{"title":"Rotation schemes and Chebyshev polynomials","authors":"J. Wesołowski","doi":"10.59170/stattrans-2023-035","DOIUrl":null,"url":null,"abstract":"There is a continuing interplay between mathematics and survey methodology\n involving different branches of mathematics, not only probability. This interplay is\n quite obvious as regards the first of the two options: probability vs. non-probability\n sampling, as proposed and discussed in Kalton (2023). There, mathematics is represented\n by probability and mathematical statistics. However, sometimes connections between\n mathematics and survey methodology are less obvious, yet still crucial and intriguing.\n In this paper we refer to such an unexpected relation, namely between rotation sampling\n and Chebyshev polynomials. This connection, introduced in Kowalski and Wesołowski\n (2015), proved fundamental for the derivation of an explicit form of the recursion for\n the BLUE µˆt of the mean on each occasion t in repeated surveys based on a cascade\n rotation scheme. This general result was obtained under two basic assumptions:\n ASSUMPTION I and ASSUMPTION II, expressed in terms of the Chebyshev polynomials.\n Moreover, in that paper, it was conjectured that these two assumptions are always\n satisfied, so the derived form of recursion is universally valid. In this paper, we\n partially confirm this conjecture by showing that ASSUMPTION I is satisfied for rotation\n patterns with a single gap of an arbitrary size.","PeriodicalId":37985,"journal":{"name":"Statistics in Transition","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Statistics in Transition","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.59170/stattrans-2023-035","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
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.
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.