make_srswor_matrix: Create a quadratic form's matrix to represent the basic...

In bschneidr/svrep: Tools for Creating, Updating, and Analyzing Survey Replicate Weights

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

Description

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.

Usage

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.

Details

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).

Value

A symmetric matrix of dimension n

bschneidr/svrep documentation built on Feb. 11, 2025, 4:24 a.m.