stratallo (version 2.2.1)

var_st: 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)

Value

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

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 \(S_1,\ldots,S_H\).

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