7 Some concluding remarks

Jae Kwang Kim and Changbao Wu

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Replication methods offer an asymptotically equivalent alternative to linearization methods but are operationally more convenient and flexible. We focused on population parameters that are smooth functions of means or totals. Our theoretical results and limited simulation studies showed that the proposed strategies for constructing sparse and efficient replication weights work well for variance estimation and confidence intervals. Nevertheless, there are a number of issues which require further investigation. First, for complex parameters such as population correlation coefficients, sparse replication variance estimators are not very stable. Second, further evidences on the effectiveness of the proposed strategies for large complex surveys in conjunction to the use of general bootstrap or jackknife weights are needed. Third, it is not clear whether the sparse replication weights will be efficient for parameters that are not smooth functions of means or totals, such as population quantiles, for which normal theory confidence intervals are known to be inefficient (Sitter and Wu 2001).

Another important issue is the potential application of the proposed methods for parameters and estimators defined through estimating equations. Let $\theta$ be defined as the solution to

$U_N(\theta )={\displaystyle \sum _{i=1}^N{u}_i(y_i,x_i;\theta )}=0.$(7.1)

Let $\widehat{\theta }$ be obtained by solving a sample-based version of (7.1) given by

$U_n(\theta )={\displaystyle \sum _{i\in S}{w}_i{u}_i(y_i,x_i;\theta )}=0.$(7.2)

Regression or logistic regression analyses using complex survey data can both be viewed special cases of the general forms given by (7.1) and (7.2). The usual sandwich-type variance of $\widehat{\theta }$ is given by

$$V(\widehat{\theta })\doteq {\left\{\frac{\partial U_N(\theta )}{\partial \theta }\right\}}^{-1}V\{U_n(\theta )\}{\left\{\frac{\partial U_N(\theta )}{\partial \theta }\right\}}^{-1}\text{ (7.3)}$$

A variance estimator $v(\widehat{\theta })$ can now be obtained if we substitute $\partial U_N(\theta )/\partial \theta$ by $\partial U_n(\theta )/\partial \theta$ at $\theta =\widehat{\theta }$ and estimate $V\{U_n(\theta )\}$ by applying replication variance estimation method to ${\widehat{U}}_n={\displaystyle {\sum }_{i\in S}{w}_i{u}_i}$ with $u_i=u_i(y_i,x_i;\widehat{\theta }).$ For detailed discussions on estimating equations and survey sampling, see, for instance, Binder (1983), Skinner (1989), and Godambe and Thompson (2009), among others.

Achieving efficient variance estimation using a limited number of sets of replication weights is an important research problem with both theoretical and practical significance. The fully efficient replication weights constructed using the procedure described in Section 2 can be treated as initial sets of weights if the sample size $n$ is large. In principle, our proposed strategies in Section 3 for producing sparse and efficient replication weights can be combined with other initial sets of replication weights, including bootstrap weights (Shao 1996) or delete-a-group jackknife (Kott 2001). One should also include as many relevant variables as possible in the calibration step, so that the final calibrated replication weights are not only sparse but also efficient in providing variance estimators for a large class of estimators. Extensions of the proposed methods to handle calibration weights or nonresponse adjustment are currently under investigation.

Acknowledgements

We thank two anonymous referees and the associate editor for their very helpful comments. This work started with initial discussions between the first author J.K. Kim and Professor Randy Sitter of Simon Fraser University who was tragically lost at sea during a kayak trip in 2007. The authors would like to dedicate this paper to the memory of Professor Sitter who was also the PhD thesis supervisor of the second author C. Wu. The research of J.K. Kim was partially supported by a Cooperative Agreement between the US Department of Agriculture Natural Resources Conservation Service and Iowa State University. The research of C. Wu was supported by grants from the Natural Sciences and Engineering Research Council of Canada and Mathematics of Information Technology and Complex Systems.

Appendix

A  Proof of Theorem 2

By assumption (4.2), we have

$\underset{1\le k\le L}{\max }{c}_k{({\widehat{t}}_y^{(k)}-{\widehat{t}}_y)}^2=O_p(L^{-1}{n}^{-1}{N}^2),$

which, combined with (4.3), implies that

$\underset{1\le k\le L}{\max }({\widehat{\mu }}_y^{(k)}-{\widehat{\mu }}_y)=o_p(1),$(A.1)

where ${\widehat{\mu }}_y^{(k)}=N^{-1}{\widehat{t}}_y^{(k)}$ and ${\widehat{\mu }}_y=N^{-1}{\widehat{t}}_y.$ Let $g({\mu }_y)=f(N{\mu }_y).$ We can write

${\widehat{\theta }}^{(k)}-\widehat{\theta }=g({\widehat{\mu }}_y^{(k)})-g({\widehat{\mu }}_y)=\dot{g}({\widehat{\mu }}_y)({\widehat{\mu }}_y^{(k)}-{\widehat{\mu }}_y)+Q_{nk}({\widehat{\mu }}_y^{(k)}-{\widehat{\mu }}_y),$

where $\dot{g}(\mu )=\partial g(\mu )/\partial \mu ,Q_{nk}=\dot{g}({\mu }_k^{*})-\dot{g}({\widehat{\mu }}_y),$ and ${\mu }_k^{*}$ is an inner point on the line segment between ${\widehat{\mu }}^{(k)}$ and $\widehat{\mu }.$ By (A.1), we have

$\underset{1\le k\le L}{\max }({\mu }_k^{*}-{\widehat{\mu }}_y)=o_p(1).$(A.2)

Define

$D_{\delta }=\left\{\mu |\underset{k}{\max }\Vert {\mu }_k^{*}-\mu \Vert <\delta \text{and}\underset{k}{\max }\Vert \dot{g}({\mu }_k^{*})-\dot{g}(\mu )\Vert >ϵ\right\}.$

By construction, we have, for any $ϵ>0$ and $\delta >0,$

$P\left\{\underset{k}{\max }\Vert \dot{g}({\mu }_k^{*})-\dot{g}({\widehat{\mu }}_y)\Vert >ϵ\right\}\le P({\widehat{\mu }}_y\in D_{\delta })+P\left(\underset{k}{\max }\Vert {\mu }_k^{*}-{\widehat{\mu }}_y\Vert \ge \delta \right).$

By the continuity of $\dot{g}(\mu )$ at $\mu ={\mu }_y$ and the fact that ${\widehat{\mu }}_y={\mu }_y+o_p(1),$ we have that, for any $ϵ>0,$ there exists a $\delta =\delta (ϵ)>0$  such that $P({\widehat{\mu }}_y\in D_{\delta })=o(1).$ This, together with (A.2), implies that

$\underset{k}{\max }\Vert \dot{g}({\mu }_k^{*})-\dot{g}({\widehat{\mu }}_y)\Vert =o_p(1).$(A.3)

Now, we have

${\displaystyle \sum _{k=1}^L{c}_k}{({\widehat{\theta }}^{(k)}-\widehat{\theta })}^2=A_n+B_n+2C_n,$(A.4)

where

$$\begin{array}{c}{A}_n={\displaystyle {\sum }_{k=1}^L{c}_k\{\dot{g}({\widehat{\mu }}_y)({\widehat{\mu }}_y^{(k)}-{\widehat{\mu }}_y)}{\}}^2,\\ B_n={\displaystyle {\sum }_{k=1}^L{c}_k\{Q_{nk}}({\widehat{\mu }}_y^{(k)}-{\widehat{\mu }}_y){\}}^2,\text{and}\\ C_n={\displaystyle {\sum }_{k=1}^L{c}_k\dot{g}({\widehat{\mu }}_y){({\widehat{\mu }}_y^{(k)}-{\widehat{\mu }}_y)}^2}{Q}_{nk}.\end{array}$$

Note that (4.4) implies

${\displaystyle \sum _{k=1}^L{c}_k}{({\widehat{\mu }}_y^{(k)}-{\widehat{\mu }}_y)}^2/V({\widehat{\mu }}_y)=1+o_p(1).$(A.5)

By standard linearization arguments, we have $A_n/V(\widehat{\theta })\to 1$ in probability. Furthermore, by (A.3) and (A.5), we have $B_n/V(\widehat{\theta })=o_p(1)$ and $C_n/V(\widehat{\theta })=o_p(1).$ This establishes (4.5).

B  Proof of Theorem 3

Combining (3.10) and (3.11) and ignoring terms of smaller order, we have

$v_0({\widehat{t}}_y)-v_C({\widehat{t}}_y)\doteq {\widehat{\beta }}^{\prime }{v}_0({\widehat{t}}_z)\widehat{\beta }-\widehat{{\beta }^{\prime }}{v}_1({\widehat{t}}_z)\widehat{\beta }\doteq {\beta }^{\prime }{v}_0({\widehat{t}}_z)\beta -{\beta }^{\prime }{v}_1({\widehat{t}}_z)\beta .$

where $\beta$ is the probability limit of $\widehat{\beta }.$ By (4.6), we have

$E^{*}\{v_0({\widehat{t}}_z)\}=v_1({\widehat{t}}_z),$(B.1)

where $E^{*}(\cdot )$ denotes expectation under random selection of the $L_0$ sets of weights conditional on the $L$ sets of weights. Similarly, by (3.11), we have

$v_1({\widehat{t}}_y)-v_C({\widehat{t}}_y)\doteq v_1({\widehat{t}}_e)-v_0({\widehat{t}}_e).$

By (4.6) again, we have

$E^{*}\{v_0({\widehat{t}}_e)\}=v_1({\widehat{t}}_e).$(B.2)

Let ${\widehat{d}}_1=v_C({\widehat{t}}_y)-v_1({\widehat{t}}_y),$ we have $E({\widehat{d}}_1)=0$ by (B.2), which proves (4.7). Furthermore, by (B.2) again, we have $\text{Cov}\{{\widehat{d}}_1,v_1({\widehat{t}}_y)\}=0.$ Thus, we have

$V\{v_C({\widehat{t}}_y)\}=V\{v_1({\widehat{t}}_y)\}+V({\widehat{d}}_1)\ge V\{v_1({\widehat{t}}_y)\}.$(B.3)

Similarly, we can also prove that $V\{v_0({\widehat{t}}_y)\}\ge V\{v_C({\widehat{t}}_y)\}.$

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