title: Beyond Zipf's Law: The Lavalette Rank Function and Its Properties.
abstract: Although Zipf's law is widespread in natural and social data, one often encounters situations where one or both ends of the ranked data deviate from the power-law function. Previously we proposed the Beta rank function to improve the fitting of data which does not follow a perfect Zipf's law. Here we show that when the two parameters in the Beta rank function have the same value, the Lavalette rank function, the probability density function can be derived analytically. We also show both computationally and analytically that Lavalette distribution is approximately equal, though not identical, to the lognormal distribution. We illustrate the utility of Lavalette rank function in several datasets. We also address three analysis issues on the statistical testing of Lavalette fitting function, comparison between Zipf's law and lognormal distribution through Lavalette function, and comparison between lognormal distribution and Lavalette distribution.
authors.affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, México, DF, México.
authors.forename: Oscar
authors.identifier: http://orcid.org/0000-0002-2381-7035
authors.initials: O
authors.lastname: Fontanelli
authors.affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, México, DF, México.|Bioinformatics Group and Interdisciplinary Center for Bioinformatics, University of Leipzig, Leipzig, Germany.
authors.forename: Pedro
authors.initials: P
authors.lastname: Miramontes
authors.affiliation: Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui, China.
authors.forename: Yaning
authors.initials: Y
authors.lastname: Yang
authors.affiliation: Instituto de Física, Universidad Nacional Autónoma de México, México, DF, México.
authors.forename: Germinal
authors.initials: G
authors.lastname: Cocho
authors.affiliation: The Robert S. Boas Center for Genomics and Human Genetics, The Feinstein Institute for Medical Research, Northwell Health, Manhasset, NY, United States of America.
authors.forename: Wentian
authors.initials: W
authors.lastname: Li