optcost | R Documentation |
Function that determines fixed strata sample sizes that minimize total cost
of the survey, under assumed level of the variance of the stratified
estimator and under optional one-sided upper bounds imposed on strata sample
sizes. Namely, the following optimization problem, formulated below in the
language of mathematical optimization, is solved by optcost()
function.
Minimize
c(x_1,\ldots,x_H) = \sum_{h=1}^H c_h x_h
subject to
\sum_{h=1}^H \frac{A^2_h}{x_h} - A_0 = V
x_h \leq M_h, \quad 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
minimization is on \mathbb R_+^H
.
The upper-bounds constraints x_h \leq M_h,\, h = 1,\ldots,H
, are
optional and can be skipped. In such a case, it is only required that
V > 0
.
optcost(V, A, A0, M = NULL, unit_costs = 1)
V |
( |
A |
( |
A0 |
( |
M |
( |
unit_costs |
( |
The algorithm that is used by optcost()
is the LRNA
and it is
described in Wójciak (2023). The allocation computed is valid for all
stratified sampling schemes for which the variance of the stratified
estimator is of the form:
\sum_{h=1}^H \frac{A^2_h}{x_h} - A_0,
where H
denotes total number of strata, x_1,\ldots,x_H
are
strata sample sizes and A_0,\, A_h > 0,\, h = 1,\ldots,H
, do not
depend on x_h,\, h = 1,\ldots,H
.
Numeric vector with optimal sample allocations in strata.
For stratified \pi
estimator of the population total and
for stratified simple random sampling without replacement design,
the population parameters are as follows:
A_h = N_h S_h, \quad 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
denotes
standard deviation of a given study variable in stratum h
.
Wójciak, W. (2023). Another Solution of Some Optimum Allocation Problem. Statistics in Transition new series, 24(5) (in press). https://arxiv.org/abs/2204.04035
rna()
, opt()
.
A <- c(3000, 4000, 5000, 2000)
M <- c(100, 90, 70, 80)
xopt <- optcost(1017579, A = A, A0 = 579, M = M)
xopt
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.