An alternative jackknife variance estimator when calibrating weights to adjust for unit nonresponse in a complex survey
Section 2. Calibration weighting

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Suppose we have a randomly drawn sample $S$ from a finite population $U.$ In the absence of nonresponse (as well as coverage error and measurement error), calibration weighting creates a set of analysis weights, $\left\{w_k|k\in S\right\},$ not dependent on the survey values of interest that

1. are close to the original inverse-probability weights, $d_k=1/{\pi }_k$  where ${\pi }_k$  is the selection probability of the $k^{\text{th}}$  selected element; and
2. satisfy a set of linear calibration equations, one for each component of $z_k,$  a vector of auxiliary variables with known population totals:

                                                                  $${\displaystyle \sum _{k\in S}{w}_k{z}_k}={\displaystyle \sum _{k\in U}{z}_k}$$

“Close” means that as the sample grows arbitrarily large, the difference between $w_k$ and $d_k$ vanishes in probability. For a more formal treatment of the assumed asymptotic structure, see Isaki and Fuller (1982).

Most surveys experience unit nonresponse beyond a statistician’s control. One is forced to assume, either explicitly or implicitly, some type of model to adjust for the nonresponse. An outcome model (also called a “prediction model”) on a survey variable of interest usually assumes the response/nonresponse mechanism, like the sampling design, is ignorable. A response model assumes the response mechanism behaves like a phase of Poisson (i.e., independent) subsampling. Double protection means that if either the prediction or response model is specified correctly, the estimator will be nearly (i.e., asymptotically) unbiased in some sense. Here we will assume a correctly specified response model.

Let $R$ be the subset of $S$ containing respondents to the survey (for simplicity, we ignore the possibility of item nonresponse). The respondent sample can be calibrated to either the full population $U:$

$${\displaystyle \sum _{k\in R}{w}_k{z}_k}={\displaystyle \sum _{k\in U}{z}_k}(2.1)$$

or to the original sample $S:$

                                                   $${\displaystyle \sum _{k\in R}{w}_k{z}_k}={\displaystyle \sum _{k\in S}{d}_k{z}_k}.(2.2)$$

We assume a response model in which the probability of response for each $k\in U,p_k,$ is an independent function having the form $p_k=p({\gamma }^T{x}_k),$ where $p(.)$ is a smooth monotonic function, and both the known vector $x_k$ and unknown parameter vector $\gamma$ have the same number of components as $z_k.$ In much of the literature $x_k$ is equal to $z_k,$ but most of the theory still follows when it does not.

If there is a vector $g$ such that inserting $w_k=d_k/p(g^T{x}_k)$ solves either the calibration equation in (2.1) or (2.2), then $g$ is a consistent estimator for $\gamma .$ Kott and Liao (2017) describe what to do when there are fewer components in $x_k$ than in $z_k.$

The function $f(g^T{x}_k)=1/p(g^T{x}_k)$ is called the weight-adjustment function. The mean-value theorem tells us that under mild conditions $f(g^T{x}_k)-f({\gamma }^T{x}_k)\approx f^{\prime }(g^T{x}_k){(g-\gamma )}^T{x}_k.$ Consequently, as the respondent sample grows arbitrarily large $f(g^T{x}_k)$ converges to $f({\gamma }^T{x}_k)=1/p_k$ and $g$ converges to $\gamma .$

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