var_st: Variance of the Stratified pi Estimator of the Population...

In stratallo: Optimum Sample Allocation in Stratified Sampling

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)

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.

Value

The value of the variance V_{st} for a given allocation vector x_1,\ldots,x_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

\insertRef

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)

stratallo documentation built on March 12, 2026, 5:06 p.m.