stratallo (version 2.2.1)

optcost: Minimum Cost Allocation in Stratified Sampling

Description

Function that determines fixed strata sample sizes that minimize total cost of the survey, under assumed level of the variance of the stratified estimator and under optional one-sided upper bounds imposed on strata sample sizes. Namely, the following optimization problem, formulated below in the language of mathematical optimization, is solved by optcost() function.

Minimize $$c(x_1,\ldots,x_H) = \sum_{h=1}^H c_h x_h$$ subject to $$\sum_{h=1}^H \frac{A^2_h}{x_h} - A_0 = V$$ $$x_h \leq M_h, \quad h = 1,\ldots,H,$$ where \(A_0,\, A_h > 0,\, c_h > 0,\, M_h > 0,\, h = 1,\ldots,H\), and \(V > \sum_{h=1}^H \frac{A^2_h}{M_h} - A_0\) are given numbers. The minimization is on \(\mathbb R_+^H\). The upper-bounds constraints \(x_h \leq M_h,\, h = 1,\ldots,H\), are optional and can be skipped. In such a case, it is only required that \(V > 0\).

Usage

optcost(V, A, A0, M = NULL, unit_costs = 1)

Value

Numeric vector with optimal sample allocations in strata.

Arguments

V

(number)
parameter \(V\) of the equality constraint. A strictly positive scalar. If M is not NULL, it is then required that V >= sum(A^2/M) - A0.

A

(numeric)
population constants \(A_1,\ldots,A_H\). Strictly positive numbers.

A0

(number)
population constant \(A_0\).

M

(numeric or NULL)
upper bounds \(M_1,\ldots,M_H\), optionally imposed on sample sizes in strata. If no upper bounds should be imposed, then M must be set to NULL.

unit_costs

(numeric)
costs \(c_1,\ldots,c_H\), of surveying one element in stratum. A strictly positive numbers. Can be also of length 1, if all unit costs are the same for all strata. In this case, the elements will be recycled to the length of bounds.

Details

The algorithm that is used by optcost() is the LRNA and it is described in Wójciak (2023). The allocation computed is valid for all stratified sampling schemes for which the variance of the stratified estimator is of the 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 \(A_0,\, A_h > 0,\, h = 1,\ldots,H\), do not depend on \(x_h,\, h = 1,\ldots,H\).

References

Wójciak, W. (2023). Another Solution of Some Optimum Allocation Problem. Statistics in Transition new series, 24(5) (in press). https://arxiv.org/abs/2204.04035

See Also

rna(), opt().

Examples

A <- c(3000, 4000, 5000, 2000)
M <- c(100, 90, 70, 80)
xopt <- optcost(1017579, A = A, A0 = 579, M = M)
xopt