Package 'stratallo'

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 Wesołowski et al. (2022), whereas the LRNA is introduced in Wójciak (2023).

• sga(): The Stenger-Gabler (SGA) algorithm, as proposed by Stenger and Gabler (2005) and described in Wesołowski et al. (2022). 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 Wójciak (2019), 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 Wesołowski et al. (2022), 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

Wójciak W (2023). “Another Solution for Some Optimum Allocation Problem.” Statistics in Transition new series, 24(5), 203-219. doi:10.59170/stattrans-2023-071.

Wesołowski J, Wieczorkowski R, Wójciak W (2022). “Optimality of the Recursive Neyman Allocation.” Journal of Survey Statistics and Methodology, 10(5), 1263-1275. ISSN 2325-0984. doi:10.1093/jssam/smab018.

Wójciak W (2019). Optimal Allocation in Stratified Sampling Schemes. Master's thesis, Warsaw University of Technology. http://home.elka.pw.edu.pl/~wwojciak/msc_wwojciech_optimum_alloc.pdf.

Stenger H, Gabler S (2005). “Combining random sampling and census strategies - Justification of inclusion probabilities equal to 1.” Metrika, 61(2), 137–156. doi:10.1007/s001840400328.

Särndal C, Swensson B, Wretman J (1992). Model Assisted Survey Sampling. Springer New York, NY. ISBN 978-0-387-40620-6.

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)

---

Summarizing the Allocation

Description

A utility that returns a simple data.frame summarizing the allocation returned by opt() or optcost().

Usage

alloc_summary(x, A, m = NULL, M = NULL)

Arguments

x	(numeric) sample allocations $x_1,\ldots,x_H$.
A	(numeric) population constants $A_1,\ldots,A_H$.
m	(numeric or NULL) optional lower bounds $m_1,\ldots,m_H$.
M	(numeric or NULL) optional upper bounds $M_1,\ldots,M_H$.

Value

A data.frame with $H + 1$ rows and up to seven variables, where $H$ is the number of strata. The first $H$ rows correspond to strata $h = 1,\ldots,H$, while the last row contains column totals (where applicable). The columns include:

A

Population constant $A_h$.

m*

Lower bound $m_h$ (if provided).

M*

Upper bound $M_h$ (if provided).

allocation

The optimized sample size $x_h$.

take_min*

Boolean indicator: $x_h = m_h$.

take_max*

Boolean indicator: $x_h = M_h$.

take_Neyman

Boolean indicator: $m_h < x_h < M_h$ (or simply the internal Neyman allocation if no bounds were violated).

See Also

opt(), optcost().

Examples

A <- c(3000, 4000, 5000, 2000)
m <- c(100, 90, 70, 80)
M <- c(200, 150, 300, 210)

xopt_1 <- opt(n = 400, A, m)
alloc_summary(xopt_1, A, m)

xopt_2 <- opt(n = 540, A, m, M)
alloc_summary(xopt_2, A, m, M)

---

Internal Diagnostic Functions for Checking Optimality of rdca() Allocations

Description

Diagnostic tools to verify the objective function and constraint satisfaction for allocations computed by the rdca() algorithm.

Usage

rdca_obj_cnstr(x, n, H_counts, N, S, rho2, J = integer(0))

rdca_cnstr_check(x, n, H_counts, N, S, rho2, J = integer(0), tol_max = 0.1)

rdca_optcond_sU(H_counts, S, rho, s, U, return_diff = FALSE)

Arguments

x	(numeric) vector of sample allocations.
n	(integerish(1)) total sample size $n$. Must satisfy ⁠0 < n <= sum(N)⁠.
H_counts	(integerish) strata counts in each domain.
N	(integerish) strata sizes $(N_{d,h},\, (d,h) \in \mathcal H)$.
S	(numeric) standard deviations $(S_{d,h},\, (d,h) \in \mathcal H)$ of surveyed variable in strata.
rho2	(numeric) the square of rho (rho^2).
J	(integerish or NULL) vector of domain indices, a subset of $\mathcal D$. Specifies domains $d \in \mathcal D$ for which values of constraint functions $g_{dh}$ are computed. If integer(0), these values are not computed. If NULL, all domains are included.
tol_max	(integerish(1)) the maximum tolerance exponent (as a power of 10).
rho	(numeric) parameters $(\rho_d,\, d \in \mathcal D)$ of the optimization problem.
s	(numeric) vector of values for function $s(\mathcal U, \boldsymbol{v}, d \mid p)$ calculated only for domains included in U that are not fully blocked by the take-max assignment.
U	(integerish or NULL) a vector of indices identifying the take-max strata, i.e., the strata $(d,h)$ for which the allocation is fixed to $x_{d,h} = N_{d,h}$. The indices refer to the positions of strata in the set $\mathcal H$, in the same order as in the input vectors (N, S, etc.). For example, if $\mathcal H = \{(1,1),\, (2,1)\}$ and stratum $(2,1)$ is a take-max stratum, then U = 2. If U contains all strata from a domain, the dimension of the D matrix is reduced accordingly. U must satisfy one of the following conditions: n > sum(N[U]), n = sum(N[U]) and n = sum(N).
return_diff	(logical(1)) If FALSE, the function returns a logical vector indicating whether the optimality condition $s(\mathcal{U},\, \boldsymbol{v},\, d \mid p) \ge \rho_d / S_{d,h}$ is satisfied for each unblocked stratum $(d,h) \in \mathcal U.$ If TRUE, the function returns the differences $s(\mathcal{U},\, \boldsymbol{v},\, d \mid p) - \rho_d / S_{d,h}$ instead of a logical vector, which can be used to assess by how much the condition is satisfied or violated.

Functions

• rdca_obj_cnstr(): Compute the value of the objective function and constraint functions for a given allocation.

• rdca_cnstr_check(): Check whether the equality and inequality constraints are satisfied for a given allocation, within a specified tolerance. The tolerance applies to equality constraints only.

• rdca_optcond_sU(): Check the optimality condition related to $s$. Specifically, verifies whether $s(\mathcal{U},\, \boldsymbol{v},\, d \mid p) \ge \rho_d / S_{d,h}$ for all $(d,h) \in \mathcal{U}$ such that $d$ is not fully blocked by $\mathcal{U}$.

Examples

H_counts <- c(2, 2) # 2 domains with 2 strata each
N <- c(140, 110, 135, 190)
S <- sqrt(c(180, 20, 5, 4))
total <- c(2, 3)
kappa <- c(0.4, 0.6)
rho <- total * sqrt(kappa)
rho2 <- total^2 * kappa
n <- 500

(x <- dca(n, H_counts, N, S, rho, rho2))

# internal functions (not exported) – examples skipped

## Not run: 
rdca_obj_cnstr(x, n, H_counts, N, S, rho2)
rdca_obj_cnstr(x, n, H_counts, N, S, rho2, 2)
rdca_obj_cnstr(x, n, H_counts, N, S, rho2, NULL)

## End(Not run)

## Not run: 
rdca_cnstr_check(x, n, H_counts, N, S, rho2)
rdca_cnstr_check(x, n, H_counts, N, S, rho2, 1)
rdca_cnstr_check(x, n, H_counts, N, S, rho2, 2)
rdca_cnstr_check(x, n, H_counts, N, S, rho2, NULL)

## End(Not run)

## Not run: 
(n <- dca_nmax(H_counts, N, S) - 1)

U <- 1
(x <- dca(n, H_counts, N, S, rho, rho2, U = U, details = TRUE))
rdca_optcond_sU(H_counts, S, rho, x$s, U) # TRUE

U <- 2
(x <- dca(n, H_counts, N, S, rho, rho2, U = U, details = TRUE))
rdca_optcond_sU(H_counts, S, rho, x$s, U) # FALSE

U <- 3
(x <- dca(n, H_counts, N, S, rho, rho2, U = U, details = TRUE))
rdca_optcond_sU(H_counts, S, rho, x$s, U) # TRUE

U <- 1:2 # domain 2 blocked
(x <- dca(n, H_counts, N, S, rho, rho2, U = U, details = TRUE))
rdca_optcond_sU(H_counts, S, rho, x$s, U) # no unblocked strata in `U`

## End(Not run)

---

Datasets with Example Populations

Description

Example datasets containing artificial populations for testing and demonstrating optimum sample allocation algorithms.

Usage

pop10s_bounds_ucost

pop507s_ucost

pop969s_ucost

pop2d4s

pop9d278s

Format

pop10s_bounds_ucost: Population with 10 strata, lower and upper bounds on sample sizes, and associated surveying costs. A matrix with 10 rows and 5 variables:

N

stratum size

S

standard deviation of study variable in the stratum.

m

lower bound for sample size in the stratum.

M

upper bound for sample size in the stratum.

unit_cost

cost of surveying one element in the stratum.

pop507s_ucost: Population with 507 strata and associated surveying costs. A matrix with 507 rows and 3 columns:

N

stratum size.

S

standard deviation of study variable in the stratum.

unit_cost

cost of surveying one element in the stratum.

pop969s_ucost: Population with 969 strata and associated surveying costs. A matrix with 969 rows and 3 columns:

N

stratum size.

S

standard deviation of study variable in the stratum.

unit_cost

cost of surveying one element in the stratum.

pop2d4s: Population with 2 domains and 4 strata. A list with the following elements:

H_counts

strata counts in each domain.

N

stratum sizes.

S

standard deviations of study variable in strata.

total

totals in domains, i.e., the sum of the study variable values for population elements in each domain.

kappa

priority weights for domains.

rho

total * sqrt(kappa).

rho2

total^2 * kappa.

n_max

See dca_nmax() or dca().

pop9d278s: Population with 9 domains and 278 strata. A list with the following elements:

H_counts

strata counts in each domain.

N

stratum sizes.

S

standard deviations of study variable in strata.

total

totals in domains, i.e., the sum of the study variable values for population elements in each domain.

kappa

priority weights for domains.

rho

total * sqrt(kappa).

rho2

total^2 * kappa.

n_max

See dca_nmax() or dca().

---

Domain-Controlled Allocation (DCA) Algorithm

Description

Functions implementing the Domain-Controlled Allocation (DCA) algorithm described in Wesołowski (2019) and Wójciak (2026). The algorithm solves the following optimum allocation problem, formulated in mathematical optimization terms:

Minimize

$f(T,\, \boldsymbol x) = T$

over $\mathbb R \times \mathbb R_+^{\lvert \mathcal H \rvert}$, subject to

$\sum_{(d,h) \in \mathcal H} x_{d,h} = n,$

$\sum_{h \in \mathcal H_d} (\frac{1}{x_{d,h}} - \frac{1}{N_{d,h}}) \frac{N_{d,h}^2 S_{d,h}^2}{\rho_d^2} = T, \qquad d \in \mathcal D,$

where:

$(T,\, \boldsymbol x) = (T,\, (x_{d,h},\, (d,h) \in \mathcal H))$

the optimization variable,

$\mathcal H \subset \mathbb N^2$

the set of domain-stratum indices,

$\mathcal D := \{d \in \mathbb N \colon\; \exists h,\, (d,h) \in \mathcal H\}$

the set of domain indices,

$\mathcal H_d := \{h \in \mathbb N \colon\; (d,h) \in \mathcal H\}$

the set of strata indices in domain $d$,

$N_{d,h} > 0$

size of stratum $(d,h)$,

$S_{d,h} > 0$

standard deviation of the study variable in stratum $(d,h)$,

$\rho_d := t_d\, \sqrt{\kappa_d}$

where $t_d$ denotes the total in domain $d$, i.e., the sum of the values of the study variable for population elements in domain $d$, and $\kappa_d$ is a priority weight for domain $d$,

$n \in (0,\, \sum_{(d,h) \in \mathcal H} N_{d,h}]$

total sample size.

Usage

dca0(n, H_counts, N, S, rho, rho2, details = FALSE)

dca(n, H_counts, N, S, rho, rho2, U = NULL, details = FALSE)

dca_nmax(H_counts, N, S)

Arguments

n	(integerish(1)) total sample size $n$. Must satisfy ⁠0 < n <= sum(N)⁠.
H_counts	(integerish) strata counts in each domain.
N	(integerish) strata sizes $(N_{d,h},\, (d,h) \in \mathcal H)$.
S	(numeric) standard deviations $(S_{d,h},\, (d,h) \in \mathcal H)$ of surveyed variable in strata.
rho	(numeric) parameters $(\rho_d,\, d \in \mathcal D)$ of the optimization problem.
rho2	(numeric) the square of rho (rho^2), provided to reduce potential loss of precision due to finite-precision arithmetic.
details	(logical(1)) whether to produce detailed debug output.
U	(integerish or NULL) a vector of indices identifying the take-max strata, i.e., the strata $(d,h)$ for which the allocation is fixed to $x_{d,h} = N_{d,h}$. The indices refer to the positions of strata in the set $\mathcal H$, in the same order as in the input vectors (N, S, etc.). For example, if $\mathcal H = \{(1,1),\, (2,1)\}$ and stratum $(2,1)$ is a take-max stratum, then U = 2. If U contains all strata from a domain, the dimension of the D matrix is reduced accordingly. U must satisfy one of the following conditions: n > sum(N[U]), n = sum(N[U]) and n = sum(N).

Details

For $n \in (0,\, n_{max})$, the optimal value satisfies $T^* > 0$, where

$n_{max} := \sum_{d \in \mathcal D} \frac{\bigl( \sum_{h \in \mathcal H_d} N_{d,h} S_{d,h} \bigr)^2}{\sum_{h \in \mathcal H_d} N_{d,h} S_{d,h}^2}.$

See Proposition 2.1 in Wesołowski (2019) or Wójciak (2026) for details. The value $n_{max}$ is less than or equal to sum(N) and can be computed with dca_nmax().

Value

If details = FALSE, the optimal $\boldsymbol x^*$ is returned. Otherwise, a list is returned containing the optimal $\boldsymbol x^*$ (element named x) along with other internal details of this algorithm. In particular, the lambda element of the list corresponds to the optimal $T^*$.

Functions

• dca0(): Domain-Controlled Allocation algorithm by Wesołowski (2019)

• dca(): Domain-Controlled Allocation algorithm by Wesołowski (2019), optionally using a set of take-max strata as described in Wójciak (2026).

• dca_nmax(): Computes the maximum total sample size $n_{max}$ such that the optimization problem solved by the Domain-Controlled Allocation (DCA) algorithm admits a strictly positive optimal value $T^*$.

Note

These functions are optimized for internal use and should typically not be called directly by users. They are designed to handle a large number of invocations, specifically recursive calls from rdca(), and, as a result, parameter assertions are minimal.

References

Wójciak W (2026). Multi-Domain Optimum Sample Allocation with Controlled-Precision under Upper-Bound Constraints. Ph.D. thesis, Warsaw University of Technology. http://home.elka.pw.edu.pl/~wwojciak/phd_wwojciech_optimum_alloc.pdf.

Wesołowski J (2019). “Multi-domain Neyman-Tchuprov optimal allocation.” Statistics in Transition new series, 20(4), 1–12. doi:10.21307/stattrans-2019-031.

Wesołowski J, Wieczorkowski R (2017). “An eigenproblem approach to optimal equal-precision sample allocation in subpopulations.” Communications in Statistics - Theory and Methods, 46(5), 2212–2231. doi:10.1080/03610926.2015.1040501.

See Also

rdca()

Examples

# Two domains with 1 and 3 strata, respectively,
# that is, H = {(1,1), (2,1), (2,2), (2,3)}.
H_counts <- c(1, 3)
N <- c(140, 110, 135, 190) # (N_{1,1}, N_{2,1}, N_{2,2}, N_{2,3})
S <- sqrt(c(180, 20, 5, 4)) # (S_{1,1}, S_{2,1}, S_{2,2}, S_{2,3})
total <- c(2, 3)
kappa <- c(0.4, 0.6)
rho <- total * sqrt(kappa) # (rho_1, rho_2)
rho2 <- total^2 * kappa
sum(N) # 575
n_max <- dca_nmax(H_counts, N, S) # 519.0416

n <- floor(n_max) - 1

dca0(n, H_counts, N, S, rho, rho2)
x0 <- dca0(n, H_counts, N, S, rho, rho2, details = TRUE)
x0$x
x0$lambda
x0$k
x0$v
x0$s

n <- ceiling(n_max) + 1
x0 <- dca0(n, H_counts, N, S, rho, rho2, details = TRUE)
x0$x
x0$lambda

n <- floor(n_max) - 1

x1 <- dca(n, H_counts, N, S, rho, rho2, details = TRUE)
x1$x
x1$x_Uc
x1$lambda
x1$s

dca(n, H_counts, N, S, rho, rho2, U = 1)
x2 <- dca(n, H_counts, N, S, rho, rho2, U = 1, details = TRUE)
x2$x
x2$x_Uc
x2$lambda
x2$s

---

DCA Algorithm for Upper-Bound Constrained Allocations (M <= N)

Description

Prototype (under testing).

Usage

dca_M(n, H_counts, N, S, rho, rho2, M = N, U = NULL)

Arguments

n	(integerish(1)) total sample size $n$. Must satisfy ⁠0 < n <= sum(N)⁠.
H_counts	(integerish) strata counts in each domain.
N	(integerish) strata sizes $(N_{d,h},\, (d,h) \in \mathcal H)$.
S	(numeric) standard deviations $(S_{d,h},\, (d,h) \in \mathcal H)$ of surveyed variable in strata.
rho	(numeric) parameters $(\rho_d,\, d \in \mathcal D)$ of the optimization problem.
rho2	(numeric) the square of rho (rho^2), provided to reduce potential loss of precision due to finite-precision arithmetic.
M	(integerish) upper bounds on sample sizes in strata. Defaults to N.
U	(integerish or NULL) a vector of indices identifying the take-max strata, i.e., the strata $(d,h)$ for which the allocation is fixed to $x_{d,h} = N_{d,h}$. The indices refer to the positions of strata in the set $\mathcal H$, in the same order as in the input vectors (N, S, etc.). For example, if $\mathcal H = \{(1,1),\, (2,1)\}$ and stratum $(2,1)$ is a take-max stratum, then U = 2. If U contains all strata from a domain, the dimension of the D matrix is reduced accordingly. U must satisfy one of the following conditions: n > sum(N[U]), n = sum(N[U]) and n = sum(N).

Examples

H_counts <- c(5, 2)
H_names <- rep(seq_along(H_counts), times = H_counts)
S <- c(154, 178, 134, 213, 124, 102, 12)
N <- c(100, 100, 100, 100, 100, 100, 100)
M <- c(80, 90, 70, 40, 10, 90, 100)
names(M) <- names(N) <- H_names
total <- c(13, 2)
kappa <- c(0.8, 0.2)
n <- 150

# experimental function (not exported) – examples skipped
## Not run: 
dca_M(n, H_counts, N, S, total, kappa, M = M, U = 5)
#         1         1         1         1         1         2         2
# 12.754880 14.742653 11.098402 17.641490 10.000000 74.945462  8.817113

## End(Not run)

---

Multi-Domain Optimum Sample Allocation with Controlled-Precision under Upper-Bound Constraints

Description

Computes the optimum allocation for the following multi-domain optimum allocation problem, formulated in mathematical optimization terms:

Minimize

$f(T,\, \boldsymbol x) = T$

over $\mathbb R \times \mathbb R_+^{\lvert \mathcal H \rvert}$, subject to

$\sum_{(d,h) \in \mathcal H} x_{d,h} = n,$

$\sum_{h \in \mathcal H_d} (\frac{1}{x_{d,h}} - \frac{1}{N_{d,h}}) \frac{N_{d,h}^2 S_{d,h}^2}{\rho_d^2} = T, \qquad d \in \mathcal D,$

$x_{d,h} \leq N_{d,h}, \qquad (d,h) \in \mathcal H,$

where:

$(T,\, \boldsymbol x) = (T,\, (x_{d,h},\, (d,h) \in \mathcal H))$

the optimization variable,

$\mathcal H \subset \mathbb N^2$

the set of domain-stratum indices,

$\mathcal D := \{d \in \mathbb N \colon\; \exists h,\, (d,h) \in \mathcal H\}$

the set of domain indices,

$\mathcal H_d := \{h \in \mathbb N \colon\; (d,h) \in \mathcal H\}$

the set of strata indices in domain $d$,

$N_{d,h} > 0$

size of stratum $(d,h)$,

$S_{d,h} > 0$

standard deviation of the study variable in stratum $(d,h)$,

$\rho_d := t_d\, \sqrt{\kappa_d}$

where $t_d$ denotes the total in domain $d$, i.e., the sum of the values of the study variable for population elements in domain $d$, and $\kappa_d$ is a priority weight for domain $d$,

$n \in (0,\, \sum_{(d,h) \in \mathcal H} N_{d,h}]$

total sample size.

Usage

dopt(n, H_counts, N, S, total, kappa, return_T = FALSE)

Arguments

n	(integerish(1)) total sample size $n$. Must satisfy ⁠0 < n <= sum(N)⁠.
H_counts	(integerish) strata counts in each domain.
N	(integerish) strata sizes $(N_{d,h},\, (d,h) \in \mathcal H)$.
S	(numeric) standard deviations $(S_{d,h},\, (d,h) \in \mathcal H)$ of surveyed variable in strata.
total	(numeric) vector of domain totals, $t_d,\, d \in \mathcal D$, i.e., the sum of the study variable over all population elements in each domain.
kappa	(numeric) vector of priority weights for the domains, $\kappa_d,\, d \in \mathcal D$.
return_T	(logical(1)) If TRUE, the function returns a list containing the optimal allocation and the optimal value of the objective function $T$. If FALSE (default), only the optimal allocation vector is returned.

Details

The dopt() function uses the RDCA algorithm implemented in rdca().

Value

If return_T = FALSE (default), a numeric vector containing the optimal sample allocations $x_{d,h}$ for each stratum $(d,h) \in \mathcal H$.

If return_T = TRUE, a list with components:

xopt

numeric vector of optimal sample allocations.

Topt

optimal value of the objective function $T$.

References

Wójciak W (2026). Multi-Domain Optimum Sample Allocation with Controlled-Precision under Upper-Bound Constraints. Ph.D. thesis, Warsaw University of Technology. http://home.elka.pw.edu.pl/~wwojciak/phd_wwojciech_optimum_alloc.pdf.

See Also

rdca(), dca(), dca_nmax(), opt(), optcost().

Examples

# Three domains with 2, 2, and 3 strata, respectively,
# that is, H = {(1,1), (1,2), (2,1), (2,2), (3,1), (3,2), (3,3)}.
H_counts <- c(2, 2, 3)
# (N_{1,1}, N_{1,2}, N_{2,1}, N_{2,2}, N_{3,1}, N_{3,2}, N_{3,3})
N <- c(140, 110, 135, 190, 200, 40, 70)
# (S_{1,1}, S_{1,2}, S_{2,1}, S_{2,2}, S_{3,1}, S_{3,2}, S_{3,3})
S <- c(180, 20, 5, 4, 35, 9, 40)
total <- c(2, 3, 5)
kappa <- c(0.5, 0.2, 0.3)
n <- 828

# Optimum allocation.
dopt(n, H_counts, N, S, total, kappa)

# Example population with 9 domains and 278 strata
p <- pop9d278s
sum(p$N)
n <- 5000
x <- dopt(n, p$H_counts, p$N, p$S, p$total, p$kappa, return_T = TRUE)
x
all(x$xopt <= p$N)
sum(x$xopt)

---

Fixed-Point Iteration Algorithm

Description

Algorithm for optimum sample allocation in stratified sampling under lower- and upper-bound constraints, based on fixed-point iteration.

Usage

fpia2(v0, Nh, Sh, mh = NULL, Mh = NULL, lambda0 = NULL, maxiter = 100)

glambda(lambda, n, Ah, mh = NULL, Mh = NULL)

philambda(lambda, n, Ah, mh = NULL, Mh = NULL)

Arguments

v0	variance
mh	(numeric or NULL) lower bounds on stratum sample sizes (optional).
Mh	(numeric or NULL) upper bounds on stratum sample sizes (optional).
lambda0	(numeric(1)) initial value of the parameter $\lambda$ (optional).
maxiter	(integerish(1)) maximum number of iterations.
lambda	(numeric(1)) $\lambda$.
tol	(numeric(1)) desired convergence tolerance.

Value

A list with elements:

nh

Vector of optimal allocation sizes.

iter

Number of iterations performed.

Functions

• fpia2(): Variant of fpia() using variance-based parametrization.

• glambda(): Helper function for the fpia()

• philambda(): Helper function for the fpia().

References

Münnich RT, Sachs EW, Wagner M (2012). “Numerical solution of optimal allocation problems in stratified sampling under box constraints.” AStA Advances in Statistical Analysis, 96(3), 435–450. doi:10.1007/s10182-011-0176-z.

---

Check for Mixed Signs in a Numeric Vector

Description

Determines whether a numeric vector contains both negative and positive values. Zero (0) is treated as neutral and does not count as either sign.

Usage

has_mixed_signs(x)

Arguments

x	(numeric) a vector to check.

Value

TRUE if the vector contains both positive and negative values, FALSE otherwise.

Examples

# internal functions (not exported) – examples skipped
## Not run: 
has_mixed_signs(1:5)
has_mixed_signs(-(1:5))
has_mixed_signs(c(-1, -2, 3))
has_mixed_signs(c(0, -1))
has_mixed_signs(c(0, 1))
has_mixed_signs(c(0, 1, -1))

## End(Not run)

---

Internal Helper Functions for dca0(), dca(), and rdca()

Description

Internal utility functions used by dca0(), dca(), and rdca() that perform operations on sets of domain–strata indices and manage the mapping between strata and domains.

Usage

H_cnt2dind(H_counts)

H_cnt2glbidx(H_counts)

H_get_strata_indices(H_counts, d)

Arguments

H_counts	(integerish) strata counts in each domain.
d	(integerish(1)) domain index. Must satisfy ⁠0 < d <= length(H_counts)⁠.

Functions

• H_cnt2dind(): Creates a vector of domain indicators from a vector of strata counts per domain; each element of the vector is the index of the domain to which the corresponding stratum belongs.

• H_cnt2glbidx(): Creates unique indices for strata across multiple domains. Returns a list of integer vectors, where the $d$-th element contains the unique indices of the strata in domain $d$.

• H_get_strata_indices(): Get the globally unique indices of strata belonging to a specific domain.

Note

These functions are internal and should typically not be called directly by users.

Examples

H_counts <- c(2, 2, 3) # three domains with 2, 2, and 3 strata respectively

# internal functions (not exported) – examples skipped
## Not run: 
H_cnt2dind(H_counts) # 1 1 2 2 3 3 3

## End(Not run)

# internal functions (not exported) – examples skipped
## Not run: 
H_cnt2glbidx(H_counts)

## End(Not run)

# internal functions (not exported) – examples skipped
## Not run: 
H_get_strata_indices(H_counts, 3) # 5 6 7

## End(Not run)

---

Check Numeric Equality within a Tolerance

Description

Compares two numeric vectors element-wise using an adaptive tolerance sequence ranging from 10^-19 to 10^tol_max. The smallest tolerance at which the values are considered equal is returned as the corresponding name in the output.

Usage

is_equal(x, y, tol_max = -1)

Arguments

x	(numeric) first vector to compare.
y	(numeric) second vector to compare.
tol_max	(integerish(1)) the maximum tolerance exponent (as a power of 10).

Value

A logical vector indicating whether each element pair is equal within the detected tolerance. The names reflect the tolerance used.

Examples

# internal functions (not exported) – examples skipped
## Not run: 
is_equal(c(3, 4), c(3, 4))
is_equal(c(3, 4), c(3.01, 4.11))
is_equal(c(3, 4), c(3.01, 4.11), tol_max = 0)

## End(Not run)

---

Object Emptiness (NULL or zero-length)

Description

Utility functions for checking whether an object is empty, where emptiness is defined as being NULL or having length 0.

Usage

is_empty(x)

is_nonempty(x)

Arguments

x	object to test.

Functions

• is_empty(): Returns TRUE if the object is NULL or has length 0, and FALSE otherwise.

• is_nonempty(): Logical negation of is_empty(). This function directly checks if length(x) > 0L for performance reasons, avoiding the extra negation step that would occur if using !is_empty(x). It is optimized for repeated use in algorithms where is_nonempty() is called many times.

Examples

# internal functions (not exported) – examples skipped

## Not run: 
is_empty(NULL)
is_empty(character(0))
is_empty(1)

## End(Not run)

## Not run: 
is_nonempty(NULL)
is_nonempty(character(0))
is_nonempty(1)

## End(Not run)

---

Optimum Sample Allocation in Stratified Sampling

Description

Computes the optimum allocation for the following optimum allocation problem, formulated in mathematical optimization terms:

Minimize

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

over $\mathbb R_+^H$, subject to

$\sum_{h=1}^H x_h = n,$

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

where $n > 0,\, A_h > 0,\, m_h > 0,\, M_h > 0$, such that $m_h < M_h,\, h = 1,\ldots,H$, and $\sum_{h=1}^H m_h \leq n \leq \sum_{h=1}^H M_h$, are given numbers. Inequality constraints are optional and may be omitted.

Inequality constraints are optional, and the user can choose whether and how they are applied to the optimization problem. This is controlled using the m and M arguments as follows:

• No inequality constraints: both m and M must be NULL (default).

• Lower bounds only ($m_1,\, \ldots,\, m_H$): specify m, and set M = NULL.

• Upper bounds only ($M_1,\, \ldots,\, M_H$): specify M, and set m = NULL.

• Box constraints ($m_h, M_h,\, h = 1,\ldots,H$): specify both m and M.

Usage

opt(n, A, m = NULL, M = NULL, M_algorithm = "rna")

Arguments

n	(integerish(1)) total sample size. Must satisfy n > 0. Additionally: If bounds_inner is not NULL, then n >= sum(bounds_inner) when bounds_inner are treated as lower bounds, or n <= sum(bounds_inner) when treated as upper bounds. If bounds_outer is not NULL, then n >= sum(bounds_outer) when bounds_outer are treated as lower bounds, or n <= sum(bounds_outer) when treated as upper bounds.
A	(numeric) population constants $A_1,\ldots,A_H$. All values must be strictly positive.
m	(numeric or NULL) optional lower bounds $m_1,\ldots,m_H$ for the stratum sample sizes. If no lower bounds are desired, set m = NULL. If M is not NULL, it is required that $m_h < M_h$ for all strata.
M	(numeric or NULL) optional upper bounds $M_1,\ldots,M_H$ for the stratum sample sizes. If no upper bounds are desired, set M = NULL. If m is not NULL, it is required that $m_h < M_h$ for all strata.
M_algorithm	(string) Name of the algorithm to use for computing the sample allocation when only upper-bound constraints are imposed. Must be one of "rna" (default), "sga", "sgaplus", or "coma". This parameter is used only when $H > 1$ and n < sum(M).

Details

The opt() function uses different allocation algorithms depending on which inequality constraints are applied. Each algorithm is implemented in a separate R function, which is generally not intended to be called directly by the end user. The algorithms are:

• Lower bounds only ($m_1,\, \ldots,\, m_H$):

  • LRNA - rna()

• Upper bounds only ($M_1,\, \ldots,\, M_H$):

  • RNA - rna()

  • SGA - sga()

  • SGAPLUS - sgaplus()

  • COMA - coma()

• Box constraints ($m_h, M_h,\, h = 1,\ldots,H$):

  • RNABOX - rnabox()

See the documentation of each specific function for more details about the corresponding algorithm.

Value

A numeric vector of the optimal sample allocations for each stratum.

Note

If no inequality constraints are applied, the allocation follows the Neyman allocation:

$x_h = A_h \frac{n}{\sum_{i=1}^H A_i}, \quad h = 1,\ldots,H.$

For a stratified $\pi$ estimator of the population total using stratified simple random sampling without replacement design, the objective function parameters $A_h$ are:

$A_h = N_h S_h, \quad h = 1,\ldots,H,$

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

References

Särndal C, Swensson B, Wretman J (1992). Model Assisted Survey Sampling. Springer New York, NY. ISBN 978-0-387-40620-6.

See Also

optcost(), rna(), sga(), sgaplus(), coma(), rnabox().

Examples

A <- c(3000, 4000, 5000, 2000)
m <- c(100, 90, 70, 50)
M <- c(300, 400, 200, 90)

# One-sided lower bounds.
opt(n = 340, A = A, m = m)
opt(n = 400, A = A, m = m)
opt(n = 700, A = A, m = m)

# One-sided upper bounds.
opt(n = 190, A = A, M = M)
opt(n = 700, A = A, M = M)

# Box-constraints.
opt(n = 340, A = A, m = m, M = M)
opt(n = 500, A = A, m = m, M = M)
x <- opt(n = 800, A = A, m = m, M = M)
x

# Variance corresponding to the allocation x.
var_st(x = x, A = A, A0 = 45000)

# Execution-time comparison of different algorithms using the microbenchmark package.
## Not run: 
N <- pop969s_ucost[, "N"]
S <- pop969s_ucost[, "S"]
A <- N * S
nfrac <- c(0.005, seq(0.05, 0.95, 0.05))
n <- setNames(as.integer(nfrac * sum(N)), nfrac)
lapply(
  n,
  function(ni) {
    microbenchmark::microbenchmark(
      RNA = opt(ni, A, M = N, M_algorithm = "rna"),
      SGA = opt(ni, A, M = N, M_algorithm = "sga"),
      SGAPLUS = opt(ni, A, M = N, M_algorithm = "sgaplus"),
      COMA = opt(ni, A, M = N, M_algorithm = "coma"),
      times = 200,
      unit = "us"
    )
  }
)

## End(Not run)

---

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 Wójciak (2023).

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

Wójciak W (2023). “Another Solution for Some Optimum Allocation Problem.” Statistics in Transition new series, 24(5), 203-219. doi:10.59170/stattrans-2023-071.

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

---

Recursive Domain-Controlled Allocation (RDCA) Algorithm

Description

Implements the Recursive Domain-Controlled Allocation (RDCA) algorithm described in Wójciak (2026). The algorithm solves the following optimum allocation problem, formulated in mathematical optimization terms:

Minimize

$f(T,\, \boldsymbol x) = T$

over $\mathbb R \times \mathbb R_+^{\lvert \mathcal H \rvert}$, subject to

$\sum_{(d,h) \in \mathcal H} x_{d,h} = n,$

$\sum_{h \in \mathcal H_d} (\frac{1}{x_{d,h}} - \frac{1}{N_{d,h}}) \frac{N_{d,h}^2 S_{d,h}^2}{\rho_d^2} = T, \qquad d \in \mathcal D,$

$x_{d,h} \leq N_{d,h}, \qquad (d,h) \in \mathcal H,$

where:

$(T,\, \boldsymbol x) = (T,\, (x_{d,h},\, (d,h) \in \mathcal H))$

the optimization variable,

$\mathcal H \subset \mathbb N^2$

the set of domain-stratum indices,

$\mathcal D := \{d \in \mathbb N \colon\; \exists h,\, (d,h) \in \mathcal H\}$

the set of domain indices,

$\mathcal H_d := \{h \in \mathbb N \colon\; (d,h) \in \mathcal H\}$

the set of strata indices in domain $d$,

$N_{d,h} > 0$

size of stratum $(d,h)$,

$S_{d,h} > 0$

standard deviation of the study variable in stratum $(d,h)$,

$\rho_d := t_d\, \sqrt{\kappa_d}$

where $t_d$ denotes the total in domain $d$, i.e., the sum of the values of the study variable for population elements in domain $d$, and $\kappa_d$ is a priority weight for domain $d$,

$n \in (0,\, \sum_{(d,h) \in \mathcal H} N_{d,h}]$

total sample size.

Usage

rdca(n, H_counts, N, S, rho, rho2 = rho^2, U = NULL, J = NULL)

Arguments

n	(integerish(1)) total sample size $n$. Must satisfy ⁠0 < n <= sum(N)⁠.
H_counts	(integerish) strata counts in each domain.
N	(integerish) strata sizes $(N_{d,h},\, (d,h) \in \mathcal H)$.
S	(numeric) standard deviations $(S_{d,h},\, (d,h) \in \mathcal H)$ of surveyed variable in strata.
rho	(numeric) parameters $(\rho_d,\, d \in \mathcal D)$ of the optimization problem.
rho2	(numeric) the square of rho (rho^2), provided to reduce potential loss of precision due to finite-precision arithmetic.
U	(integerish or NULL) a vector of indices identifying the take-max strata, i.e., the strata $(d,h)$ for which the allocation is fixed to $x_{d,h} = N_{d,h}$. The indices refer to the positions of strata in the set $\mathcal H$, in the same order as in the input vectors (N, S, etc.). For example, if $\mathcal H = \{(1,1),\, (2,1)\}$ and stratum $(2,1)$ is a take-max stratum, then U = 2. If U contains all strata from a domain, the dimension of the D matrix is reduced accordingly. U must satisfy one of the following conditions: n > sum(N[U]), n = sum(N[U]) and n = sum(N).
J	(integerish or NULL) vector of domain indices, subset of $\mathcal D$. Specifies the domains for which allocated sample sizes must not exceed the corresponding strata sizes N. For domains not included in J, allocations may exceed strata sizes. Must not be empty. If J is NULL, it is treated as containing all domains. U must not contain any strata from domains included in J.

Details

The upper-bound constraints $x_{d,h} \le N_{d,h}$ are guaranteed to be preserved only if domain $d$ is in J. The parameter J is used in the recursion. The specified optimization problem is solved when J = NULL, i.e., when J contains all domains.

Note

This function is optimized for internal use and should typically not be called directly by users. It is designed to handle a large number of invocations, specifically recursive invocations of rdca(), and, as a result, parameter assertions are minimal.

References

Wójciak W (2026). Multi-Domain Optimum Sample Allocation with Controlled-Precision under Upper-Bound Constraints. Ph.D. thesis, Warsaw University of Technology. http://home.elka.pw.edu.pl/~wwojciak/phd_wwojciech_optimum_alloc.pdf.

See Also

dca()

Examples

# Three domains with 2, 2, and 3 strata, respectively,
# that is, H = {(1,1), (1,2), (2,1), (2,2), (3,1), (3,2), (3,3)}.
H_counts <- c(2, 2, 3)
# (N_{1,1}, N_{1,2}, N_{2,1}, N_{2,2}, N_{3,1}, N_{3,2}, N_{3,3})
N <- c(140, 110, 135, 190, 200, 40, 70)
# (S_{1,1}, S_{1,2}, S_{2,1}, S_{2,2}, S_{3,1}, S_{3,2}, S_{3,3})
S <- c(180, 20, 5, 4, 35, 9, 40)
total <- c(2, 3, 5)
kappa <- c(0.5, 0.2, 0.3)
rho <- total * sqrt(kappa) # (rho_1, rho_2, rho_3)
rho2 <- total^2 * kappa
sum(N)
n <- 828

# Optimum allocation.
rdca(n, H_counts, N, S, rho, rho2)

# Upper bounds enforced only for domain 1.
rdca(n, H_counts, N, S, rho, rho2, J = 1)

---

Iterative RDCA Implementation

Description

Iterative implementation of the Recursive Domain-Controlled Allocation (RDCA) algorithm. Not tested.

Usage

rdca_iter(n, H_counts, N, S, rho, rho2 = NULL, ref_domain = 1L)

Arguments

n	(integerish(1)) total sample size $n$. Must satisfy ⁠0 < n <= sum(N)⁠.
H_counts	(integerish) strata counts in each domain.
N	(integerish) strata sizes $(N_{d,h},\, (d,h) \in \mathcal H)$.
S	(numeric) standard deviations $(S_{d,h},\, (d,h) \in \mathcal H)$ of surveyed variable in strata.
rho	(numeric) parameters $(\rho_d,\, d \in \mathcal D)$ of the optimization problem.
rho2	(numeric) the square of rho (rho^2), provided to reduce potential loss of precision due to finite-precision arithmetic.
ref_domain	(integerish(1)) reference domain (denoted by j in the thesis).

Examples

H_counts <- c(2, 2, 3)
N <- c(140, 110, 135, 190, 200, 40, 70)
S <- sqrt(c(180, 20, 5, 4, 35, 9, 40))
total <- c(2, 3, 5)
kappa <- c(0.5, 0.2, 0.3)
rho <- total * sqrt(kappa)
(n <- dca_nmax(H_counts, N, S) - 1)

# experimental function (not exported) – examples skipped
## Not run: 
rdca_iter(n, H_counts, N, S, rho)
# 140.0000 103.6139 132.1970 166.4127 195.9701  19.8750  70.0000

## End(Not run)

---

RNA – Experimental Versions

Description

Experimental variants of the Recursive Neyman Algorithm (RNA).

Usage

rna_rec(
  total_cost,
  A,
  bounds = NULL,
  unit_costs = rep(1, length(A)),
  cmp = .Primitive(">=")
)

rna_prior(
  total_cost,
  A,
  bounds = NULL,
  check = NULL,
  cmp = .Primitive(">="),
  details = FALSE
)

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.
check	(integer) indices of strata for which allocation constraints may be violated. For all other strata, constraint violations are not checked.
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?

Functions

• rna_rec(): Recursive implementation of the RNA.

• rna_prior(): A variant of the Recursive Neyman Algorithm (RNA) that uses prior information about strata for which allocation constraints may be violated. For all other strata, allocations are assumed to satisfy the bounds.

Note

This code has not been extensively tested and may change in future releases.

Examples

A <- c(3000, 4000, 5000, 2000)
M <- c(100, 90, 70, 80) # upper bounds

# experimental function (not exported) – examples skipped
## Not run: 
rna_rec(total_cost = 190, A = A, bounds = M)
rna_rec(total_cost = 312, A = A, bounds = M)
rna_rec(total_cost = 339, A = A, bounds = M)
rna_rec(total_cost = 340, A = A, bounds = M)

## End(Not run)

---

Recursive Neyman Algorithm for Optimum Sample Allocation under Box Constraints (RNABOX)

Description

Implements the Recursive Neyman Algorithm for Optimum Sample Allocation under Box Constraints (RNABOX), as proposed in Wesołowski et al. (2024). The algorithm solves the following optimum allocation problem, formulated in mathematical optimization terms:

Minimize

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

over $\mathbb R_+^H$, subject to

$\sum_{h=1}^H x_h = n,$

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

where $n > 0,\, A_h > 0,\, m_h > 0,\, M_h > 0$, such that $m_h < M_h,\, h = 1,\ldots,H$, and $\sum_{h=1}^H m_h \leq n \leq \sum_{h=1}^H M_h$, are given numbers. Inequality constraints are optional and may be omitted.

Usage

rnabox(
  n,
  A,
  bounds_inner = NULL,
  bounds_outer = NULL,
  cmp_inner = .Primitive(">="),
  cmp_outer = .Primitive("<=")
)

Arguments

n	(integerish(1)) total sample size. Must satisfy n > 0. Additionally: If bounds_inner is not NULL, then n >= sum(bounds_inner) when bounds_inner are treated as lower bounds, or n <= sum(bounds_inner) when treated as upper bounds. If bounds_outer is not NULL, then n >= sum(bounds_outer) when bounds_outer are treated as lower bounds, or n <= sum(bounds_outer) when treated as upper bounds.
A	(numeric) population constants $A_1,\ldots,A_H$. All values must be strictly positive.
bounds_inner	(numeric or NULL) optional bounds on sample sizes in strata for the interim (inner) allocation phase of rnabox(). These can be either lower bounds $m_1,\ldots,m_H$ or upper bounds $M_1,\ldots,M_H$, depending on the value of cmp_inner. If bounds_inner is NULL, no bounds are imposed during the interim allocation phase. If both bounds_inner and bounds_outer are not NULL, the following element-wise relationships must hold: If bounds_inner are treated as lower bounds, then bounds_inner < bounds_outer. If bounds_inner are treated as upper bounds, then bounds_inner > bounds_outer. bounds_inner is passed by rnabox() to rna() as the bounds argument, serving as an interim allocation step applied before enforcing the outer bounds_outer.
bounds_outer	(numeric or NULL) optional bounds on sample sizes in strata, checked during the violation check (outer) phase of the iterative rnabox() loop. These can be either lower bounds $m_1,\ldots,m_H$ or upper bounds $M_1,\ldots,M_H$, depending on the value of cmp_outer. If bounds_outer is NULL, no bounds are imposed during the outer phase. If both bounds_outer and bounds_inner are not NULL, the following element-wise relationships must hold: If bounds_outer are treated as lower bounds, then bounds_outer < bounds_inner. If bounds_outer are treated as upper bounds, then bounds_outer > bounds_inner.
cmp_inner	(function) a binary comparison operator used to check for violations of bounds_inner. Must be either .Primitive("<=") or .Primitive(">="). This operator determines how bounds_inner is handled: .Primitive("<=") treats bounds_inner as lower bounds and causes rnabox() to apply the LRNA algorithm as an inner allocation step. .Primitive(">=") treats bounds_inner as upper bounds and causes rnabox() to apply the RNA algorithm as an inner allocation step. The value of this argument has no effect if bounds_inner is NULL. cmp_inner is passed by rnabox() to rna() as the cmp argument, serving as an interim allocation step applied before enforcing the outer bounds_outer.
cmp_outer	(function) a binary comparison operator used to check for violations of bounds_outer. It must be the logical complement of cmp_inner (up to equality): if cmp_inner = .Primitive("<="), then cmp_outer = .Primitive(">="), and vice versa. It determines how bounds_outer is handled: .Primitive("<=") treats bounds_outer as lower bounds. .Primitive(">=") treats bounds_outer as upper bounds. The value of this argument has no effect if bounds_outer is NULL. cmp_outer is provided solely for computational efficiency, as its value is fully determined by cmp_inner.

Value

A numeric vector of optimum sample allocations in strata.

Note

The rnabox() function is optimized for internal use and should typically not be called directly by users. Use opt() instead.

References

Wesołowski J, Wieczorkowski R, Wójciak W (2024). “Recursive Neyman algorithm for optimum sample allocation under box constraints on sample sizes in strata.” Survey Methodology, 50(2), 487–511. ISSN 1492-0921. https://www150.statcan.gc.ca/n1/en/catalogue/12-001-X201200111682.

See Also

opt(), optcost(), rna(), sga(), sgaplus(), coma()

Examples

N <- c(454, 10, 116, 2500, 2240, 260, 39, 3000, 2500, 400)
S <- c(0.9, 5000, 32, 0.1, 3, 5, 300, 13, 20, 7)
A <- N * S
m <- c(322, 3, 57, 207, 715, 121, 9, 1246, 1095, 294) # lower bounds
M <- N # upper bounds

# Regular allocation.
n <- 6000
opt_regular <- rnabox(n, A, M, m)

# Vertex allocation.
n <- 4076
opt_vertex <- rnabox(n, A, M, m)

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