stratallo (version 3.0.1)

alg_1sided: 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)

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.

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).

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 WWW;textualstratallo, whereas the LRNA is introduced in WojciakLRNA;textualstratallo.

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

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 \(h\).

References

WojciakLRNAstratallo

WWWstratallo

WojciakMscstratallo

SGstratallo

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)