optcost {stratallo}

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)

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.

Value

Numeric vector with optimal sample allocations in strata.

Note

For stratified \pi estimator of the population total and for stratified simple random sampling without replacement design, the population parameters are as follows:

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 denotes standard deviation of a given study variable in stratum 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