4 Validity

Jae Kwang Kim and Changbao Wu

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In this section we provide some general discussion on the validity of the replication variance estimator. Let $\theta =f(t_y)$ be a finite population parameter, which is a smooth function of the population total $t_y={\displaystyle {\sum }_{i=1}^N{y}_i}.$ We assume that $\widehat{\theta }=f({\widehat{t}}_y)$ is used to estimate $\theta ,$ where ${\widehat{t}}_y$ is the Horvitz-Thompson estimator of $t_y$ defined in (2.1). The replication variance estimator of $\widehat{\theta }$ is constructed by

$v_R(\widehat{\theta })={\displaystyle \sum _{k=1}^L{c}_k}{({\widehat{\theta }}^{(k)}-\widehat{\theta })}^2,$(4.1)

where ${\widehat{\theta }}^{(k)}=f({\widehat{t}}_y^{(k)})$ and ${\widehat{t}}_y^{(k)}$ is the $k^{\text{th}}$ replicate of ${\widehat{t}}_y.$

To explore the asymptotic properties of the replication variance estimator (4.1), we assume a sequence of the finite populations and the survey samples, as described in Isaki and Fuller (1982). The finite populations and the sampling designs satisfy following regularity conditions.

C1. For any population characteristics $u_i$ with bounded second moments,

${\displaystyle \sum _{i\in S}{w}_i{u}_i{u^{\prime }}_i-{\displaystyle \sum _{i=1}^N{u}_i{u^{\prime }}_i=O_p(n^{-1/2}N).}}$

C2. The design weights are uniformly bounded. That is, $K_1<N^{-1}n{w}_i<K_2$ for all $i$ and any $n,$ where $K_1$ and $K_2$ are fixed constants.

C3. $nV(N^{-1}{\widehat{t}}_y)$ is bounded.

C4. For any $y$ with bounded fourth moments, the replication variance estimator $v_R({\widehat{t}}_y)={\displaystyle {\sum }_{k=1}^L{c}_k}{({\widehat{t}}_y^{(k)}-{\widehat{t}}_y)}^2$ satisfies

$E\left[{\{c_k{({\widehat{t}}_y^{(k)}-{\widehat{t}}_y)}^2\}}^2\right]<K{L}^{-2}{\{V({\widehat{t}}_y)\}}^2$(4.2)

for some $K,$ uniformly in $k=1,\dots ,L,$

$\underset{k}{\max }{c}_k^{-1}=O(L),$(4.3)

and

$$E\left[{\left\{\frac{v_R({\widehat{t}}_y)}{V({\widehat{t}}_y)}-1\right\}}^2\right]=o(1).\text{ (4.4)}$$

Condition (4.2) ensures that no single replicate dominate the others. Condition (4.3) controls the order of the factor $c_k.$ Condition (4.4) implies that $v_R({\widehat{t}}_y)$ is a consistent estimator of $V({\widehat{t}}_y).$ Conditions (4.2) - (4.4) were also used in Kim, Navarro and Fuller (2006).

Using the above regularity conditions, the following theorem proves the consistency of the replication variance estimator in the form of (4.1).

Theorem 2. Let $\theta =f(t_y)$ be the parameter of interest and $\widehat{\theta }=f({\widehat{t}}_y),$ where $f(\cdot )$ is a smooth function with derivative continuous at $t_y\text{}.$ Under the regularity conditions described above, the variance estimator $v_R(\widehat{\theta })$ in (4.1) satisfies

$$\frac{v_R(\widehat{\theta })}{V(\widehat{\theta })}=1+o_p(1).\text{ (4.5)}$$

Proof. See Appendix A.

We now prove the validity of the improved variance estimator $v_C({\widehat{t}}_y)$ proposed in Section 3.2. For simplicity, we assume that $v_1({\widehat{t}}_y)$ is a fully efficient estimator of the variance $V({\widehat{t}}_y)$ for ${\widehat{t}}_y={\displaystyle {\sum }_{i\in S}{w}_i{y}_i}.$ We also assume that $v_0({\widehat{t}}_y),$ defined in (3.4), satisfies

$E^{*}\{v_0({\widehat{t}}_y)\}=v_1({\widehat{t}}_y),$(4.6)

where $E^{*}(\cdot )$ denotes expectation under the random selection of the $L_0$ replicates from the $L$ sets of fully efficient replication weights, as discussed in Section 3.1. If $v_1({\widehat{t}}_y)$ is asymptotically unbiased, then $v_0({\widehat{t}}_y)$ is also asymptotically unbiased by (4.6). For the delete-a-group jackknife, condition (4.6) can be understood as $E\{v_0({\widehat{t}}_y)\}=E\{v_1({\widehat{t}}_y)\}$ and $V\{v_0({\widehat{t}}_y)\}\ge V\{v_1({\widehat{t}}_y)\}.$

Theorem 3. Assume that the initial variance estimator $v_0({\widehat{t}}_y)$ defined in (3.4) satisfies (4.6). Assume that the improved variance estimator $v_C({\widehat{t}}_y)={\displaystyle {\sum }_{k=1}^{L_0}{c}_{k0}}{({\widehat{t}}_{yc}^{(k)}-{\widehat{t}}_y)}^2$ is computed using the calibrated replication weights as described in Section 3.2, with the choice of ${\tau }_i$ satisfying $\text{Cov}({\widehat{t}}_e,{\widehat{t}}_z)\doteq 0.$ By ignoring smaller order terms, we have

$E\{v_C({\widehat{t}}_y)\}=E\{v_1({\widehat{t}}_y)\}$(4.7)

and

$V\{v_1({\widehat{t}}_y)\}\le V\{v_C({\widehat{t}}_y)\}\le V\{v_0({\widehat{t}}_y)\}.$(4.8)

Proof. See Appendix B.

For a general parameter $\theta =f(t_y),$ we let ${\widehat{\theta }}_c^{(k)}=f({\widehat{t}}_{yc}^{(k)})$ and compute $v_C(\widehat{\theta })={\displaystyle {\sum }_{k=1}^{L_0}{c}_{k0}}{({\widehat{\theta }}_c^{(k)}-\widehat{\theta })}^2.$ Validity of $v_C(\widehat{\theta })$ can be established by combining results from Theorem 2 and Theorem 3.

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