var_st {stratallo}

Variance of the Stratified Estimator

Description

Compute the value of the variance function V of the stratified estimator, which is of the following generic form:

\sum_{h=1}^H \frac{A^2_h}{x_h} - A_0,

where H denotes total number of strata, x_1,\ldots,x_H are strata sample sizes and A_0,\, A_h > 0,\, h = 1,\ldots,H, are population constants.

Usage

var_st(x, A, A0)

var_st_tsi(x, N, S)

Arguments

x

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

A

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

A0

(number)
population constant A_0.

N

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

S

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

Value

Value of the variance V for a given allocation vector x_1,\ldots,x_H.

Functions

• var_st_tsi(): computes value of variance V for the case of stratified \pi estimator of the population total and stratified simple random sampling without replacement design. This particular case yields:

  A_h = N_h S_h, \quad h = 1,\ldots,H,

  A_0 = \sum_{h=1}^H N_h S_h^2,

  where N_h is the size of stratum h, and S_h is stratum standard deviation of a study variable, h = 1,\ldots,H.

References

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling, Chapter 3.7 Stratified Sampling, Springer, New York.

Examples

N <- c(3000, 4000, 5000, 2000)
S <- rep(1, 4)
M <- c(100, 90, 70, 80)
xopt <- opt(n = 190, A = N * S, M = M)
var_st_tsi(x = xopt, N, S) # 1017579