A successive-difference variance estimator can be represented as a quadratic form. This function determines the matrix of the quadratic form.
make_sd_matrix(n, f = 0, type = "SD1")A matrix of dimension n
Number of rows or columns for the matrix
A single number between 0 and 1,
representing the sampling fraction. Default value is 0.
Either "SD1" or "SD2". See the "Details" section for definitions.
Ash (2014) describes each estimator as follows: $$ \hat{v}_{SD1}(\hat{Y}) = (1-f) \frac{n}{2(n-1)} \sum_{k=2}^n\left(\breve{y}_k-\breve{y}_{k-1}\right)^2 $$ $$ \hat{v}_{SD2}(\hat{Y}) = \frac{1}{2}(1-f)\left[\sum_{k=2}^n\left(\breve{y}_k-\breve{y}_{k-1}\right)^2+\left(\breve{y}_n-\breve{y}_1\right)^2\right] $$ where \(\breve{y}_k\) is the weighted value \(y_k/\pi_k\) of unit \(k\) with selection probability \(\pi_k\), and \(f\) is the sampling fraction \(\frac{n}{N}\).
Ash, S. (2014). "Using successive difference replication for estimating variances." Survey Methodology, Statistics Canada, 40(1), 47–59.