@misc{rnabox_perf,
	author = {Jacek Wesołowski and Robert Wieczorkowski and Wojciech Wójciak},
	title = {{Numerical Performance of the RNABOX Algorithm}},
	version = {2.0.0},
	month = jun,
	year = {2023},
	url = {https://github.com/rwieczor/recursive_Neyman_rnabox}
}

@misc{stratallo_src,
	author = {Jacek Wesołowski and Robert Wieczorkowski and Wojciech Wójciak},
	title = {R package stratallo - source code},
	version = {3.0.0},
	month = jun,
	year = {2026},
	url = {https://github.com/wwojciech/stratallo},
}

@Book{Sarndal,
	author = {Carl-Erik {S\"arndal} and Bengt Swensson and Jan Wretman},
	title = "{Model Assisted Survey Sampling}",
	year = {1992},
	publisher = {Springer New York, NY},
	isbn = {978-0-387-40620-6}
}

@mastersthesis{WojciakMsc,
	author  = {Wojciech Wójciak},
	title = "{Optimal Allocation in Stratified Sampling Schemes}",
	school = {Warsaw University of Technology},
	year = {2019},
	month = {jul},
	url = {http://home.elka.pw.edu.pl/~wwojciak/msc_wwojciech_optimum_alloc.pdf},
}

@phdthesis{WojciakPhD,
	author = {Wojciech Wójciak},
	title = {Multi-Domain Optimum Sample Allocation with Controlled-Precision under Upper-Bound Constraints},
	school = {Warsaw University of Technology},
	year = {2026},
	url = {http://home.elka.pw.edu.pl/~wwojciak/phd_wwojciech_optimum_alloc.pdf},
}

@Manual{microbenchmark,
	title = "{{\tt microbenchmark}: Accurate Timing Functions}",
	author = {Olaf Mersmann},
	year = {2021},
	note = {R package version 1.4.9},
	url = {https://CRAN.R-project.org/package=microbenchmark},
}

@Article{Choudhry,
	author = {G. Hussain Choudhry and Mike Hidiroglou and J.N.K. Rao},
	title = "{On sample allocation for efficient domain estimation}",
	journal = {Survey Methodology},
	year = {2012},
	month = {jun},
	volume = {38},
	number = {1},
	pages = {23--29},
	publisher = {Statistics Canada},
	url = {https://www150.statcan.gc.ca/n1/en/catalogue/12-001-X201200111682},
}

@misc{Cont,
	title={Optimal rounding under integer constraints},
	author={Rama Cont and Massoud Heidari},
	year={2014},
	eprint={1501.00014},
	archivePrefix={arXiv},
	primaryClass={cs.DS},
	url={https://arxiv.org/abs/1501.00014},
}

@Article{Friedrich,
	author = {Ulf Friedrich and Ralf Münnich and Sven {de Vries} and Matthias Wagner},
	title = "{Fast integer-valued algorithms for optimal allocations under constraints in stratified sampling}",
	journal = {Computational Statistics \& Data Analysis},
	volume = {92},
	pages = {1-12},
	year = {2015},
	issn = {0167-9473},
	doi = {10.1016/j.csda.2015.06.003},
	url = {https://doi.org/10.1016/j.csda.2015.06.003},
	keywords = {German Census 2011, Non-linear discrete optimization, Optimal allocation, Greedy algorithm, Polymatroid, Box
	constraints},
	abstract = {In stratified random sampling, minimizing the variance of a total estimate leads to the optimal allocation.
	However, in practice, this original method is scarcely appropriate since in many applications additional constraints have to
	be considered. Three optimization algorithms are presented that solve the integral allocation problem with upper and lower
	bounds. All three algorithms exploit the fact that the feasible region is a polymatroid and share the important feature of
	computing the globally optimal integral solution, which generally differs from a solution obtained by rounding. This is in
	contrast to recent references which, in general, treat the continuous relaxation of the optimization problem. Two algorithms
	are of polynomial complexity and all of them are fast enough to be applied to complex problems such as the German
	Census 2011 allocation problem with almost 20,000 strata.},
}

@Article{GGM,
	abstract = {In stratified random sampling without replacement boundary conditions, such as the sample sizes within strata
	shall not exceed the population sizes in the respective strata, have to be considered. Stenger and Gabler (Metrika,
	61:137--156, 2005) have shown a solution that satisfies upper boundaries of sample fractions within the strata. However, in
	modern applications one may wish to guarantee also minimal sampling fractions within strata in order to allow for
	reasonable separate estimations. Within this paper, an optimal allocation in the Neyman-Tschuprov sense is developed
	which satisfies upper and lower bounds of the sample sizes within strata. Further, a stable algorithm is given which ensures
	optimality. The resulting sample allocation enables users to bound design weights within stratified random sampling while
	considering optimality in allocation.},
	author = {Siegfried Gabler and Matthias Ganninger and Ralf Münnich},
	date = {2012/02/01},
	date-added = {2023-03-19 23:27:24 +0100},
	date-modified = {2023-03-19 23:27:24 +0100},
	doi = {10.1007/s00184-010-0319-3},
	id = {Gabler2012},
	isbn = {1435-926X},
	journal = {Metrika},
	number = {2},
	pages = {151--161},
	title = "{Optimal allocation of the sample size to strata under box constraints}",
	url = {https://doi.org/10.1007/s00184-010-0319-3},
	volume = {75},
	year = {2012},
}

@Article{MSW,
	abstract = {Modern sampling designs in survey statistics, in general, are constructed in order to optimize the accuracy of
	estimators such as totals, means and proportions. In stratified random sampling a variance minimal solution was introduced
	by Neyman and Tschuprov. However, practical constraints may lead to limitations of the domain of sampling fractions
	which have to be considered within the optimization process. Special attention on the complexity of numerical solutions has
	to be paid in cases with many strata or when the optimal allocation has to be applied repeatedly, such as in iterative
	solutions of stratification problems. The present article gives an overview of recent numerical algorithms which allow
	adequate inclusion of box constraints in the numerical optimization process. These box constraints may play an important
	role in statistical modeling. Furthermore, a new approach through a fixed point iteration with a finite termination property is
	presented.},
	author = {Ralf T. Münnich and Ekkehard W. Sachs and Matthias Wagner},
	date = {2012/07/01},
	date-added = {2023-03-19 23:53:44 +0100},
	date-modified = {2023-03-19 23:53:44 +0100},
	doi = {10.1007/s10182-011-0176-z},
	isbn = {1863-818X},
	journal = {AStA Advances in Statistical Analysis},
	number = {3},
	pages = {435--450},
	title = "{Numerical solution of optimal allocation problems in stratified sampling under box constraints}",
	url = {https://doi.org/10.1007/s10182-011-0176-z},
	volume = {96},
	year = {2012},
}

@Article{Neyman,
	author = {Jerzy Neyman},
	title = "{On the Two Different Aspects of the Representative Method: the Method of Stratified Sampling and the Method of
	Purposive Selection}",
	journal = {Journal of the Royal Statistical Society},
	year = {1934},
	volume = {97},
	number = {4},
	pages = {558--625},
	publisher = {Blackwell Publishing for the Royal Statistical Society},
}

@Article{SG,
	abstract = {Very often values of a size variable are known for the elements of a population we want to sample. For example,
	the elements may be clusters, the size variable denoting the number of units in a cluster. Then, it is quite usual to base the
	selection of elements on inclusion probabilities which are proportionate to the size values. To estimate the total of all values
	of an unknown variable for the units in the population of interest (i.e. for the units contained in the clusters) we may use
	weights, e.g. inverse inclusion probabilities. We want to clarify these ideas by the minimax principle. Especially, we will
	show that the use of inclusion probabilities equal to 1 is recommendable for units with high values of the size measure.},
	author = {Horst Stenger and Siegfried Gabler},
	date = {2005/04/01},
	date-added = {2023-03-20 00:14:36 +0100},
	date-modified = {2023-03-20 00:14:36 +0100},
	doi = {10.1007/s001840400328},
	id = {Stenger2005},
	isbn = {1435-926X},
	journal = {Metrika},
	number = {2},
	pages = {137--156},
	title = "{Combining random sampling and census strategies - Justification of inclusion probabilities equal to 1}",
	url = {https://doi.org/10.1007/s001840400328},
	volume = {61},
	year = {2005},
}

@Article{Tschuprow1923a,
	author = {Alexander Alexandrovich Tschuprow},
	title = "{On the mathematical expectation of the moments of frequency distributions in the case of correlated observation
	(Chapters 1-3)}",
	journal = {Metron},
	year = {1923},
	volume = {2},
	number = {3},
	pages = {461--493},
}

@Article{Tschuprow1923b,
	author = {Alexander Alexandrovich Tschuprow},
	title = "{On the mathematical expectation of the moments of frequency distributions in the case of correlated observation
	(Chapters 4-6)}",
	journal = {Metron},
	year = {1923},
	volume = {2},
	number = {4},
	pages = {636--680},
}

@Article{WJWR2017,
	author = {Jacek Wesołowski and Robert Wieczorkowski},
	title = {An eigenproblem approach to optimal equal-precision sample allocation in subpopulations},
	journal = {Communications in Statistics - Theory and Methods},
	volume = {46},
	number = {5},
	pages = {2212--2231},
	year = {2017},
	publisher = {Taylor \& Francis},
	doi = {10.1080/03610926.2015.1040501},
	url = {https://doi.org/10.1080/03610926.2015.1040501},
}

@Article{Wesolowski,
	author = {Jacek Wesołowski},
	title = "{Multi-domain Neyman-Tchuprov optimal allocation}",
	journal = {Statistics in Transition new series},
	year = {2019},
	month = {dec},
	volume = {20},
	number = {4},
	pages = {1--12},
	publisher = {Polish Statistical Association and Statistics Poland},
	doi = {10.21307/stattrans-2019-031},
	url = {https://doi.org/10.21307/stattrans-2019-031},
}

@Article{WWW,
	author = {Jacek Wesołowski and Robert Wieczorkowski and Wojciech Wójciak},
	title = "{Optimality of the Recursive Neyman Allocation}",
	journal = {Journal of Survey Statistics and Methodology},
	volume = {10},
	number = {5},
	pages = {1263-1275},
	year = {2022},
	month = {nov},
	abstract = {We derive a formula for the optimal sample allocation in a general stratified scheme under upper bounds on the
	sample strata sizes. Such a general scheme includes SRSWOR within strata as a special case. The solution is given in terms
	of V-allocation with V being the set of take-all strata. We use V-allocation to give a formal proof of optimality of the popular
	recursive Neyman algorithm, rNa. We also propose a quick proof of optimality of the algorithm of Stenger and Gabler, SGa,
	as well as of our proposed modification, coma. Finally, we compare running times of rNa, SGa, and coma. Ready-to-use
	R-implementations of these algorithms are available on CRAN repository at
	https://cran.r-project.org/web/packages/stratallo.},
	issn = {2325-0984},
	doi = {10.1093/jssam/smab018},
	url = {https://doi.org/10.1093/jssam/smab018},
}

@Article{rnabox,
	author = {Jacek Wesołowski and Robert Wieczorkowski and Wojciech Wójciak},
	title = "{Recursive Neyman algorithm for optimum sample allocation under box constraints on sample sizes in strata}",
	journal = {Survey Methodology},
	volume = {50},
	number = {2},
	pages = {487--511},
	year = {2024},
	month = {dec},
	issn = {1492-0921},
	url = {https://www150.statcan.gc.ca/n1/en/catalogue/12-001-X201200111682},
}

@Article{WojciakLRNA,
	author = {Wojciech Wójciak},
	title = "{Another Solution for Some Optimum Allocation Problem}",
	journal = {Statistics in Transition new series},
	year = {2023},
	month = {dec},
	volume = {24},
	number = {5},
	pages = {203-219},
	publisher = {Polish Statistical Association and Statistics Poland},
	doi = {10.59170/stattrans-2023-071},
}
