An alternative jackknife variance estimator when calibrating weights to adjust for unit nonresponse in a complex survey
Section 3. Linearization-based variance estimation

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When calibrating the respondent sample to the full sample with (2.2), the calibration estimator for a population total, $t={\displaystyle {\sum }_R{w}_k{y}_k}$ can be expressed as

$$\begin{array}{ll}t\hfill & ={\displaystyle \sum _{k\in S}{d}_k{z}_k^{T}b}+{\displaystyle \sum _{k\in R}{d}_{k}f(g^T{x}_k)(y_k-z_k^{T}b)}\hfill \\ \hfill & \approx {\displaystyle \sum _{k\in S}{d}_k{z}_k^{T}b}+{\displaystyle \sum _{k\in R}{d}_{k}f({\gamma }^T{x}_k)(y_k-z_k^{T}b)}+{\displaystyle \sum _{k\in R}{d}_k{f}^{\prime }(g^T{x}_k)\left[{(g-\gamma )}^T{x}_k\right]}(y_k-z_k^{T}b)\hfill \\ \hfill & ={\displaystyle \sum _{k\in S}{d}_k{z}_k^{T}b}+{\displaystyle \sum _{k\in R}{d}_{k}f({\gamma }^T{x}_k)(y_k-z_k^{T}b)}+{(g-\gamma )}^T{\displaystyle \sum _{k\in R}{d}_k{f}^{\prime }(g^T{x}_k)x_k(y_k-z_k^{T}b)}\hfill \\ \hfill & ={\displaystyle \sum _{k\in S}{d}_k{z}_k^{T}b}+{\displaystyle \sum _{k\in R}{d}_k{p}_k^{-1}(y_k-z_k^{T}b)},(3.1)\hfill \end{array}$$

where

$$b={\left[{\displaystyle \sum _{k\in R}{d}_k{f}^{\prime }(g^T{x}_k)x_k{z}_k^T}\right]}^{-1}{\displaystyle \sum _{k\in R}{d}_k{f}^{\prime }(g^T{x}_k)x_k{y}_k}.$$

The key step here is that $b$ has been defined so that ${\displaystyle {\sum }_R{d}_k}{f}_k^{\prime }(g^T{x}_k)x_k(y_k-z_k^{T}b)=0.$ Observe that $f^{\prime }(.)$ in $b$ is the derivative of the weighting-adjustment function.

Let $b^{*}$ be the probability limit of $b$ as the respondent sample (of PSUs) grows arbitrarily large. The variance of $t$ under the original design and the selection model is nearly equivalent to the variance of ${\displaystyle {\sum }_S{d}_k{q}_k^{*}},$ where

                                                       $$q_k^{*}=z_k^T{b}^{*}+p_k^{-1}(y_k-z_k^T{b}^{*})I_k,$$

and $I_k=1$ when $k$ is a unit respondent and 0 otherwise.

For many designs, $q_k^{*}$ can be approximated by replacing $b^{*}$ with $b$ and $p_k^{-1}$ with $f(g^T{x}_k),$ and the variance of ${\displaystyle {\sum }_S{d}_k{q}_k}$ estimated under the original design as if the $q_k=z_k^{T}b+f(g^T{x}_k)(y_k-z_k^{T}b)I_k$ were constants. When calibrating the respondent sample to the population with equation (2.1), the ${\displaystyle {\sum }_S{d}_k{z}_k^{T}b}$ in equation (3.1) is replaced by ${\displaystyle {\sum }_U{z}_k^{T}b},$ which does not contribute to the variance, so $q_k=f(g^T{x}_k)(y_k-z_k^{T}b)I_k.$ Either way, replacing $q_k^{*}$ with $q_k$ tends to underestimate variances with finite samples (the replacement is asymptotically ignorable) because $e_k={(y_k-z_k^{T}b)}^2$ tends to be smaller than $e_k^{*}={(y_k-z_k^T{b}^{*})}^2.$

Given a stratified multistage probability sample with $n_h$ sampled PSUs in each of $H$ strata, let $S_{hj}$ denote the subsample of elements within each PSU $j$ in stratum $h.$ A nearly unbiased linearization-based estimator for the variance of $t$ is

$$v(t)={\displaystyle \sum _{h=1}^H\frac{n_h}{n_h-1}}\left[{\displaystyle \sum _{j=1}^{n_h}{\left({\displaystyle \sum _{k\in S_{hj}}{d}_k{q}_k}\right)}^2-\frac{1}{n_h}{\left({\displaystyle \sum _{a=1}^{n_h}{\displaystyle \sum _{\kappa \in S_{ha}}{d}_{\kappa }{q}_{\kappa }}}\right)}^2}\right],(3.2)$$

where $q_k=\alpha z_k^{T}b+f(g^T{x}_k)e_k{I}_k,$ and $\alpha =1$ when the respondent sample is calibrated to the original sample and 0 when the respondent sample is calibrated to the population. As is common in practice and continued here, equation (3.2) assumes that the little is lost by treating the PSU selection within strata as if it had been drawn with replacement, obviating the need for finite population correction.

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