{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T06:54:56Z","timestamp":1763535296562,"version":"build-2065373602"},"reference-count":7,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2014,7,18]],"date-time":"2014-07-18T00:00:00Z","timestamp":1405641600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>One dimensional exponential families on finite sample spaces are studied using the geometry of the simplex \u0394n\u00b0-1  and that of a transformation Vn-1 of its interior. This transformation is the natural parameter space associated with the family of multinomial distributions. The space Vn-1  is partitioned into cones that are used to find one dimensional families with desirable properties for modeling and inference. These properties include the availability of uniformly most powerful tests and estimators that exhibit optimal properties in terms of variability and unbiasedness.<\/jats:p>","DOI":"10.3390\/e16074088","type":"journal-article","created":{"date-parts":[[2014,7,18]],"date-time":"2014-07-18T10:30:52Z","timestamp":1405679452000},"page":"4088-4100","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Using Geometry to Select One Dimensional Exponential Families That Are Monotone Likelihood Ratio in the Sample Space, Are Weakly Unimodal and Can Be Parametrized by a Measure of Central Tendency"],"prefix":"10.3390","volume":"16","author":[{"given":"Paul","family":"Vos","sequence":"first","affiliation":[{"name":"Department of Biostatistics, East Carolina University, Greenville, NC 27858, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karim","family":"Anaya-Izquierdo","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Bath, Bath BA27AY, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,7,18]]},"reference":[{"key":"ref_1","first-page":"162","article-title":"Two Generalizationsof the Binomial Distribution","volume":"27","author":"Altham","year":"1978","journal-title":"J. R. Stat. Soc. Ser. C (Appl. Stat.)"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1016\/S0167-7152(97)00088-6","article-title":"An alternative representation of Altham\u2019s multiplicative-binomial distribution","volume":"36","author":"Lovison","year":"1998","journal-title":"Stat. Probab. Lett"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Brown, L. (1986). Fundamentals of Statistical Exponential Families: With Applications in Statistical Decision Theory; IMS Lecture Notes;, Institute of Mathematical Statistics.","DOI":"10.1214\/lnms\/1215466757"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1080\/01621459.1986.10478327","article-title":"Double Exponential Families and Their Use in Generalized Linear Regression","volume":"81","author":"Efron","year":"1986","journal-title":"J. Am. Stat. Assoc"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Aitchison, J. (1986). The Statistical Analysis of Compositional Data, Chapman and Hall.","DOI":"10.1007\/978-94-009-4109-0"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1016\/j.jspi.2012.01.002","article-title":"Decomposition of Kullback\u2013Leibler risk and unbiasedness for parameter-free estimators","volume":"142","author":"Wu","year":"2012","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Vos, P., and Wu, Q. (2014). Maximum Likelihood Estimators Uniformly Minimize Distribution Variance among Distribution Unbiased Estimators in Exponential Families.. Bernoulli, submitted.","DOI":"10.3150\/14-BEJ637"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/7\/4088\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:13:47Z","timestamp":1760217227000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/7\/4088"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,7,18]]},"references-count":7,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2014,7]]}},"alternative-id":["e16074088"],"URL":"https:\/\/doi.org\/10.3390\/e16074088","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,7,18]]}}}