{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:41:47Z","timestamp":1760035307489,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,6]],"date-time":"2025-07-06T00:00:00Z","timestamp":1751760000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["PNURSP2025R376"],"award-info":[{"award-number":["PNURSP2025R376"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>It is frequent for real-life count data to show inflation in lower values; however, most of the well-known count distributions cannot capture such a feature. The present paper introduces a new distribution for modeling inflated count data in small values based on a conflation of distributions approach. The new distribution inherits some properties from Poisson distribution (PD) and logarithmic distribution (LD), making it a powerful modeling tool. It can serve as an alternative to PD, LD, and zero-truncated distributions. The new distribution is worth considering theoretically, as it belongs to the weighted PD family. With zero as a support point, two additional models are suggested for the new distribution. These modifications yield distributions that demonstrate overdispersion models comparable to the negative binomial distribution (NBD) while retaining essential PD properties, making them suitable for accurately representing count data with frequent events of low frequency and high variance. Furthermore, we discuss the superior performance of three new distributions in modeling real count data compared to traditional count distributions such as PD and NBD, as well as other discrete distributions. This paper examines the key statistical properties of the proposed distributions. A comparison of the novel and other distributions in the literature is shown employing real-life data from some domains. All of the computations shown in this study are generated using the R programming language.<\/jats:p>","DOI":"10.3390\/axioms14070518","type":"journal-article","created":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T06:03:13Z","timestamp":1751868193000},"page":"518","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Conflation of Poisson and Logarithmic Distributions with Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-1139","authenticated-orcid":false,"given":"Abdulhamid A.","family":"Alzaid","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-7199-1604","authenticated-orcid":false,"given":"Anfal A.","family":"Alqefari","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, Qassim University, P.O. Box 6644, Buraydah 51482, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7343-098X","authenticated-orcid":false,"given":"Najla","family":"Qarmalah","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"255","DOI":"10.2307\/2341080","article-title":"An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents","volume":"83","author":"Greenwood","year":"1920","journal-title":"J. R. Stat. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"145","DOI":"10.2307\/2529053","article-title":"The Discrete Poisson-Lindley Distribution","volume":"26","author":"Sankaran","year":"1970","journal-title":"Biometrics"},{"key":"ref_3","first-page":"146","article-title":"On Poisson-Akash distribution and its applications","volume":"3","author":"Shanker","year":"2016","journal-title":"Biom. Biostat. Int. J."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Shanker, R., Fesshaye, H., Shanker, R., Leonida, T.A., and Sium, S. (2017). On discrete Poisson-Shanker distribution and its applications. Biom. Biostat. Int. J., 5.","DOI":"10.15406\/bbij.2017.05.00121"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1287\/opre.10.1.51","article-title":"Network dispatching by the shortest-operation discipline","volume":"10","author":"Conway","year":"1962","journal-title":"Oper. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1111\/j.1467-9876.2005.00474.x","article-title":"A useful distribution for fitting discrete data: Revival of the Conway\u2013Maxwell\u2013Poisson distribution","volume":"54","author":"Shmueli","year":"2005","journal-title":"J. R. Stat. Soc. Ser. C Appl. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1111\/j.1469-1809.1934.tb02105.x","article-title":"The effect of methods of ascertainment upon the estimation of frequencies","volume":"6","author":"Fisher","year":"1934","journal-title":"Ann. Eugen."},{"key":"ref_8","first-page":"311","article-title":"On discrete distributions arising out of methods of ascertainment","volume":"27","author":"Rao","year":"1965","journal-title":"Sankhy\u0101 Indian J. Stat. Ser. A"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Johnson, N.L., Kemp, A.W., and Kotz, S. (2005). Univariate Discrete Distributions, John Wiley & Sons. [3rd ed.].","DOI":"10.1002\/0471715816"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3351","DOI":"10.1090\/S0002-9947-2011-05340-7","article-title":"Conflations of probability distributions","volume":"363","author":"Hill","year":"2011","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Alqefari, A.A., Alzaid, A.A., and Qarmalah, N. (2024). On the Conflation of Negative Binomial and Logarithmic Distributions. Axioms, 13.","DOI":"10.3390\/axioms13100707"},{"key":"ref_12","unstructured":"Abramowitz, M., and Stegun, I.A. (1968). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, US Government Printing Office. [10th ed.]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0304-4076(89)90079-1","article-title":"Using incomplete moments to measure inequality","volume":"42","author":"Butler","year":"1989","journal-title":"J. Econom."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1287\/mnsc.19.3.290","article-title":"The determination of partial moments","volume":"19","author":"Winkler","year":"1972","journal-title":"Manag. Sci."},{"key":"ref_15","unstructured":"Karlin, S. (1968). Total Positivity, Stanford University Press."},{"key":"ref_16","unstructured":"Barlow, R.E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Sellers, K.F. (2023). The Conway-Maxwell-Poisson Distribution, Cambridge University Press. [1st ed.].","DOI":"10.1017\/9781108646437"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1007\/s00199-004-0514-4","article-title":"Log-concave probability and its applications","volume":"26","author":"Bagnoli","year":"2005","journal-title":"Econ. Theory"},{"key":"ref_19","unstructured":"Bertin, E.M., Cuculescu, I., and Theodorescu, R. (2013). Unimodality of Probability Measures, Springer. [1st ed.]."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Shaked, M., and Shanthikumar, J.G. (2007). Stochastic Orders, Springer. [1st ed.].","DOI":"10.1007\/978-0-387-34675-5"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Dennis, J.E., and Schnabel, R.B. (1996). Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM.","DOI":"10.1137\/1.9781611971200"},{"key":"ref_22","first-page":"1","article-title":"On zero-truncation of Poisson and Poisson-Lindley distributions and their applications","volume":"2","author":"Shanker","year":"2015","journal-title":"Biom. Biostat. Int. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"65","DOI":"10.2307\/3001903","article-title":"The distribution of European corn borer larvae Pyrausta nubilalis (Hbn.), in field corn","volume":"13","author":"McGuire","year":"1957","journal-title":"Biometrics"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"167","DOI":"10.2478\/stattrans-2022-0036","article-title":"Poisson area-biased Ailamujia Distribution and its applications in environmental and medical sciences","volume":"23","author":"Aijaz","year":"2022","journal-title":"Stat. Transit. New Ser."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/518\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:05:22Z","timestamp":1760033122000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/518"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,6]]},"references-count":24,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070518"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070518","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,6]]}}}