{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T23:15:33Z","timestamp":1790723733108,"version":"4.1.0"},"reference-count":11,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,5]],"date-time":"2023-01-05T00:00:00Z","timestamp":1672876800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Sciences and Engineering Researh Council of Canada","award":["RGPIN 2017-05179"],"award-info":[{"award-number":["RGPIN 2017-05179"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The two-parameter gamma distribution is one of the most commonly used distributions in analyzing environmental, meteorological, medical, and survival data. It has a two-dimensional minimal sufficient statistic, and the two parameters can be taken to be the mean and shape parameters. This makes it closely comparable to the normal model, but it differs substantially in that the exact distribution for the minimal sufficient statistic is not available. A Bartlett-type correction of the log-likelihood ratio statistic is proposed for the one-sample gamma mean problem and extended to testing for homogeneity of k\u22652 independent gamma means. The exact correction factor, in general, does not exist in closed form. In this paper, a simulation algorithm is proposed to obtain the correction factor numerically. Real-life examples and simulation studies are used to illustrate the application and the accuracy of the proposed method.<\/jats:p>","DOI":"10.3390\/e25010111","type":"journal-article","created":{"date-parts":[[2023,1,5]],"date-time":"2023-01-05T03:12:15Z","timestamp":1672888335000},"page":"111","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Comparing Several Gamma Means: An Improved Log-Likelihood Ratio Test"],"prefix":"10.3390","volume":"25","author":[{"given":"Augustine","family":"Wong","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,5]]},"reference":[{"key":"ref_1","unstructured":"Fraser, D.A.S. (1968). The Structure of Inference, Wiley."},{"key":"ref_2","first-page":"41","article-title":"Exponential tranformation models","volume":"39","author":"Blasild","year":"1982","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_3","first-page":"135","article-title":"Inference for the mean of a gamma distribution with unknown shape parameter","volume":"13","author":"Jensen","year":"1986","journal-title":"Scand. J. Stat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"91","DOI":"10.2307\/3315359","article-title":"Simple and accurate inference for the mean of the gamma model","volume":"25","author":"Fraser","year":"1997","journal-title":"Can. J. Stat."},{"key":"ref_5","unstructured":"Gross, A.J., and Clark, V.A. (1975). Survival Distributions: Reliability Applications in the Biomedical Sciences, Wiley."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Cox, D.R., and Hinkley, D.V. (1979). Theoretical Statistics, CRC Press.","DOI":"10.1201\/b14832"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Barndorff-Nielsen, O.E., and Cox, D.R. (1994). Inference and Asymptotics, Chapman and Hall.","DOI":"10.1007\/978-1-4899-3210-5"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1515","DOI":"10.1080\/03610929108830579","article-title":"Saddlepoint approximation to exact test and improved likelihood ratio tests for the gamma distribution","volume":"20","author":"Jensen","year":"1991","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1098\/rspa.1937.0109","article-title":"Properties of sufficiency and statistical tests","volume":"160","author":"Bartlett","year":"1937","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_10","first-page":"33","article-title":"Ancillaries and third order significance","volume":"7","author":"Fraser","year":"1995","journal-title":"Util. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Cox, D.R., and Snell, E.J. (1981). Applied Statistics: Principles and Examples, Chapman and Hall.","DOI":"10.1007\/978-94-009-5838-8"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/111\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:59:26Z","timestamp":1760119166000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/111"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,5]]},"references-count":11,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010111"],"URL":"https:\/\/doi.org\/10.3390\/e25010111","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,5]]}}}