{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T09:41:38Z","timestamp":1761644498893,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T00:00:00Z","timestamp":1761523200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A statistical manifold M can be defined as a Riemannian manifold each of whose points is a probability distribution on the same support. In fact, statistical manifolds possess a richer geometric structure beyond the Fisher information metric defined on the tangent bundle TM. Recognizing that points in M are distributions and not just generic points in a manifold, TM can be extended to a Hilbert bundle HM. This extension proves fundamental when we generalize the classical notion of a point estimate\u2014a single point in M\u2014to a function on M that characterizes the relationship between observed data and each distribution in M. The log likelihood and score functions are important examples of generalized estimators. In terms of a parameterization \u03b8:M\u2192\u0398\u2282Rk, \u03b8^ is a distribution on \u0398 while its generalization g\u03b8^=\u03b8^\u2212E\u03b8^ as an estimate is a function over \u0398 that indicates inconsistency between the model and data. As an estimator, g\u03b8^ is a distribution of functions. Geometric properties of these functions describe statistical properties of g\u03b8^. In particular, the expected slopes of g\u03b8^ are used to define \u039b(g\u03b8^), the \u039b-information of g\u03b8^. The Fisher information I is an upper bound for the \u039b-information: for all g, \u039b(g)\u2264I. We demonstrate the utility of this geometric perspective using the two-sample problem.<\/jats:p>","DOI":"10.3390\/e27111110","type":"journal-article","created":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T09:16:07Z","timestamp":1761642967000},"page":"1110","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Geometry of Statistical Manifolds"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9996-5627","authenticated-orcid":false,"given":"Paul W.","family":"Vos","sequence":"first","affiliation":[{"name":"Department of Public Health, Brody School of Medicine, East Carolina University, Greenville, NC 27834, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lauritzen, S.L. (1987). Chapter 4: Statistical Manifolds. 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Geom."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/0360-3016(84)90054-3","article-title":"Stage T3 squamous cell carcinoma of the glottic larynx treated with surgery and\/or radiation therapy","volume":"10","author":"Mendenhall","year":"1984","journal-title":"Int. J. Radiat. Oncol. Biol. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1002\/sim.8829","article-title":"Fay Keith Lumbard. Confidence Intervals for Difference in Proportions for Matched Pairs Compatible with Exact McNemars or Sign Tests","volume":"40","author":"Michael","year":"2021","journal-title":"Stat. Med."},{"key":"ref_15","unstructured":"R Core Team (2025). 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