alg_1sided: Algorithms for Optimum Sample Allocation Under One-Sided...

In stratallo: Optimum Sample Allocation in Stratified Sampling

Algorithms for Optimum Sample Allocation Under One-Sided Bound Constraints

Description

Functions implementing selected optimum allocation algorithms for solving the optimum sample allocation problem, formulated as follows:

Minimize

f(x_1,\ldots,x_H) = \sum_{h=1}^H \frac{A^2_h}{x_h}

over \mathbb R_+^H, subject to

\sum_{h=1}^H c_h x_h = c,

and either

x_h \leq M_h, \qquad h = 1,\ldots,H,

or

x_h \geq m_h, \qquad h = 1,\ldots,H,

where c > 0,\, c_h > 0,\, A_h > 0,\, m_h > 0,\, M_h > 0,\, h = 1,\ldots,H, are given numbers.

The following is a list of all available algorithms along with the functions that implement them:

• RNA - rna(),

• LRNA - rna(),

• SGA - sga(),

• SGAPLUS - sgaplus(),

• COMA - coma().

See the documentation of a specific function for details.

The inequality constraints are optional. The user can choose whether and how they are imposed in the optimization problem, depending on the chosen algorithm:

• Lower bounds m_1, \ldots, m_H can be specified only for the LRNA algorithm (by setting cmp = .Primitive("<=") for rna()).

• Upper bounds M_1, \ldots, M_H are supported by all other algorithms.

• Simultaneous constraints (both lower and upper bounds) are not supported by these functions.

The costs c_1, \ldots, c_H of surveying one element in a stratum can be specified by the user only for the RNA and LRNA algorithms. For the remaining algorithms, these costs are fixed at 1, i.e., c_h = 1,\, h = 1,\ldots,H.

Usage

rna(
  total_cost,
  A,
  bounds = NULL,
  unit_costs = 1,
  cmp = .Primitive(">="),
  details = FALSE
)

sga(total_cost, A, M)

sgaplus(total_cost, A, M)

coma(total_cost, A, M)

Arguments

total_cost	(numeric(1)) total survey cost c. Must be strictly positive. Additionally: If one-sided lower bounds m_1, \ldots, m_H are imposed, it is required that c \geq \sum_{h=1}^H c_h m_h, i.e. total_cost >= sum(unit_costs * bounds). If one-sided upper bounds M_1, \ldots, M_H are imposed, it is required that c \leq \sum_{h=1}^H c_h M_h, i.e. total_cost <= sum(unit_costs * bounds).
A	(numeric) population constants A_1,\ldots,A_H. All values must be strictly positive.
bounds	(numeric or NULL) optional lower bounds m_1,\ldots,m_H, or upper bounds M_1,\ldots,M_H, or NULL to indicate that no inequality constraints are imposed. If not NULL, bounds is interpreted as: lower bounds, if cmp = .Primitive("<="), or upper bounds, if cmp = .Primitive(">="). See also total_cost.
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.
cmp	(function) a binary comparison operator used to check for violations of bounds. Must be either .Primitive("<=") (treating bounds as lower bounds and invoking the LRNA algorithm) or .Primitive(">=") (treating bounds as upper bounds and invoking the RNA algorithm). The value of this argument has no effect if bounds is NULL.
details	(logical(1)) should detailed information on stratum assignments (either take-Neyman or take-bound), values of the set function s, and the number of iterations be included in the output?
M	(numeric or NULL) upper bounds M_1,\ldots,M_H optionally imposed on sample sizes in strata. Set to NULL if no upper bounds are imposed. Otherwise, it is required that total_cost <= sum(unit_costs * M).

Details

If no inequality constraints are imposed, the allocation is given by the Neyman allocation:

x_h = \frac{A_h}{\sqrt{c_h}} \frac{c}{\sum_{i=1}^H A_i \sqrt{c_i}}, \qquad h = 1,\ldots,H.

For the stratified \pi-estimator of the population total under stratified simple random sampling without replacement design, the parameters of the objective function f are

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

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

Value

A numeric vector of optimum sample allocations in strata. In the case of rna() only, the return value may also be a list containing the optimum allocations and strata assignments.

Functions

• rna(): Implements the Recursive Neyman Algorithm (RNA) and its counterpart, the Lower Recursive Neyman Algorithm (LRNA), designed for the optimum allocation problem with one-sided lower-bound constraints. The RNA is described in \insertCiteWWW;textualstratallo, whereas the LRNA is introduced in \insertCiteWojciakLRNA;textualstratallo.

• sga(): The Stenger-Gabler (SGA) algorithm, as proposed by \insertCiteSG;textualstratallo and described in \insertCiteWWW;textualstratallo. This algorithm solves the optimum allocation problem with one-sided upper-bound constraints. It assumes unit costs are constant and equal to 1, i.e., c_h = 1,\, h = 1,\ldots,H.

• sgaplus(): A modified Stenger-Gabler-type algorithm, described in \insertCiteWojciakMsc;textualstratallo, implemented as the Sequential Allocation (version 1) algorithm. This algorithm solves the optimum allocation problem with one-sided upper-bound constraints. It assumes unit costs are constant and equal to 1, i.e., c_h = 1,\, h = 1,\ldots,H.

• coma(): The Change of Monotonicity Algorithm (COMA), described in \insertCiteWWW;textualstratallo, solves the optimum allocation problem with one-sided upper-bound constraints. It assumes unit costs are constant and equal to 1, i.e., c_h = 1,\, h = 1,\ldots,H.

Note

These functions are optimized for internal use and should typically not be called directly by users. Use opt() or optcost() instead.

References

\insertRef

WojciakLRNAstratallo

\insertRef

WWWstratallo

\insertRef

WojciakMscstratallo

\insertRef

SGstratallo

\insertRef

Sarndalstratallo

See Also

opt(), optcost(), rnabox().

Examples

A <- c(3000, 4000, 5000, 2000)
m <- c(50, 40, 10, 30) # lower bounds
M <- c(100, 90, 70, 80) # upper bounds

rna(total_cost = 190, A = A, bounds = M)

rna(total_cost = 190, A = A, bounds = m, cmp = .Primitive("<="))

rna(total_cost = 300, A = A, bounds = M)

sga(total_cost = 190, A = A, M = M)

sgaplus(total_cost = 190, A = A, M = M)

coma(total_cost = 190, A = A, M = M)

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