make_srswor_matrix function

Create a quadratic form's matrix to represent the basic variance estimator for a total under simple random sampling without replacement

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)

Arguments

• n: Sample size
• f: A single number between 0 and 1, representing the sampling fraction. Default value is 0.

Returns

A symmetric matrix of dimension n

Details

The basic variance estimator of a total for simple random sampling without replacement is as follows:

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 $1$).