Appendix

Jun Shao, Eric Slud, Yang Cheng, Sheng Wang, and Carma Hogue

Previous

Proof of Theorem 1. Under PPS sampling, ${\pi }_i=n_j{p}_{ij}$ for unit $i\in U_j,$ and on each with-replacement draw, the sampled index $i_t\in U_j\mathrm{,}t=1,\dots \mathrm{,}{n}_j$ has $P\left(i_t=i\right)=p_{ij}$ for each $i\in U_j.$ By calculating the means and variances (under repeated sampling) of ${\widehat{N}}_j,$ ${\widehat{X}}_j,$ ${\widehat{Y}}_j,$ $N_j^{-1}{\displaystyle {\sum }_{i\in S_j}}{x}_i{y}_i/{\pi }_i$ and $N_j^{-1}{\displaystyle {\sum }_{i\in S_j}}{x}_i^2/{\pi }_i,$ we find the variances to be of order $n_j^{-1}$ by means of the limits in (C2)-(C3) and the bounds in (C4). The assertions in part (a) follow directly.

For assertion (b), we have by definition of $\widehat{\beta }$ that

$$\begin{array}{lll}\widehat{\beta }\hfill & =\hfill & \frac{{\displaystyle {\sum }_{j=1}^2{\displaystyle {\sum }_{i\in S_j}\left(x_i-{\widehat{\mu }}_{x\mathrm{,}j}+{\widehat{\mu }}_{x\mathrm{,}j}-\left({\widehat{X}}_1+{\widehat{X}}_2\right)/\left({\widehat{N}}_1+{\widehat{N}}_2\right)\right)y_i/{\pi }_i}}}{{\displaystyle {\sum }_{j=1}^2{\displaystyle {\sum }_{i\in S_j}{\left(x_i-{\widehat{\mu }}_{x\mathrm{,}j}+{\widehat{\mu }}_{x\mathrm{,}j}-\left({\widehat{X}}_1+{\widehat{X}}_2\right)/\left({\widehat{N}}_1+{\widehat{N}}_2\right)\right)}^2/{\pi }_i}}}\hfill \\ \hfill & =\hfill & \frac{N^{-1}\left({\displaystyle {\sum }_{j=1}^2{\widehat{\beta }}_j{\widehat{\sigma }}_{xj}^2{\widehat{N}}_j+\left({\widehat{N}}_1{\widehat{N}}_2/\left({\widehat{N}}_1+{\widehat{N}}_2\right)\right)\left({\widehat{\mu }}_{x1}-{\widehat{\mu }}_{x2}\right)\left({\widehat{\mu }}_{y1}-{\widehat{\mu }}_{y2}\right)}\right)}{N^{-1}\left({\displaystyle \sum _{j=1}^2}{\widehat{\sigma }}_{xj}^2{\widehat{N}}_j+\left({\widehat{N}}_1{\widehat{N}}_2/\left({\widehat{N}}_1+{\widehat{N}}_2\right)\right){\left({\widehat{\mu }}_{x1}-{\widehat{\mu }}_{x2}\right)}^2\right)}\mathrm{,}\hfill \end{array}$$

from which the equality (2.1) in (b) follows immediately by substituting the limits in part (a) along with the limits $N_j/N\to {\omega }_j.$

Let ${\Sigma }_N$ be the block diagonal matrix with two diagonal blocks $D_{N_1}$ and $D_{N_2},$ and for $j=1,2,$let

$$\begin{array}{llll}{\Omega }_{1j}\hfill & =\hfill & \frac{1}{N_j\sqrt{n_j}}{\displaystyle \sum _{i\in S_j}}\left(\frac{1}{p_{ij}}-N_j\right)\mathrm{,}{\Omega }_{2j}=\frac{1}{N_j\sqrt{n_j}}{\displaystyle \sum _{i\in S_j}}\left(\frac{x_i}{p_{ij}}-X_j\right)\mathrm{,}\hfill & (\text{A}\text{.1})\hfill \\ {\Omega }_{3j}\hfill & =\hfill & \frac{1}{N_j\sqrt{n_j}}{\displaystyle \sum _{i\in S_j}}\left(\frac{y_i}{p_{ij}}-Y_j\right)\mathrm{,}{\Omega }_{4j}=\frac{1}{N_j\sqrt{n_j}}{\displaystyle \sum _{i\in S_j}}\frac{x_i-{\mu }_{x\mathrm{,}j}}{p_{ij}}\left(y_i-{\alpha }_j-{\beta }_j{x}_i\right).\hfill & \hfill \end{array}$$

Since $S_1$ and $S_2$ are independent, ${\left\{{\Omega }_{k1}\right\}}_{k=1}^4$ is independent of ${\left\{{\Omega }_{k2}\right\}}_{k=1}^4.$ Note that, here and throughout this proof, sums over $i\in S_j$ used to define ${\widehat{X}}_j,$ ${\widehat{Y}}_j,$ ${\Omega }_{kj},$ and variance estimators should be understood as sums with multiplicity in view of the with-replacement PPS sampling framework. Condition (C4) makes Liapounov's Central Limit Theorem applicable to show that

$${\Sigma }_N^{-1/2}{\left[{\Omega }_{11}\mathrm{,}{\Omega }_{21}\mathrm{,}{\Omega }_{31}\mathrm{,}{\Omega }_{12}\mathrm{,}{\Omega }_{22}\mathrm{,}{\Omega }_{32}\right]}^T{\to }_{{}_d}N\left(\mathrm{0,}{I}_6\right)\mathrm{,}{\Omega }_{4j}{\to }_{{}_d}N\left(\mathrm{0,}{\sigma }_{xe\mathrm{,}j}^2\right)\mathrm{,}(\text{A}.2)$$

where $I_6$ is the $6\times 6$ identity matrix, and ${\sigma }_{xe\mathrm{,}j}^2$ is given in the statement of (d). The limits defining the asymptotic variances in (A.2) exist according to (C3).

Proof of (c). It is straightforward to check from the definition that

$$\left(\begin{array}{l}{\widehat{\beta }}_j-{\beta }_j\hfill \\ {\widehat{\alpha }}_j-{\alpha }_j\hfill \end{array}\right)=\frac{1}{{\widehat{N}}_j{\widehat{\sigma }}_{xj}^2}{\displaystyle \sum _{i\in S_j}}\left(\begin{array}{c}{x}_i-{\widehat{\mu }}_{xj}\\ {\widehat{\sigma }}_{xj}^2-(x_i-{\widehat{\mu }}_{xj}){\widehat{\mu }}_{xj}\end{array}\right)\frac{y_i-{\alpha }_j-{\beta }_j{x}_i}{{\pi }_i}\mathrm{.}$$

Since it was established in (a) that ${\widehat{\sigma }}_{xj}^2{\to }_{{}_P}{\sigma }_{xj}^2$ and ${\widehat{N}}_j/N_j{\to }_{{}_P}1,$ it follows that the limiting distribution of $\sqrt{n_j}\left({\widehat{\beta }}_j-{\beta }_j\right)$ is the same as that of

$${\sqrt{n}}_j{\left(N_j{\sigma }_{xj}^2\right)}^{-1}{\displaystyle \sum _{i\in S_j}}\left(x_i-{\mu }_{xj}\right)\left(y_i-{\alpha }_j-{\beta }_j{x}_i\right)/{\pi }_i\mathrm{,}$$

which is clearly the same as that of ${\sigma }_{xj}^{-2}{\Omega }_{4j}$ in (A.1). The first assertion of (c) follows immediately from (A.2). The consistency of ${\widehat{\sigma }}_{xe\mathrm{,}j}^2$ follows by noting by (a) that

$${\widehat{\sigma }}_{xe\mathrm{,}j}^2-N_j^{-2}{\displaystyle \sum _{i\in S_j}}\frac{{\left(x_i-{\mu }_{xj}\right)}^2}{{\pi }_i{p}_{ij}}{\left(y_i-{\alpha }_j-{\beta }_j{x}_i\right)}^2{\to }_{{}_P}0.(\text{A}.3)$$

The second term on the left-hand side of (A.3) has PPS with-replacement sampling variance calculated to be bounded by $1/n_j$ according to (C4), and by (C3) has expectation converging to ${\sigma }_{xe\mathrm{,}j}^2.$

Proof of (d). From (1.2) and (a), $\left({\widehat{Y}}_{\text{reg}\mathrm{,2}}-Y\right)/N{\to }_{{}_P}0,$ which can also be seen from the representation

$$\begin{array}{lll}\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,2}}-Y\right)/N\hfill & =\hfill & \frac{\sqrt{n}}{N}{\displaystyle \sum _{j=1}^2}\left[\frac{N_j{\widehat{Y}}_j}{{\widehat{N}}_j}-Y_j+{\widehat{\beta }}_j\left(X_j-\frac{N_j{\widehat{X}}_j}{{\widehat{N}}_j}\right)\right]\hfill \\ \hfill & =\hfill & \frac{\sqrt{n}{N}_1^2}{\sqrt{n_1}N{\widehat{N}}_1}\left[\left(-{\overline{Y}}_1+{\widehat{\beta }}_1{\overline{X}}_1\right){\Omega }_{11}-{\widehat{\beta }}_1{\Omega }_{21}+{\Omega }_{31}\right]\hfill \\ \hfill & +\hfill & \frac{\sqrt{n}{N}_2^2}{\sqrt{n_1}N{\widehat{N}}_2}\left[\left(-{\overline{Y}}_2+{\widehat{\beta }}_2{\overline{X}}_2\right){\Omega }_{12}-{\widehat{\beta }}_2{\Omega }_{22}+{\Omega }_{32}\right]\hfill \\ \hfill & =\hfill & d_{n1}^T{\overline{\Omega }}_1+d_{n2}^T{\overline{\Omega }}_2\mathrm{,}\hfill \end{array}$$

where the second equality follows from the notational definitions of ${\Omega }_{kj}$ along with ${\pi }_i=n_j{p}_{ij},$ ${\widehat{Y}}_j={\displaystyle {\sum }_{i\in S_j}}{y}_i/{\pi }_i\mathrm{,}{\widehat{X}}_j={\displaystyle {\sum }_{i\in S_j}}{x}_i/{\pi }_i,$ and the third from

$$d_{nj}=\frac{\sqrt{n}{N}_j^2}{\sqrt{n_j}N{\widehat{N}}_j}{\left[-{\overline{Y}}_j+{\widehat{\beta }}_j{\overline{X}}_j\mathrm{,}-{\widehat{\beta }}_j\mathrm{,1}\right]}^T\mathrm{,}{\overline{\Omega }}_1={\left[{\Omega }_{11}\mathrm{,}{\Omega }_{21}\mathrm{,}{\Omega }_{31}\right]}^T\mathrm{,}{\overline{\Omega }}_2={\left[{\Omega }_{21}\mathrm{,}{\Omega }_{22}\mathrm{,}{\Omega }_{32}\right]}^T\mathrm{.}$$

By (A.2), ${\overline{\Omega }}_1=O_p(1)$ and ${\overline{\Omega }}_2=O_p\left(1\right).$ By condition (C2), $d_{nj}^T=a_{2j}^T+o_p\left(1\right).$ Therefore, by (A.2), condition (C3) and the delta method,

$$\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,2}}-Y\right)/N=a_{21}^T{\overline{\Omega }}_1+a_{22}^T{\overline{\Omega }}_2+o_p\left(1\right){\to }_{d}N\left(\mathrm{0,}{\sigma }_2^2\right)\mathrm{,}$$

where the asymptotic variance ${\sigma }_2^2={\displaystyle {\sum }_{j=1}^2}{a}_{2j}^T{D}_j{a}_{2j}$ is consistently estimated by

$$\frac{n}{N^2}{\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i\in S_j}}\frac{1}{{\pi }_i^2}{\left(y_i-{\widehat{\beta }}_j{x}_i-\left({\widehat{Y}}_j-{\widehat{\beta }}_j{\widehat{X}}_j\right)/{\widehat{N}}_j\right)}^2\mathrm{,}$$

which agrees with formula (9) of Cheng et al. (2010). The proof that $\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,1}}-Y\right)/N{\to }_{d}N\left(\mathrm{0,}{\sigma }_1^2\right)$ is similar.

Proof of Theorem 2. By Theorem 1 conclusion (c),

$$\sqrt{n}\left({\widehat{\beta }}_2-{\widehat{\beta }}_1-{\beta }_2+{\beta }_1\right){\to }_{d}N\left(\mathrm{0,}{\displaystyle \sum _{j=1}^2}{\sigma }_{xe\mathrm{,}j}^2/\left({\phi }_j^2{\sigma }_{xj}^4\right)\right)\mathrm{.} (\text{A}.4)$$

The conclusion (2.4) in (a) of this Theorem follows immediately.

In the proof of Theorem 1, we showed that

$$\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,2}}-Y\right)/N=a_{21}^T{\overline{\Omega }}_1+a_{22}^T{\overline{\Omega }}_2+o_p\left(1\right)\mathrm{,} (\text{A}.5)$$

where the constant vectors $a_{kj}$ (and ${\mu }_x\mathrm{,}{\mu }_y$) were defined in Theorem 1 (d). Similarly,

$$\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,1}}-Y\right)/N=a_{11}^T{\overline{\Omega }}_1+a_{12}^T{\overline{\Omega }}_2+o_p\left(1\right)\mathrm{.} (\text{A}.6)$$

When (2.3) holds, ${\beta }_j=\beta$ (by Theorem 1 (b)) and ${\mu }_y-\beta {\mu }_x={\displaystyle {\sum }_{j=1}^2}{\omega }_j\left({\mu }_{yj}-{\beta }_j{\mu }_{xj}\right)={\mu }_{y2}-\beta {\mu }_{x2},$ so that $a_{1j}=a_{2j}$ for $j=1,2.$ It follows immediately from (A.5)-(A.6) that $\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,1}}-{\widehat{Y}}_{\text{reg}\mathrm{,2}}\right)/N{\to }_{{}_P}0,$ and therefore that the estimators ${\widehat{Y}}_{\text{reg}\mathrm{,}k}$ have the same asymptotic distribution, which was shown to be normal in Theorem 1 (d). Finally, the definition of ${\widehat{Y}}_{\text{dec}}$ implies that $P\left({\widehat{Y}}_{\text{dec}}={\widehat{Y}}_{\text{reg}\mathrm{,1}}\text{or}{\widehat{Y}}_{\text{reg}\mathrm{,2}}\right)=1$ and (A.5)-(A.6) imply

$$\sqrt{n}\left({\widehat{Y}}_{\text{dec}}-Y\right)/N=a_{21}^T{\overline{\Omega }}_1+a_{22}^T{\overline{\Omega }}_2+o_p\left(1\right)\mathrm{,}(\text{A}.7)$$

which completes the proof of (2.5) in (a).

Proof of (b). If ${\beta }_1\ne {\beta }_2,$ then (A.4) implies that $P\left({\widehat{Y}}_{\text{dec}}={\widehat{Y}}_{\text{reg}\mathrm{,2}}\right)\to 1,$ i.e., that the t-test for equality of ${\widehat{\beta }}_j$ rejects with certainty in the limit. Then (A.7) continues to hold, and the asymptotic distribution of ${\widehat{Y}}_{\text{dec}}$ is still as same as that of ${\widehat{Y}}_{\text{reg}\mathrm{,2}}.$

Proof of Theorem 3. In this Theorem, the hypotheses (C2)-(C4) are replaced by the assumptions that the iid triples $\left(y_i\mathrm{,}{x}_i\mathrm{,}{z}_i\right)$ satisfy moment conditions and the model (2.7). The assertions in (C2)-(C4) are then results holding with probability tending to 1 with large $n\mathrm{,}N$ which are established with the aid of the (strong) law of large numbers.

Beyond the conclusions of Theorems 1-2, it remains to show that ${\widehat{Y}}_{\text{reg}\mathrm{,2}}$ has a smaller asymptotic variance than ${\widehat{Y}}_{\text{reg}\mathrm{,1}}.$ Let $\vartheta =\left({\vartheta }_1\mathrm{,}{\vartheta }_2\right)$ and

$$F_j\left(\vartheta \right)=\left[-{\vartheta }_1\mathrm{,}-{\vartheta }_2\mathrm{,1}\right]D_j{\left[-{\vartheta }_1\mathrm{,}-{\vartheta }_2\mathrm{,1}\right]}^T.$$

According to the definition of ${\sigma }_1^2$ and ${\sigma }_2^2$ in (2.2), it suffices to show that $F_j\left(\vartheta \right)$ has its minimum value at $\vartheta =\left({\alpha }_j\mathrm{,}{\beta }_j\right).$ We now prove this for $j=1.$ The proof for $j=2$ is similar. Let $m_{i{i}^{\prime }}$ be the $\left(i\mathrm{,}{i}^{\prime }\right)$ element of $D_1.$ Since $D_1$ is symmetric and positive definite under condition (C3), $m_{12}=m_{21}$ and there exists a unique ${\theta }^{\ast }=\left({\theta }_1^{\ast }\mathrm{,}{\theta }_2^{\ast }\right)$ such that $F_1\left({\theta }^{\ast }\right)={\min }_{\vartheta }{F}_1\left(\vartheta \right)$ and $\partial F_1\left(\vartheta \right)/{\partial {\vartheta }^T|}_{\vartheta ={\theta }^{\ast }}=0.$ This implies that ${\theta }^{\ast }$ is the solution to the following two equations:

$$m_{11}{\vartheta }_1+m_{12}{\vartheta }_2=m_{13}\mathrm{,}{m}_{12}{\vartheta }_1+m_{22}{\vartheta }_2=m_{23} (\text{A}.8)$$

Therefore, it suffices to show that ${\theta }^{\ast }=\left({\alpha }_1\mathrm{,}{\beta }_1\right).$ Since $D_1$ is positive definite, the equation system (A.8) has a unique solution. By the definition of $D_1,$

$$\begin{array}{lll}{m}_{11}{\alpha }_1+m_{12}{\beta }_1\hfill & =\hfill & \underset{N_1\to \infty }{{\displaystyle \lim }}\frac{1}{N_1^2}\left[{\displaystyle \sum _{i\in U_1}}{\left(\frac{1}{p_{i1}}-N_1\right)}^2{p}_{i1}{\alpha }_1+{\displaystyle \sum _{i\in U_1}}\left(\frac{1}{p_{i1}}-N_1\right)\left(\frac{x_i}{p_{i1}}-X_1\right)p_{i1}{\beta }_1\right]\hfill \\ \hfill & =\hfill & \underset{N_1\to \infty }{{\displaystyle \lim }}\frac{1}{N_1^2}\left[{\displaystyle \sum _{i\in U_1}}\left(\frac{1}{p_{i1}}-N_1\right)\left({\alpha }_1-N_1{\alpha }_1{p}_{i1}+{\beta }_1{x}_i-{\beta }_1{p}_{i1}{X}_1\right)\right]\mathrm{,}\hfill \end{array}$$

and

$$\begin{array}{lll}{m}_{13}\hfill & =\hfill & \underset{N_1\to \infty }{{\displaystyle \lim }}\frac{1}{N_1^2}\left[{\displaystyle \sum _{i\in U_1}}\left(\frac{1}{p_{i1}}-N_1\right)\left(\frac{y_i}{p_{i1}}-Y_1\right)p_{i1}\right]\hfill \\ \hfill & =\hfill & \underset{N_1\to \infty }{{\displaystyle \lim }}\frac{1}{N_1^2}\left[{\displaystyle \sum _{i\in U_1}}\left(\frac{1}{p_{i1}}-N_1\right)\left({\alpha }_1+{\beta }_1{x}_i+{\epsilon }_i-N_1{\alpha }_1{p}_{i1}-{\beta }_1{p}_{i1}{X}_1\right)\right]\hfill \\ \hfill & =\hfill & \underset{N1\to \infty }{{\displaystyle \lim }}\frac{1}{N_1^2}\left[{\displaystyle \sum _{i\in U_1}}\left(\frac{1}{p_{i1}}-N_1\right)\left({\alpha }_1-N_1{\alpha }_1{p}_{i1}+{\beta }_1{x}_i-{\beta }_1{p}_{i1}{X}_1\right)\right]\mathrm{,}\hfill \end{array}$$

where the last equality follows from the assumption that ${\epsilon }_i$ is independent of $x_i$ and $z_i$ and has mean 0 and a finite variance, and each of the sequences $z_i,$ $1/z_i,$ and $x_i/z_i$ is iid with finite expectation. Therefore, $m_{11}{\alpha }_1+m_{12}{\beta }_1=m_{13}.$ Similarly one proves that $m_{12}{\alpha }_1+m_{22}{\beta }_2=m_{23}.$ Therefore, $\left({\alpha }_1\mathrm{,}{\beta }_1\right)$ is the unique solution to equation system (A.8), i.e., $F_1\left(\vartheta \right)$ achieves its minimum value at $\vartheta =\left({\alpha }_1\mathrm{,}{\beta }_1\right).$ Hence, ${\sigma }_2^2\text{<}{\sigma }_1^2.$ This finishes the proof of Theorem 3.

Acknowledgements

This paper describes research and analysis of the authors, and is released to inform interested parties and encourage discussion. Results and conclusions are the authors' and have not been endorsed by the Census Bureau. We would like to thank three referees and an associate editor for helpful comments and suggestions which improved the paper. Jun Shao's research was partially supported by the NSF Grant DMS-1007454.

References

Bancroft, T., and Han, C.-P. (1977). Inference based on conditional specifications: A note and a bibliography. International Statistical Review, 45, 117-127.

Cheng, Y., Corcoran, C., Barth, J. and Hogue, C. (2009). An estimation procedure for the new public employment survey design. Washington, DC: American Statistical Association. Survey Research Methods Section, American Statistical Association, 3032-3046.

Cheng, Y., Slud, E. and Hogue, C. (2010). Variance estimation for decision-based estimators with application to the annual survey of public employment and payroll. Government Statistics Section of the American Statistical Association. Vancouver: American Statistical Association.

Deville, J.-C., and Särndal, C.-E. (1992). Calibration estimators in survey sampling. Journal of the American Statistical Association, 87, 376-382.

Fuller, W.A. (2009). Sampling Statistics. New York: John Wiley & Sons, Inc.

Isaki, C., and Fuller, W. (1982). Survey design under the regression superpopulation model. Journal of the American Statistical Association, 77, 89-96.

Rao, J.N.K., and Ramachandran, V. (1974). Comparison of the separate and combined ratio estimators. Sankhyã, C, 36, 151-156.

Saleh, A.K. Md. (2006). Theory of Preliminary Test and Stein-type Estimation, with Applications. Hoboken: Wiley-Interscience.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling. New York: Springer-Verlag.

Shao, J., and Tu, D. (1995) The Jackknife and Bootstrap. New York: Springer.

Slud, E.V. (2012). Moderate-sample behavior of adaptively pooled stratified regression estimators. U.S. Census Bureau preprint.

Previous