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We show the relationship between the ESS expressions used in the literature and two entropy families, the R\u00e9nyi and Tsallis entropy. The R\u00e9nyi entropy is connected to the Huggins-Roy\u2019s ESS family introduced in Huggins and Roy (2015). We prove that that all the ESS functions included in the Huggins-Roy\u2019s family fulfill all the desirable theoretical conditions. We analyzed and remark the connections with several other fields, such as the Hill numbers introduced in ecology, the Gini inequality coefficient employed in economics, and the Gini impurity index used mainly in machine learning, to name a few. Finally, by numerical simulations, we study the performance of different ESS expressions contained in the previous ESS families in terms of approximation of the theoretical ESS definition, and show the application of ESS formulas in a variable selection problem.<\/jats:p>","DOI":"10.1007\/s00180-025-01665-8","type":"journal-article","created":{"date-parts":[[2025,7,29]],"date-time":"2025-07-29T12:32:50Z","timestamp":1753792370000},"page":"5433-5464","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Effective sample size approximations as entropy measures"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7611-6558","authenticated-orcid":false,"given":"L.","family":"Martino","sequence":"first","affiliation":[]},{"given":"V.","family":"Elvira","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,29]]},"reference":[{"issue":"2","key":"1665_CR1","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1109\/78.978374","volume":"50","author":"MS Arulumpalam","year":"2002","unstructured":"Arulumpalam MS, Maskell S, Gordon N, Klapp T (2002) A tutorial on particle filters for online nonlinear\/non-Gaussian Bayesian tracking. 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