optcost: Minimum-Cost Allocation in Stratified Sampling

In stratallo: Optimum Sample Allocation in Stratified Sampling

Minimum-Cost Allocation in Stratified Sampling

Description

Computes stratum sample sizes that minimize the total survey cost for a given target variance of a stratified estimator, optionally subject to one-sided upper bounds on the stratum sample sizes. Specifically, the function solves the following optimization problem:

Minimize

c(x_1,\ldots,x_H) = \sum_{h=1}^H c_h x_h

over \mathbb R_+^H, subject to

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

x_h \leq M_h, \qquad 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 upper-bound constraints x_h \leq M_h are optional. If they are not imposed, it is only required that V > 0.

Usage

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

Arguments

V	(number) parameter V in the variance constraint. If upper bounds are imposed (M is not NULL), it must satisfy V > sum(A^2/M) - A_0. Otherwise, V > 0.
A	(numeric) population constants A_1,\ldots,A_H. All values must be strictly positive.
A0	(number) population constant A_0.
M	(numeric or NULL) optional upper bounds M_1,\ldots,M_H on the stratum sample sizes. If no upper bounds are imposed, set M = NULL.
unit_costs	(numeric) costs c_1,\ldots,c_H of surveying one element in each stratum. Strictly positive values. May also be of length 1, in which case the value is recycled to match the length of bounds.

Details

The allocation is computed using the LRNA algorithm, described in \insertCiteWojciakLRNA;textualstratallo.

The solution is valid for stratified sampling designs in which the variance V_{st} of the stratified estimator can be expressed as

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

where H is the number of strata, x_1,\ldots,x_H are the stratum sample sizes, and A_0,\, A_h > 0 do not depend on x_h.

Value

A numeric vector containing the optimal sample allocation for each stratum.

Note

For the stratified \pi-estimator of the population total under stratified simple random sampling without replacement design, the parameters take the form

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 is the size of stratum h and S_h is the standard deviation of the study variable in stratum h.

References

\insertRef

WojciakLRNAstratallo

See Also

rna(), opt().

Examples

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

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