_{K'}\int _{\mathbb {R}} w\left[ yf'(y)t_2'(w|y) +f'(y)t(w|y)+f(y)t'_2(w|y)\right] dw+o\left( \frac{1}{Nh}\right) ,\nonumber \\ \end{aligned}$$

(A.18)

where $t'_2(u|v)=\partial t(u|v)/\partial v$. Substituting by (A.18) in (A.17), we get

$$\begin{aligned} \frac{1}{N}\mathrm {E}_{\xi }(W_2)= & {} \frac{1}{Nh^3}\beta yf(y)d_{K'} \int _{\mathbb {R}}xt(x|y)dx\nonumber \\&+\frac{1}{Nh}\beta c^{*}_{K'}\int _{\mathbb {R}}x\left[ yf'(y)t_2'(x|y)+f'(y) t(x|y)+f(y)t'_2(x|y)\right] dx\nonumber \\&+O\left( \frac{1}{N^{\frac{3}{2}}h^4}\right) +o\left( \frac{1}{Nh}\right) . \end{aligned}$$

(A.19)

We now consider $W_1$. Observe that using (A.16), we have

$$\begin{aligned} \frac{1}{N}\mathrm {E}_{\xi }(W_1)= & {} \frac{1}{Nh^4}\mathrm {E}_{\xi } \left\{ \beta ^2_{\text {U}}X^2_i\left[ K'\left( \frac{Y_i-y}{h}\right) \right] ^2\right\} \nonumber \\= & {} \frac{1}{Nh^4}\mathrm {E}_{\xi }\left\{ \left\{ \beta +O_p\left( N^{-1/2}\right) \right\} ^2X^2_i\left[ K'\left( \frac{Y_i-y}{h}\right) \right] ^2\right\} \nonumber \\= & {} \frac{1}{Nh^4}\mathrm {E}_{\xi }\left\{ \beta ^2 X^2_i\left[ K' \left( \frac{Y_i-y}{h}\right) \right] ^2\right\} +O\left( \frac{1}{N^{\frac{3}{2}}h^4}+\frac{1}{N^2h^4}\right) . \end{aligned}$$

(A.20)

But

$$\begin{aligned} \frac{1}{Nh^4}\mathrm {E}_{\xi }\left\{ X^2_i\left[ K'\left( \frac{Y_i-y}{h}\right) \right] ^2\right\}= & {} \frac{1}{Nh^3}f(y)d_{K'}\int _{\mathbb {R}}w^2t(w|y)dw\nonumber \\&+\frac{1}{Nh}c^{*}_{K'}f'(y)\int _{\mathbb {R}}w^2t_2'(w|y)dw\nonumber \\&+o\left( \{Nh\}^{-1}\right) . \end{aligned}$$

(A.21)

Substituting (A.21) in (A.20) gives

$$\begin{aligned} \frac{1}{N}\mathrm {E}_{\xi }(W_1)= & {} \frac{1}{Nh^3}\beta ^2 d_{K'}f(y)\int _{\mathbb {R}}w^2t(w|y)dw+\frac{1}{Nh}\beta ^2 c^{*}_{K'}f'(y)\int _{\mathbb {R}}w^2t_2'(w|y)dw\nonumber \\&+O\left( \frac{1}{N^{\frac{3}{2}}h^4}+\frac{1}{N^2h^4}\right) +o\left( \frac{1}{Nh}\right) . \end{aligned}$$

(A.22)

Now, we use (A.15), (A.19) and (A.22) in (A.14) to get

$$\begin{aligned} J_{11}= & {} \left( \frac{1}{N}\sum _{i \in U}\Delta _{i}\right) \left\{ \frac{1}{Nh^3}\left[ \beta ^2 f(y)\int _{\mathbb {R}}x^2t(x|y)dx-2\beta yf(y)\right. \right. \nonumber \\&\quad \left. \int _{\mathbb {R}}xt(x|y)dx+y^2f(y)\right] d_{K'}\nonumber \\&\left. +\frac{1}{Nh}\left[ \beta ^2 f'(y)\int _{\mathbb {R}}x^2t_2'(x|y)dx-2\beta \int _{\mathbb {R}}x[yf'(y)t_2'(x|y)\right. \right. \nonumber \\&\qquad +f'(y)t(x|y)+f(y)t'_2(x|y)]dx\nonumber \\&\left. \left. \qquad +f(y)+2yf'(y) \right] c^{*}_{K'}+O\left( \frac{1}{N^{\frac{3}{2}}h^4}+\frac{1}{N^2h^4}\right) +o\left( \frac{1}{Nh}\right) \right\} +o\left( \frac{1}{Nh^3}\right) .\nonumber \\ \end{aligned}$$

(A.23)

If we keep only terms of order $O\left( \{Nh^3\}^{-1}\right)$, (A.23) reduces to

$$\begin{aligned} J_{11}= & {} \left( \frac{1}{N}\sum _{i \in U}\Delta _{i}\right) \frac{1}{Nh^3}\left[ \beta ^2f(y)\int _{\mathbb {R}}x^2t(x|y)dx-2\beta yf(y) \int _{\mathbb {R}}xt(x|y)dx+y^2f(y)\right] d_{K'}\nonumber \\&+\,o\left( \{Nh^3\}^{-1}\right) . \end{aligned}$$

(A.24)

Next, using a Taylor expansion for $K_h(\hat{Y}_{\text {U}l}-y)$ (see (A.12)), we can write

$$\begin{aligned} J_{12}= & {} \frac{1}{N^2}\underset{i,j\in U, i\not = j}{\sum \sum }\Delta _{ij}\mathrm {E}_{\xi }\left\{ K_{h}(Y_i-y)K_{h}(Y_j-y) -2K_{h}(Y_i-y)K_{h}(\hat{Y}_{\text {U}j}-y)\right. \nonumber \\&\left. +K_{h}(\hat{Y}_{\text {U}i}-y)K_{h} (\hat{Y}_{\text {U}j}-y)\right\} \nonumber \\= & {} \frac{1}{N^2}\underset{i,j\in U, i\not = j}{\sum \sum }\Delta _{ij}\mathrm {E}_{\xi } \left\{ \frac{1}{h^2}K\left( \frac{Y_i-y}{h}\right) K\left( \frac{Y_j-y}{h}\right) -\frac{2}{h^2}K\left( \frac{Y_i-y}{h}\right) K\left( \frac{Y_j-y}{h}\right) \right. \nonumber \\&\left. -\frac{2}{h^3}(\hat{Y}_{\text {U}j}-Y_j)K \left( \frac{Y_i-y}{h}\right) K'\left( \frac{Y_j-y}{h}\right) +\frac{1}{h^2}K \left( \frac{Y_i-y}{h}\right) \right. \nonumber \\&\left. \times K\left( \frac{Y_j-y}{h}\right) +\frac{2}{h^3}(\hat{Y}_{\text {U}j}-Y_j)K\left( \frac{Y_i-y}{h}\right) K' \left( \frac{Y_j-y}{h}\right) \right. \nonumber \\&\left. +\frac{1}{h^4}(\hat{Y}_{\text {U}i}-Y_i) (\hat{Y}_{\text {U}j}-Y_j)K'\left( \frac{Y_i-y}{h}\right) K'\left( \frac{Y_j-y}{h} \right) \right\} +o\left( \frac{1}{nh^2}\right) \nonumber \\= & {} \frac{1}{N^2}\underset{i,j\in U, i\not = j}{\sum \sum }\Delta _{ij} \mathrm {E}_{\xi }\left\{ \frac{1}{h^4}\left( \hat{Y}_{\text {U}i} \hat{Y}_{\text {U}j}-2Y_i\hat{Y}_{\text {U}j}+Y_iY_j\right) K'\left( \frac{Y_i-y}{h} \right) K'\left( \frac{Y_j-y}{h}\right) \right\} +o\left( \frac{1}{nh^2}\right) \nonumber \\= & {} \frac{1}{N^2}\underset{i,j\in U, i\not = j}{\sum \sum }\Delta _{ij} \mathrm {E}_{\xi }\left\{ \frac{1}{h^4}\hat{Y}_{\text {U}i}\hat{Y}_{\text {U}j} K'\left( \frac{Y_i-y}{h}\right) K'\left( \frac{Y_j-y}{h}\right) -\frac{2}{h^4}Y_i \hat{Y}_{\text {U}j}K'\left( \frac{Y_i-y}{h}\right) \right. \nonumber \\&\left. \times K'\left( \frac{Y_j-y}{h}\right) +\frac{1}{h^4} Y_iY_jK'\left( \frac{Y_i-y}{h}\right) K'\left( \frac{Y_j-y}{h}\right) \right\} +o \left( \frac{1}{nh^2}\right) \nonumber \\=: & {} \frac{1}{N^2}\underset{i,j\in U, i\not = j}{\sum \sum }\Delta _{ij} \mathrm {E}_{\xi }\left\{ H_1-2H_2+H_3\right\} +o\left( \frac{1}{nh^2}\right) . \end{aligned}$$

(A.25)

Starting with $H_3$, note that

$$\begin{aligned} \frac{1}{n}\mathrm {E}_{\xi }(H_3)= & {} \frac{1}{nh^4}\mathrm {E}_{\xi }\left[ Y_iY_jK'\left( \frac{Y_i-y}{h}\right) K'\left( \frac{Y_j-y}{h}\right) \right] \nonumber \\&\overset{iid}{=}&\frac{1}{nh^4}\left\{ \mathrm {E}_{\xi }\left[ Y_1K'\left( \frac{Y_1-y}{h}\right) \right] \right\} ^2\nonumber \\= & {} \frac{1}{nh^4}\left\{ \int _{\mathbb {R}}y_1K'\left( \frac{y_1-y}{h}\right) f(y_1)dy_1\right\} ^2\nonumber \\= & {} \frac{1}{nh^4}\left\{ h\int _{\mathbb {R}}(y+hz)K'(z)f(y+hz)dz\right\} ^2\nonumber \\= & {} \frac{1}{nh^2}\left\{ yf(y)\int _{\mathbb {R}}K'(z)dz+h[f(y)+yf'(y)]\int _{\mathbb {R}}zK'(z)dz+o(h)\right\} ^2\nonumber \\= & {} \frac{1}{nh^2}\left\{ h[f(y)+yf'(y)]\int _{\mathbb {R}}zK'(z)dz+o(h)\right\} ^2=O\left( \frac{1}{n}\right) , \end{aligned}$$

(A.26)

where the first equality in (A.26) follows from the fact that $\int _{\mathbb {R}}K'(z)dz=0$, by the assumptions on K.

Next, consider $H_1$ and note that

$$\begin{aligned} \frac{1}{n}\mathrm {E}_{\xi }(H_1)= & {} \frac{1}{nh^4}\mathrm {E}_{\xi }\left[ \beta ^2_{\text {U}}X_iX_jK'\left( \frac{Y_i-y}{h}\right) K'\left( \frac{Y_j-y}{h}\right) \right] \nonumber \\&\overset{iid}{=}&\frac{1}{nh^4}\left\{ \mathrm {E}_{\xi }\left[ \{\beta +O_p(N^{-1/2})\}^2X_1K'\left( \frac{Y_1-y}{h}\right) \right] \right\} ^2\nonumber \\= & {} \frac{\beta ^2}{nh^4}\left\{ \int \int _{\mathbb {R}}x_1K'\left( \frac{y_1-y}{h}\right) t(x_1,y_1)dx_1dy_1\right\} ^2+O(\{nNh^2\}^{-1})\nonumber \\= & {} \frac{\beta ^2}{nh^4}\left\{ h\int \int _{\mathbb {R}}wK'(z)t(w,y+hz)dwdz\right\} ^2+O(\{nNh^2\}^{-1})\nonumber \\= & {} \frac{\beta ^2}{nh^2}\left\{ \int \int _{\mathbb {R}}wK'(z)\left\{ f(y)+hzf'(y)+o(h)\right\} \right. \nonumber \\&\qquad \left. \left\{ t(w|y)+hzt'_2(w|y)+o(h)\right\} dwdz\right\} ^2\nonumber \\&+O(\{nNh^2\}^{-1})\nonumber \\= & {} \frac{\beta ^2}{nh^2}\left\{ f(y)\left( \int _{\mathbb {R}}K'(z)dz\right) \left( \int _{\mathbb {R}}m(w)t(w|y)dw\right) +h\left( \int _{\mathbb {R}}zK'(z)dz\right) \right. \nonumber \\&\times \left. \left( f(y)\int _{\mathbb {R}}m(w)t'_2(w|y)dw+f'(y)\int _{\mathbb {R}}m(w)t(w|y)dw\right) +o(h)\right\} ^2\nonumber \\&+O\left( \frac{1}{nNh^2}\right) \nonumber \\= & {} O(n^{-1}). \end{aligned}$$

(A.27)

Similar calculations show that the model expectation of $H_2$ is O(1). Further, note that Assumption A4 implies

$$\begin{aligned} \frac{n}{N^2}\sum _{i \in U}\Delta _{i}= & {} \frac{n}{N^2}\sum _{i \in U}\frac{(1-\pi _i)}{\pi _i}\le \frac{n}{N^2}\sum _{i \in U}\frac{(1-\lambda )}{\lambda }=\frac{n}{N}\frac{(1-\lambda )}{\lambda }=O(1) \end{aligned}$$

and

$$\begin{aligned} \frac{n}{N^2}\underset{i,j \in U, \ i\not = j}{\sum \sum }\Delta _{ij}= & {} \frac{n}{N^2}\underset{i,j \in U, \ i\not = j}{\sum \sum }\frac{(\pi _{ij}-\pi _i\pi _j)}{\pi _i\pi _j}\\\le & {} \frac{1}{N^2}\underset{i,j \in U, \ i\not = j}{\sum \sum }\frac{n \ \underset{i\not =j}{\max }|\pi _{ij}-\pi _i\pi _j|}{\lambda ^2}=O(1). \end{aligned}$$

Consequently,

$$\begin{aligned} J_{12}= & {} O(n^{-1}). \end{aligned}$$

(A.28)

Finally, using (A.24) and (A.28) in (A.11), we get

$$\begin{aligned} J_1= & {} \left( \frac{n}{N^2}\sum _{i \in U}\Delta _{i}\right) \frac{1}{nh^3}\left[ \beta ^2 f(y)\int _{\mathbb {R}}x^2t(x|y)dx-2\beta yf(y)\int _{\mathbb {R}}xt(x|y)dx+y^2f(y)\right] d_{K'}\nonumber \\&+\,o\left( \{Nh^3\}^{-1}\right) . \end{aligned}$$

(A.29)

Now, use (A.8) and (A.29) in (A.7) to get

$$\begin{aligned} \mathrm {Var}_C\left[ \hat{f}_{\text {par}}(y)\right]= & {} \left( \frac{n}{N^2}\sum _{i \in U}\Delta _{i}\right) \frac{1}{nh^3}\left[ \beta ^2 f(y)\int _{\mathbb {R}}x^2t(x|y)dx-2\beta yf(y)\right. \nonumber \\&\quad \left. \int _{\mathbb {R}}xt(x|y)dx+y^2f(y)\right] d_{K'} +o\left( \{Nh^3\}^{-1}\right) . \end{aligned}$$

(A.30)

Integrating (A.30) over y gives the following integrated variance of $\hat{f}_{\text {par}}(\cdot )$:

$$\begin{aligned} {\mathrm {IVar}}_C\left[ \hat{f}_{\text {par}}(\cdot )\right]= & {} \left( \frac{n}{N^2}\sum _{i \in U}\Delta _{i}\right) \frac{1}{nh^3}\left[ \beta ^2\iint _{\mathbb {R}}x^2t(x|y)f(y) dxdy\right. \nonumber \\&\left. -2\beta \iint _{\mathbb {R}}yxt(x|y)f(y)dxdy+\int _{\mathbb {R}}y^2f(y)dy\right] d_{K'}+o \left( \frac{1}{Nh^3}\right) \nonumber \\= & {} \left( \frac{n}{N^2}\sum _{i \in U}\Delta _{i}\right) \frac{1}{nh^3} \left[ \beta ^2\mu _{_{X^2}}- 2\beta \mu _{_{XY}}+\mu _{_{Y^2}}\right] d_{K'}+o\left( \frac{1}{Nh^3}\right) .\nonumber \\ \end{aligned}$$

(A.31)

Integrating the squared bias, see (2.10), gives the second part of the MISE of $\hat{f}_{\text {par}}(\cdot )$:

$$\begin{aligned} {\mathrm {ISB}}_C\left[ \hat{f}_{\text {par}}(\cdot )\right]= & {} \frac{1}{4}h^4c_K \int _{\mathbb {R}}\left\{ f''(y)\right\} ^2dy+o(h^4)\nonumber \\= & {} \frac{1}{4}h^4c_Kd_{f''}+o(h^4). \end{aligned}$$

(A.32)

Adding (A.31) to (A.32) completes the proof.

Proof of Lemma 2.1

First, rewrite the pseudo density estimator in (2.9) as follows

$$\begin{aligned} \tilde{f}_{\text {par}}(y;h)= & {} \frac{1}{N}\left[ \sum _{i\in s}d_i(u_i(h)-\tilde{v}_i(h))+\sum _{i\in U}\tilde{v}_i(h)\right] , \end{aligned}$$

where $u_i(h)=K_h(y-y_i)$ and $\tilde{v}_i(h)~{:=}~K_h(y-\beta _{\text {U}}x_i)$. Under SRS, $d_i=N/n$ for all $i\in U$. Hence,

$$\begin{aligned} \tilde{f}_{\text {par}}(y;h)= & {} \frac{1}{n}\sum _{i\in s}(u_i(h)-\tilde{v}_i(h))+\frac{1}{N}\sum _{i\in U}\tilde{v}_i(h)=\frac{1}{n}\sum _{i\in s}w_i(h), \end{aligned}$$

(A.33)

where $w_i(h)~{:=}~u_i(h)-\tilde{v}_i(h)+(1/N)\sum _{j\in U}\tilde{v}_j(h)$. Therefore, the design-variance of $\tilde{f}_{\text {par}}$ is

$$\begin{aligned} \Gamma _{\mathcal {D}}=\left( 1-\frac{n}{N}\right) \frac{\left[ \sum _{i\in U}(u_i(h)-\tilde{v}_i(h))^2-N^{-1}\{\sum _{i\in U}(u_i(h)-\tilde{v}_i(h))\}^2\right] }{n(N-1)}.\qquad \quad \end{aligned}$$

(A.34)

Since $\tilde{f}_{\text {par}}$ is a sample mean as shown in (A.33), to conclude that

$$\begin{aligned} \frac{\tilde{f}_{\text {par}}(y;h)-f_{\text {U}}(y;h)}{\Gamma ^{1/2}_{\mathcal {D}}} \overset{\mathcal {L}_{\mathcal {D}}}{\longrightarrow }\mathcal {N}(0,1), \end{aligned}$$

(A.35)

we need to verify the following Lyapunov’s condition (e.g., Thompson (1997, p. 59)):

$$\begin{aligned} \left( 1-\frac{n}{N}\right) \sum _{i\in s}\mathrm {E}_{\mathcal {D}}\big |w_i(h) -\mathrm {E}_{\mathcal {D}}[w_i(h)]\big |^{2+\eta }=o\left( \left[ n^2\left( \frac{N-1}{N}\right) \Gamma _{\mathcal {D}}\right] ^{(2+\eta )/2}\right) .\nonumber \\ \end{aligned}$$

(A.36)

Note that,

$$\begin{aligned} \mathrm {E}_{\mathcal {D}}[w_i(h)]= & {} \frac{1}{N}\sum _{i \in U}w_i(h)\\= & {} \frac{1}{N}\sum _{i \in U}[u_i(h)-\tilde{v}_i(h)+\frac{1}{N}\sum _{i\in U}\tilde{v}_j(h)]\\= & {} \frac{1}{N}\sum _{i\in U}u_i(h)=\frac{1}{N}\sum _{i\in U}K_h(y-y_i)=f_{\text {U}}(y;h). \end{aligned}$$

Therefore, by the boundedness of the kernel ($|K(u)|\le K^*<\infty$ for all u), we have

$$\begin{aligned} \left| w_i(h)-\mathrm {E}_{\mathcal {D}}[w_i(h)]\right|= & {} \left| u_i(h)-\tilde{v}_i(h) +\frac{1}{N}\sum _{i\in U}\tilde{v}_j(h)-\frac{1}{N}\sum _{i\in U}u_i(h)\right| \\= & {} \left| (u_i(h)-\tilde{v}_i(h))-\frac{1}{N}\sum _{j\in U}(u_j(h)-\tilde{v}_j(h)) \right| \\\le & {} \left| \left( \frac{K^*}{h}-0\right) -\frac{1}{N}\sum _{i\in U}\left( 0 -\frac{K^*}{h}\right) \right| =\frac{2K^*}{h}. \end{aligned}$$

Consequently,

$$\begin{aligned}&\sum _{i\in s}\mathrm {E}_{\mathcal {D}}\big |w_i(h)-\mathrm {E}_{\mathcal {D}}[w_i(h)]\big |^3\\&\quad \le \sum _{i\in s}\mathrm {E}_{\mathcal {D}}\big |w_i(h)-\mathrm {E}_{\mathcal {D}}[w_i(h)]\big |^2 \underset{i\in s}{\max }\big |w_i(h)-\mathrm {E}_{\mathcal {D}}[w_i(h)]\big |\\&\quad =\frac{2K^*}{h}\sum _{i\in s}\mathrm {E}_{\mathcal {D}}\left[ w_i(h)-\mathrm {E}_{\mathcal {D}} \{w_i(h)\}\right] ^2\\&\quad =\frac{2K^*}{h}\sum _{i\in s}\mathrm {E}_{\mathcal {D}}\left[ (u_i(h)-\tilde{v}_i(h)) -\frac{1}{N}\sum _{j\in U}(u_j(h)-\tilde{v}_j(h))\right] ^2\\&\quad =\frac{2K^*}{h}\frac{n}{N}\sum _{i\in U}\left[ (u_i(h)-\tilde{v}_i(h)) -\frac{1}{N}\sum _{j\in U}(u_j(h)-\tilde{v}_j(h))\right] ^2\\&\quad =\frac{2K^*}{h}n^2\left( \frac{N-1}{N}\right) \left( 1-\frac{n}{N}\right) ^{-1} \Gamma _{\mathcal {D}}. \end{aligned}$$

Thus,

$$\begin{aligned} \left( 1-\frac{n}{N}\right) \sum _{i\in s}\mathrm {E}_{\mathcal {D}}\big |w_i(h) -\mathrm {E}_{\mathcal {D}}[w_i(h)]\big |^3\le & {} \frac{2K^*}{h}n^2 \left( \frac{N-1}{N}\right) \Gamma _{\mathcal {D}}, \end{aligned}$$

(A.37)

and

$$\begin{aligned}&\frac{\left( 1-\frac{n}{N}\right) \sum _{i\in s}\mathrm {E}_{\mathcal {D}}\big |w_i(h) -\mathrm {E}_{\mathcal {D}}[w_i(h)]\big |^3}{\left[ n^2\left( \frac{N-1}{N}\right) \Gamma _{\mathcal {D}}\right] ^{3/2}}\nonumber \\&\quad \le \frac{2K^*}{h}n^2\left( \frac{N-1}{N}\right) \Gamma _{\mathcal {D}} \left[ n^2\left( \frac{N-1}{N}\right) \Gamma _{\mathcal {D}}\right] ^{-3/2}\nonumber \\&\quad =\frac{2K^*}{nh}\left( \frac{N-1}{N}\right) ^{-1/2} \Gamma ^{-1/2}_{\mathcal {D}}\nonumber \\&\quad =\frac{2K^*}{\sqrt{n}h}\left( 1-\frac{n}{N}\right) ^{-1/2} \left[ \frac{\sum _{i\in U}(u_i(h)-\tilde{v}_i(h))^2-N^{-1}\{\sum _{i\in U} (u_i(h)-\tilde{v}_i(h))\}^2}{N}\right] ^{-1/2}\nonumber \\&\quad \rightarrow 0, \end{aligned}$$

(A.38)

because $\sqrt{n}h\rightarrow \infty$ by assumption. Therefore, Lyapunov’s condition (A.36) holds with $\eta =1$ and the asymptotic result in (A.35) is proven. Since by Lemma A2, $\hat{f}_{\text {par}}(y;h)$ and $\tilde{f}_{\text {par}}(y;h)$ have the same limiting distribution in the design space, it follows from (A.35) and Assumption A5 that

$$\begin{aligned} \frac{\hat{f}_{\text {par}}(y;h)-f_{\text {U}}(y;h)}{\Gamma ^{1/2}_{\mathcal {D}}} \overset{\mathcal {L}_{\mathcal {D}}}{\longrightarrow }\mathcal {N}(0,1). \end{aligned}$$

(A.39)

To complete the proof of the lemma, it remains to show that $\hat{\Gamma }_{\mathcal {D}}$ is a design-consistent estimator for $\Gamma _{\mathcal {D}}$, or equivalently,

$$\begin{aligned} \big |\hat{\Gamma }_{\mathcal {D}}-\Gamma _{\mathcal {D}}\big | \overset{\mathbb {P}_{\mathcal {D}}}{\longrightarrow }0, \ \text {as} \ n \ \text {increases}. \end{aligned}$$

Consider using

$$\begin{aligned} \tilde{\sigma }^2_{\mathcal {D}}=\frac{1}{n-1}\sum _{i\in s}\left[ (u^*_i(h)-\tilde{v}^*_i(h))-\frac{1}{n}\sum _{i\in s}(u^*_i(h)-\tilde{v}^*_i(h))\right] ^2 \end{aligned}$$

(A.40)

as an estimator for

$$\begin{aligned} \sigma ^2_{\mathcal {D}}=\frac{1}{N}\sum _{i\in U}\left[ (u^*_i(h)-\tilde{v}^*_i(h))-\frac{1}{N}\sum _{i\in U}(u^*_i(h)-\tilde{v}^*_i(h))\right] ^2, \end{aligned}$$

(A.41)

where $u^*_i(h)~{:=}~hu_i(h)=K(\{y-y_i\}/h)$ and $\tilde{v}^*_i(h)~{:=}~h\tilde{v}_i(h)=K(\{y-\beta _{\text {U}} x_i\}/h)$. Note that by the boundedness of K,

$$\begin{aligned} 0<\sigma ^2_{\mathcal {D}}\le \frac{1}{N}\sum _{i\in U}\left[ (K^*-0)-\frac{1}{N}\sum _{i\in U}(0-K^*)\right] ^2=4(K^*)^2<\infty . \end{aligned}$$

(A.42)

Similarly,

$$\begin{aligned}&\underset{i\in U}{\max }\left[ (u^*_i(h)-\tilde{v}^*_i(h))-\frac{1}{N}\sum _{i\in U}(u^*_i(h)-\tilde{v}^*_i(h))\right] ^2/N\sigma ^2_{\mathcal {D}}\nonumber \\&\quad \le \frac{4(K^*)^2}{N\sigma ^2_{\mathcal {D}}}\rightarrow 0 \ \text {as} \ \ N\rightarrow \infty . \end{aligned}$$

(A.43)

The bounds in (A.42) and (A.43) imply that conditions (2.10) and (2.12) on page 294 of Sen (1988) are satisfied. Thus, using the result in (2.19) of Sen (1988), we have

$$\begin{aligned} \big |\tilde{\sigma }^2_{\mathcal {D}}-\sigma ^2_{\mathcal {D}}\big | \overset{\mathbb {P}_{\mathcal {D}}}{\longrightarrow }0, \quad \text {as} \ n \ \text {increases}. \end{aligned}$$

(A.44)

Consequently, taking $\tilde{\Gamma }_{\mathcal {D}}=(1-n/N)\tilde{\sigma }^2_{\mathcal {D}}/nh^2$, we have

$$\begin{aligned} \big |\tilde{\Gamma }_{\mathcal {D}}-\Gamma _{\mathcal {D}}\big |= & {} \frac{1}{nh^2} \left( 1-\frac{n}{N}\right) \Big |\tilde{\sigma }^2_{\mathcal {D}}-\frac{N}{(N-1)}\sigma ^2_{\mathcal {D}}\Big |\overset{\mathbb {P}_{\mathcal {D}}}{\longrightarrow }0, \ \text {as} \ n \ \text {increases}.\qquad \qquad \end{aligned}$$

(A.45)

Define

$$\begin{aligned} \hat{\sigma }^2_{\mathcal {D}}=\frac{1}{n-1}\sum _{i\in s}\left[ (u^*_i(h)-v^*_i(h))-\frac{1}{n}\sum _{i\in s}(u^*_i(h)-v^*_i(h))\right] ^2, \end{aligned}$$

where $v^*_i(h)~{:=}~hv_i(h)=K(\{y-\hat{\beta }x_i\}/h)$, and observe that the only difference between $\hat{\sigma }^2_{\mathcal {D}}$ and $\tilde{\sigma }^2_{\mathcal {D}}$ is that $\hat{\sigma }^2_{\mathcal {D}}$ depends on $\hat{\beta }$ through $v^*_i(h)$ while $\tilde{\sigma }^2_{\mathcal {D}}$ depends on $\beta _{\text {U}}$ through $\tilde{v}^*_i(h)$. Since, in the design space, $\hat{\beta }$ consistently estimates $\beta _{\text {U}}$, as we discussed in Sect. 2.1, the results of Randles (1982) imply that $\hat{\sigma }^2_{\mathcal {D}}$ and $\tilde{\sigma }^2_{\mathcal {D}}$ share the same limiting properties. Additionally, it is readily seen that $\hat{\Gamma }_{\mathcal {D}}=(1-n/N)\hat{\sigma }^2_{\mathcal {D}}/nh^2$. Therefore, (A.45) implies that

$$\begin{aligned} \big |\hat{\Gamma }_{\mathcal {D}}-\Gamma _{\mathcal {D}}\big |= & {} \frac{1}{nh^2} \left( 1-\frac{n}{N}\right) \Big |\hat{\sigma }^2_{\mathcal {D}}-\frac{N}{(N-1)} \sigma ^2_{\mathcal {D}}\Big |\overset{\mathbb {P}_{\mathcal {D}}}{\longrightarrow }0, \ \text {as} \ n \ \text {increases}.\nonumber \\ \end{aligned}$$

(A.46)

The proof of the lemma is complete upon using the results in (A.35) and (A.46) in conjunction with Slutsky’s theorem. $\square$

Proof of Theorem 2.3

First, note that

$$\begin{aligned} \sqrt{nh^3}[\hat{f}_{\text {par}}(y;h)-\mathrm {E}_{\xi }\{f_{\text {U}}(y;h)\}]= & {} \sqrt{nh^3}[\hat{f}_{\text {par}}(y;h)-f_{\text {U}}(y;h)]\nonumber \\&+\sqrt{nh^3}[f_{\text {U}}(y;h)-\mathrm {E}_{\xi }\{f_{\text {U}}(y;h)\}].\nonumber \\ \end{aligned}$$

(A.47)

Now, we have

$$\begin{aligned} \sqrt{nh^3}[f_{\text {U}}(y;h)-\mathrm {E}_{\xi }\{f_{\text {U}}(y;h)\}]= & {} h\sqrt{\frac{n}{N}}\sqrt{Nh}[f_{\text {U}}(y;h)-\mathrm {E}_{\xi } \{f_{\text {U}}(y;h)\}]\overset{\mathbb {P}_{\xi }}{\longrightarrow }0,\nonumber \\ \end{aligned}$$

(A.48)

where (A.48) follows by using the assumptions $h\rightarrow 0$ and $n/N\rightarrow \pi$, and the fact that $\sqrt{Nh}[f_{\text {U}}(y;h)-\mathrm {E}_{\xi }\{f_{\text {U}}(y;h)\}]$ converges to $\mathcal {N}(0,f(y)d_K)$ in the model space [note that $f_{\text {U}}$ is the standard KDE from an iid sample where the sample is the entire finite population and see Parzen (1962, p. 1069)]. Moreover, from (A.39), $[\hat{f}_{\text {par}}(y;h)-f_{\text {U}}(y;h)]\Gamma ^{-1/2}_{\mathcal {D}}$ converges to $\mathcal {N}(0,1)$ distribution in the design space where $\Gamma _{\mathcal {D}}$ is defined in (A.34). Using (A.34) and (A.41), we write $\Gamma _{\mathcal {D}}=(1-n/N)[N/(N-1)]\sigma ^2_{\mathcal {D}}/nh^2$. Using results from the proof of Theorem 2.2, see Eqs. (A.24) and (A.28), it can be shown that

$$\begin{aligned} \mathrm {E}_{\xi }\left[ \frac{1}{nh^2}\sigma ^2_{\mathcal {D}}\right]= & {} \frac{1}{nh^3}\left[ \beta ^2\mu _{_{X^2|y}}-2\beta \mu _{_{X|y}}y+y^2\right] f(y)d_{K'}+o\left( \frac{1}{Nh^3}\right) . \end{aligned}$$

Thus,

$$\begin{aligned} \mathrm {E}_{\xi }(\Gamma _{\mathcal {D}})= & {} \left( 1-\frac{n}{N}\right) \frac{N}{(N-1)}\frac{1}{nh^3}\left[ \beta ^2\mu _{_{X^2|y}}-2\beta \mu _{_{X|y}}y+y^2\right] f(y)d_{K'}+o\left( \frac{1}{Nh^3}\right) \\= & {} \frac{N}{(N-1)}\mathrm {Var}_C\{\hat{f}_{\text {par}}(y;h)\}. \end{aligned}$$

Applying Theorem 5.1 of Bleuer and Kratina (2005) followed by an application of Slutsky’s theorem completes the proof.