The usual variance estimator for simple random sampling without replacement can be represented as a quadratic form. This function determines the matrix of the quadratic form.
make_srswor_matrix(n, f = 0)A symmetric matrix of dimension n
Sample size
A single number between 0 and 1,
representing the sampling fraction. Default value is 0.
The basic variance estimator of a total for simple random sampling without replacement is as follows:
$$
\hat{v}(\hat{Y}) = (1 - f)\frac{n}{n - 1} \sum_{i=1}^{n} (y_i - \bar{y})^2
$$
where \(f\) is the sampling fraction \(\frac{n}{N}\).
If \(f=0\), then the matrix of the quadratic form has all non-diagonal elements equal to \(-(n-1)^{-1}\),
and all diagonal elements equal to \(1\). If \(f > 0\), then each element
is multiplied by \((1-f)\).
If \(n=1\), then this function returns a \(1 \times 1\) matrix whose sole element equals \(0\)
(essentially treating the sole sampled unit as a selection made with probability \(1\)).