Rotation schemes and Chebyshev polynomials

Q4 Mathematics
J. Wesołowski
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引用次数: 0

Abstract

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 µˆt 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.
旋转格式与Chebyshev多项式
数学和调查方法之间有持续的相互作用,涉及不同的数学分支,而不仅仅是概率。Kalton(2023)提出并讨论了两种选择中的第一种:概率与非概率抽样,这种相互作用非常明显。在那里,数学是由概率和数理统计来表示的。然而,有时数学和调查方法之间的联系不那么明显,但仍然是至关重要和有趣的。在本文中,我们提到了这样一种意想不到的关系,即旋转采样与切比雪夫多项式之间的关系。在Kowalski和Wesołowski(2015)中介绍的这种联系,证明了在基于级联旋转方案的重复调查中,每次t的平均值的BLUE μ t递归的显式形式推导的基础。这个一般结果是在两个基本假设下得到的:假设1和假设2,用切比雪夫多项式表示。此外,文中还推测这两个假设总是满足的,因此推导出的递归形式是普遍有效的。本文通过证明假设1对于任意大小的单间隙旋转模式是满足的,部分地证实了这一猜想。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Statistics in Transition
Statistics in Transition Decision Sciences-Statistics, Probability and Uncertainty
CiteScore
1.00
自引率
0.00%
发文量
0
审稿时长
9 weeks
期刊介绍: Statistics in Transition (SiT) is an international journal published jointly by the Polish Statistical Association (PTS) and the Central Statistical Office of Poland (CSO/GUS), which sponsors this publication. Launched in 1993, it was issued twice a year until 2006; since then it appears - under a slightly changed title, Statistics in Transition new series - three times a year; and after 2013 as a regular quarterly journal." The journal provides a forum for exchange of ideas and experience amongst members of international community of statisticians, data producers and users, including researchers, teachers, policy makers and the general public. Its initially dominating focus on statistical issues pertinent to transition from centrally planned to a market-oriented economy has gradually been extended to embracing statistical problems related to development and modernization of the system of public (official) statistics, in general.
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