3 Variance estimation

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

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It is common practice to report a variance estimate or standard error for each survey estimate. Variance estimation is also crucial for statistical inference when setting a confidence interval for an unknown parameter of interest.

The asymptotic results in Section 2 suggest a variance estimator for ${\widehat{Y}}_{\text{reg}\mathrm{,}k}$ by substituting into (2.2) estimators for unknown quantities in ${\sigma }_k^2.$ Since the total variance is a sum of $H$ within-stratum variances, without loss of generality we consider one stratum $\left(H=1\right).$ For $j=1,2,$ let

$${\widehat{D}}_{n_j}={\displaystyle \sum _{i\in S_j}}\frac{{\widehat{b}}_{ij}{\widehat{b}}_{ij}^T}{\left(n_j-1\right){\widehat{N}}_j}\mathrm{,}{\widehat{b}}_{ij}={\left[1/p_{ij}-{\widehat{N}}_j\mathrm{,}{x}_i/p_{ij}-{\widehat{X}}_j\mathrm{,}{y}_i/p_{ij}-{\widehat{Y}}_j\right]}^T\text{}\mathrm{,}i\in S_j\mathrm{,}$$

$${\widehat{a}}_{1j}=\frac{{\widehat{N}}_j{n}^{1/2}}{\widehat{N}{n}_j^{1/2}}{\left[-\left({\overline{y}}_j-{\widehat{\beta }}_j{\overline{x}}_j\right)\mathrm{,}-{\widehat{\beta }}_j\mathrm{,1}\right]}^T\mathrm{,}{\widehat{a}}_{2j}=\frac{{\widehat{N}}_j{n}^{1/2}}{\widehat{N}{n}_j^{1/2}}{\left[-\left(\overline{y}-\widehat{\beta }\overline{x}\right)\mathrm{,}-\widehat{\beta }\mathrm{,1}\right]}^T\mathrm{,}$$

$${\overline{y}}_j={\widehat{Y}}_j/{\widehat{N}}_j\mathrm{,}{\overline{x}}_j={\widehat{X}}_j/{\widehat{N}}_j\mathrm{,}\overline{y}={\displaystyle \sum _{j=1}^2}{\widehat{Y}}_j/\left({\widehat{N}}_1+{\widehat{N}}_2\right)\mathrm{,}\overline{x}={\displaystyle \sum _{j=1}^2}{\widehat{X}}_j/\left({\widehat{N}}_1+{\widehat{N}}_2\right)\mathrm{.}$$

Then, under the conditions in Theorem 1,

$${\widehat{\sigma }}_k^2={\displaystyle \sum _{j=1}^2}{\widehat{a}}_{kj}^T{\widehat{D}}_{n_j}{\widehat{a}}_{kj}{\to }_{{}_P}{\sigma }_k^2\mathrm{,}k=1,2.$$

That is, ${\widehat{\sigma }}_k^2$ is consistent for ${\sigma }_k^2.$ The results in Theorems 2 and 3 also show that ${\widehat{\sigma }}_2^2$ is a consistent variance estimator for the decision-based estimator ${\widehat{Y}}_{\text{dec}},$ because we have either ${\sigma }_1^2={\sigma }_2^2$ or $P\left({\widehat{Y}}_{\text{dec}}={\widehat{Y}}_{\text{reg}\mathrm{,2}}\right)\to 1.$

These substitution variance estimators, however, may not perform well when one of $n_1$ and $n_2$ is moderate (see Section 4). An alternative method is the bootstrap as suggested by Cheng et al. (2010). Let $\widehat{\theta }$ be the estimator under consideration. Its bootstrap variance estimator can be obtained as follows.

1. Draw a bootstrap sample $S_j^{\ast }$ as a simple random sample of size $n_j$ with replacement from $S_j,$ where $S_1^{\ast }$ and $S_2^{*}$ are independently obtained. If there are $k_j$ self-representing units in $S_j,$ as discussed in Section 4.1 below, then with-replacement samples of sizes $n_j-k_j$ are drawn, $j=1,2.$

2. The survey weights and observed data from the original data set are used to form a bootstrap data set $S_1^{*}\cup S_2^{*}.$ From this dataset, calculate the bootstrap analog ${\widehat{\theta }}^{*}$ of $\widehat{\theta }.$

3. Independently repeat the previous steps $B$ times to obtain ${\widehat{\theta }}^{*1}\mathrm{,}\dots \mathrm{,}{\widehat{\theta }}^{*B}.$ The sample variance of ${\widehat{\theta }}^{*1}\mathrm{,}\dots \mathrm{,}{\widehat{\theta }}^{*B}$ is the bootstrap variance estimator for $\widehat{\theta }.$

Under the conditions in Theorems 1-2, the bootstrap variance estimators for ${\widehat{Y}}_{\text{reg}\mathrm{,1}},$ ${\widehat{Y}}_{\text{reg}\mathrm{,2}}$ and ${\widehat{Y}}_{\text{dec}}$are consistent estimators. The proof for the bootstrap is similar to the proofs of the theorems and is omitted.

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