rnabox: 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 rnabox;textualstratallo. 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("

Value

A numeric vector of optimum sample allocations in strata.

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.

References

rnaboxstratallo

Examples

Run this code

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)