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However, this interpretation is only valid before observing the sample. After observing the sample, any confidence sets have probability zero or one to contain the parameter value. In this paper, we provide a confidence set analysis for an observed sample based on fuzzy set theory by using the concept of membership functions. We show that the traditional ad hoc thresholds (the confidence and significance levels) can be attained from a general membership function. The applicability of the newly proposed theory is demonstrated by using well-known examples from the statistical literature and an application in the context of contingency tables.<\/jats:p>","DOI":"10.3390\/e18060211","type":"journal-article","created":{"date-parts":[[2016,5,30]],"date-time":"2016-05-30T10:26:58Z","timestamp":1464604018000},"page":"211","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Confidence Set Analysis for Observed Samples: A Fuzzy Set Approach"],"prefix":"10.3390","volume":"18","author":[{"given":"Jos\u00e9","family":"Gonz\u00e1lez","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica y Estad\u00edstica, Universidad de Playa Ancha, Valpara\u00edso 2340000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luis","family":"Castro","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica and CI<sup>2<\/sup>MA, Universidad de Concepci\u00f3n, Concepci\u00f3n 4030000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"V\u00edctor","family":"Lachos","sequence":"additional","affiliation":[{"name":"Departamento de Estat\u00edstica, Universidade Estadual de Campinas, Campinas 13083-859, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1327-3473","authenticated-orcid":false,"given":"Alexandre","family":"Patriota","sequence":"additional","affiliation":[{"name":"Departamento de Estat\u00edstica, IME, Universidade de S\u00e3o Paulo, S\u00e3o Paulo 05508-090, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,5,30]]},"reference":[{"key":"ref_1","first-page":"309","article-title":"On the mathematical foundations of theoretical statistics","volume":"222","author":"Fisher","year":"1922","journal-title":"Philos. 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