An alternative jackknife variance estimator when calibrating weights to adjust for unit nonresponse in a complex survey
Section 4. Jackknife variance estimation

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Let $S_{h+}$  be all the sampled elements in stratum $h.$  The conventional $h{j}^{\text{th}}$  (delete-1) jackknife replicate for an estimated total $t={\displaystyle {\sum }_R{w}_k{y}_k}$  is

                                 $$t^{(hj)}={\displaystyle \sum _{k\in R}{w}_k^{(hj)}{y}_k}={\displaystyle \sum _{k\in S}{d}_k^{(hj)}f(g^{(hj)T}{x}_k)}{I}_k{y}_k,(4.1)$$

and the $g^{(hj)}$ solve the replicate calibration equation:

${\displaystyle \sum _{k\in R}{d}_k^{(hj)}f(g^{(hj)T}{x}_k)z_k}={\displaystyle \sum _{k\in U}{z}_k}$  when the respondent sample is calibrated to $U,$  or

${\displaystyle \sum _{k\in R}{d}_k^{(hj)}f(g^{(hj)T}{x}_k)z_k}={\displaystyle \sum _{k\in S}{d}_k^{(hj)}{z}_k}$  when the respondent sample is calibrated to $S$

for each $hj.$ Observe that $t^{(hj)}$ is an estimate of the population total ${\displaystyle {\sum }_U{y}_k}$ with the PSU $hj$ removed.

The delete-1 jackknife variance estimator for $t$ is

$${\text{var}}_J(t)={\displaystyle \sum _{h=1}^H\frac{n_h-1}{n_h}}{\displaystyle \sum _{j=1}^{n_h}{(t^{(hj)}-t)}^2}.(4.2)$$

Let $e_k^{*}=(y_k-z_k^T{b}^{*}).$ Consequently,

$$\begin{array}{ll}{t}^{(hj)}-t\hfill & ={\displaystyle \sum _{k\in S}\left\{(d_k^{(hj)}-d_k)\alpha z_k^T{b}^{*}+\left[d_k^{(hj)}f(g^{(hj)T}{x}_k)-d_{k}f(g^T{x}_k)\right]I_k{e}_k^{*}\right\}}\hfill \\ \hfill & \approx {\displaystyle \sum _{k\in S}(d_k^{(hj)}-d_k)\left[\alpha z_k^T{b}^{*}+f(g^T{x}_k)I_k{e}_k^{*}\right].}\hfill \end{array}$$

From which we can conclude that ${\mathrm{var}}_J(t)$ is approximately equal to the delete-1 jackknife for ${\displaystyle {\sum }_S{d}_k{q}_k^{*}},$ where $q_k^{*}=\alpha z_k^T{b}^{*}+f(g^T{x}_k)I_k{e}_k^{*},$ based on a stratified multistage sample with $H$ strata and $n_h$ PSUs in stratum $h.$ A little algebra will show that the delete-1 jackknife for ${\displaystyle {\sum }_S{d}_k{q}_k^{*}}$ is equal to $\mathrm{var}(t)$ in equation (3.2) with $q_k^{*}$ replacing $q_k$ because

$${\displaystyle \sum _{k\in S}(d_k^{(hj)}-d_k)}{q}_k^{*}={\displaystyle \sum _{h=1}^H{\displaystyle \sum _{k\in S_{h+}}(d_k^{(hj)}-d_k)}}{q}_k^{*},$$

where

$$\begin{array}{ll}{\displaystyle \sum _{k\in S_{h+}}(d_k^{(hj)}-d_k)q_k^{*}}\hfill & ={\displaystyle \sum _{k\in S_{h+}}\frac{1}{n_h-1}}{d}_k{q}^{*}-\frac{n_h}{n_h-1}{\displaystyle \sum _{k\in S_{hj}}{d}_k{q}^{*}}\hfill \\ \hfill & =-\frac{n_h}{n_h-1}\left({\displaystyle \sum _{k\in S_{hj}}{d}_k{q}^{*}}-\frac{1}{n_h}{\displaystyle \sum _{k\in S_{h+}}{d}_k{q}^{*}}\right).\hfill \end{array}$$

Note that the contribution to the jackknife variance estimator for the $h{j}^{\text{th}}$ replicate comes mostly from the $h{j}^{\text{th}}$ PSU.

Observe that the small downward bias in finite samples caused by $q_k$ replacing $q_k^{*}$ in $\mathrm{var}(t)$ does not apply to ${\mathrm{var}}_J(t)$ in equation (4.2). The latter may have a slight tendency to be upwardly biased in finite samples because $g^{(hj)}$ and $g,$ while both consistent estimators for $\gamma ,$ need not be exactly equal.

There is sometimes a problem with computing the jackknife variance estimator ${\mathrm{var}}_J(t)$ in practice. That problem occurs when $f(.)$ is such that while there is a $g$ satisfying the calibration equation in (2.1) or (2.2), no $g^{(hj)}$ satisfies its analogue for at least one $hj$ jackknife replicate. When that happens, one can follow a suggestion in Kott (2006) and compute the $w_k^{(hj)}$ in equation (4.1) with this alternative:

$${\tilde{w}}_k^{(hj)}=d_k^{(hj)}f(g^T{x}_k)+\left[c^{(hj)}-{\displaystyle \sum _{\kappa \in R}{d}_{\kappa }^{(hj)}f(g^T{x}_{\kappa })z_{\kappa }^T}\right]{\left[{\displaystyle \sum _{\kappa \in R}{d}_{\kappa }^{(hj)}{f}^{\prime }(g^T{x}_{\kappa })x_{\kappa }{z}_{\kappa }^T}\right]}^{-1}{d}_k^{(hj)}{f}^{\prime }(g^T{x}_k)x_k,(4.3)$$

where $c^{(hj)}$ is the calibration target for the $h{j}^{\text{th}}$ replicate:

$c^{(hj)}\text{=}{\displaystyle \sum _{\kappa \in U}{z}_{\kappa }^T}$ when the respondent sample is calibrated to $U,$

and

$c^{(hj)}={\displaystyle \sum _{\kappa \in S}{d}_{\kappa }^{(hj)}{z}_{\kappa }^T}$ when the respondent sample is calibrated to $S.$

By design ${\displaystyle {\sum }_R{\tilde{w}}_k^{(hj)}{z}_k}=c^{(hj)}.$

Letting $e_k=y_k-z_k^{T}b,$ one can see that with ${\tilde{t}}^{(hj)}={\displaystyle {\sum }_R{\tilde{w}}_k^{(hj)}{y}_k},$

$$\begin{array}{ll}{\tilde{t}}^{(hj)}-t\hfill & ={\displaystyle \sum _{k\in S}(d_k^{(hj)}-d_k)}\left[\alpha z_k^{T}b+f(g^T{x}_k)I_k{e}_k\right]\hfill \\ \hfill & \approx {\displaystyle \sum _{k\in S}(d_k^{(hj)}-d_k)}\left[\alpha z_k^T{b}^{*}+f(g^T{x}_k)I_k{e}_k^{*}\right]\hfill \end{array}$$

so the alternative jackknife variance estimator ${\mathrm{var}}_{AJ}(t)$ computed with ${\tilde{t}}^{(hj)}$ in place of $t^{(hj)}$ is nearly unbiased. Observe that the only possible restriction on the computation of ${\tilde{w}}_k^{(hj)}$ is that ${\displaystyle {\sum }_R{d}_k^{(hj)}{f}^{\prime }(g^T{x}_k)x_k{z}_k^T}$ be non-singular.

Observe that equation (4.3) can be rewritten as

$${\tilde{w}}_k^{(hj)}={\tilde{d}}_k^{(hj)}\tilde{f}({\tilde{g}}^{(hj)T}{\tilde{x}}_k)(4.4)$$

where ${\tilde{d}}_k^{(hj)}=d_k^{(hj)}f(g^T{x}_k),$

$$\tilde{f}({\tilde{g}}^{(hj)T}{\tilde{x}}_k)=1+{\tilde{g}}^{(hj)T}{\tilde{x}}_k,$$         

$${\tilde{g}}^{(hj)T}=\left[c-{\displaystyle \sum _{\kappa \in R}{\tilde{d}}_{\kappa }^{(hj)}{z}_{\kappa }^T}\right]{\left[{\displaystyle \sum _{\kappa \in R}{\tilde{d}}_{\kappa }^{(hj)}{\tilde{x}}_{\kappa }{z}_{\kappa }^T}\right]}^{-1},$$

and

$${\tilde{x}}_k=\frac{f^{\prime }(g^T{x}_k)}{f(g^T{x}_k)}{x}_k.$$               

This equation treats the $h{j}^{\text{th}}$ replicate as the full sample. The weight-adjustment function $\tilde{f}(.)$ is linear, and the ${\tilde{f}}_k$ are not restricted to positive values even when the $f_k$ are. In addition, observe that even when $x_k=z_k,{\tilde{x}}_k$ will not equal $z_k$ unless $f(.)=\exp (.).$

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