This kernel-based variance estimator was proposed by Breidt, Opsomer, and Sanchez-Borrego (2016),
for use with samples selected using systematic sampling or where only a single
sampling unit is selected from each stratum (sometimes referred to as "fine stratification").
Suppose there are \(n\) sampled units, and
for each unit \(i\) there is a numeric population characteristic \(x_i\)
and there is a weighted total \(\hat{Y}_i\), where
\(\hat{Y}_i\) is only observed in the selected sample but \(x_i\)
is known prior to sampling.
The variance estimator has the following form:
$$
\hat{V}_{ker}=\frac{1}{C_d} \sum_{i=1}^n (\hat{Y}_i-\sum_{j=1}^n d_j(i) \hat{Y}_j)^2
$$
The terms \(d_j(i)\) are kernel weights given by
$$
d_j(i)=\frac{K(\frac{x_i-x_j}{h})}{\sum_{j=1}^n K(\frac{x_i-x_j}{h})}
$$
where \(K(\cdot)\) is a symmetric, bounded kernel function
and \(h\) is a bandwidth parameter. The normalizing constant \(C_d\)
is computed as:
$$
C_d=\frac{1}{n} \sum_{i=1}^n(1-2 d_i(i)+\sum_{j=1}^H d_j^2(i))
$$
If \(n=2\), then the estimator is simply the estimator
used for simple random sampling without replacement.
If \(n=1\), then the matrix simply has an entry equal to 0.