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Jacek Wesołowski

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Wesołowski J.
Images. The International Journal of European Film, Performing Arts and Audiovisual Communication
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2014
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vol. 14
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issue 23
219-243
2
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Rotation schemes and Chebyshev polynomials

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Wesołowski J.
Statistics in Transition new series
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2023
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vol. 24
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issue 3
47-59
EN
There is a continuing interplay between mathematics and survey methodology involving different branches of mathematics, not only probability. This interplay is quite obvious as regards the first of the two options: probability vs. non-probability sampling, as proposed and discussed in Kalton (2023). There, mathematics is represented by probability and mathematical statistics. However, sometimes connections between mathematics and survey methodology are less obvious, yet still crucial and intriguing. In this paper we refer to such an unexpected relation, namely between rotation sampling and Chebyshev polynomials. This connection, introduced in Kowalski and Wesołowski (2015), proved fundamental for the derivation of an explicit form of the recursion for the BLUE µ^t of the mean on each occasion t in repeated surveys based on a cascade rotation scheme. This general result was obtained under two basic assumptions: ASSUMPTION I and ASSUMPTION II, expressed in terms of the Chebyshev polynomials. Moreover, in that paper, it was conjectured that these two assumptions are always satisfied, so the derived form of recursion is universally valid. In this paper, we partially confirm this conjecture by showing that ASSUMPTION I is satisfied for rotation patterns with a single gap of an arbitrary size.
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Multi-domain Neyman-Tchuprov optimal allocation

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Wesołowski J.
Statistics in Transition new series
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2019
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vol. 20
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issue 4
1-12
EN
The eigenproblem solution of the multi-domain efficient allocation is identified as a direct generalization of the classical Neyman-Tchuprov optimal allocation in stratified SRSWOR. This is achieved through analysis of eigenvalues and eigenvectors of a suitable population-based matrix D. Such a solution is an analytical companion to NLP approaches, which are often used in applications, see, e.g. Choudhry, Rao and Hidiroglou (2012). In this paper we are interested rather in the structure of the optimal allocation vector and relative variance than in such purely numerical tools (although the eigenproblem solution provides also numerical solutions, see, e.g. Wesołowski and Wieczorkowski (2017)). The domain-wise optimal allocation and the respective optimal variance of the estimator are determined by the unique I direction (defined in terms of the positive eigenvector of matrix D) in the space R , where I is the number of domains in the population.
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