stratallo (version 3.0.1)

var_st: Variance of the Stratified \(\pi\) Estimator of the Population Total

Description

Computes the value of the variance function of the stratified \(\pi\) estimator of the population total, which has the following generic form: $$V_{st} = \sum_{h=1}^H \frac{A_h^2}{x_h} - A_0,$$ where \(H\) denotes the total number of strata, \(x_1,\ldots,x_H\) are the stratum sample sizes, and \(A_0\) and \(A_h > 0\), for \(h = 1,\ldots,H\), are population constants that do not depend on the \(x_h\).

Usage

var_st(x, A, A0)
var_stsi(x, N, S)

Value

The value of the variance \(V_{st}\) for a given allocation vector \(x_1,\ldots,x_H\).

Arguments

x

(numeric)
sample allocations \(x_1,\ldots,x_H\).

A

(numeric)
population constants \(A_1,\ldots,A_H\).

A0

(numeric(1))
population constant \(A_0\).

N

(integerish)
strata sizes \(N_1,\ldots,N_H\).

S

(numeric)
strata standard deviations of a given study variable \(S_1,\ldots,S_H\).

Functions

• var_st(): The value of the variance \(V_{st}\).

• var_stsi(): The value of the variance \(V_{st}\) for the case of simple random sampling without replacement design within each stratum.

  This particular case yields: $$A_h = N_h S_h, \qquad h = 1,\ldots,H,$$ $$A_0 = \sum_{h=1}^H N_h S_h^2,$$ where \(N_h\) denotes the size of stratum \(h\) and \(S_h\) is the corresponding stratum standard deviation of the study variable, for \(h = 1,\ldots,H\).

References

Sarndalstratallo

Examples

N <- c(300, 400, 500, 200)
S <- c(2, 5, 3, 1)
x <- c(27, 88, 66, 9)
A <- N * S
A0 <- sum(N * S^2)

var_st(x, A, A0)

N <- c(3000, 4000, 5000, 2000)
S <- rep(1, 4)
M <- c(100, 90, 70, 80)
x <- opt(n = 320, A = N * S, M = M)

var_stsi(x = x, N, S)