Acta Scientiarum http://periodicos.uem.br/ojs ISSN on-line: 1807-8664 Doi: 10.4025/actascitechnol.v45i1.61519 STATISTICS Acta Scientiarum. Technology, v. 45, e61519, 2023 Normal-tangent-logarithm-(  ,  ): a class of probabilistic distributions depending on two baselines Natália Moraes Cordeiro 1 , Frank Gomes-Silva 1* , Cícero Carlos Ramos de Brito 2 , Jader da Silva Jale 1 and Josimar Mendes de Vasconcelos 1 1 Departamento de Estatística e Informática, Universidade Federal Rural de Pernambuco, Rua Dom Manuel de Medeiros, s/n, 52171-900, Dois Irmãos, Recife, Pernambuco, Brazil. 2 Instituto Federal de Educação, Ciência e Tecnologia de Pernambuco, Recife, Pernambuco, Brazil. *Author for correspondence. E-mail: franksinatrags@gmail.com ABSTRACT. Based on the normal distribution, a new generator of continuous distributions is presented using the monotonic functions tan((/2)G 1 ) and log (1 − G 2 ) , such that G 1 and G 2 are the baselines. A study of identifiability of the proposed class is exhibited as well as the series expansions for its cumulative distribution function and probability density function. Additionally, some mathematical properties of the class are discussed, namely, the raw moments, the central moments, the moment generating function, the characteristic function, the derivatives of the log-likelihood function, and a study of the support. A numerical analysis comprising a simulation study and an application to real data is presented. Comparisons between the proposed model and other well-known models evince its potentialities and modeling benefits. Keywords: normal distribution; goodness-of-fit; identifiability; maximum likelihood; Monte Carlo simulation. Received on November 11, 2021. Accepted on August 22, 2022. Introduction Probability distributions play statistics a fundamental role. Probability models are important tools to deal with real problems since they can provide powerful models to describe natural and social phenomena. In recent times, several methods to generate new distributions have been presented in order to create distributions with higher flexibility than the classical ones. A review of some relevant methods is presented in Lee, Famoye, and Alzaatreh (2013) and a detailed list of generalized classes of continuous distributions widely found in the statistical literature is cited in Tahir and Nadarajah (2015). Some notable examples are the − family of continuous distributions (Alzaatreh, Lee, & Famoye, 2013), the beta-G (Eugene, Lee, & Famoye, 2002), the McDonald-G (Alexander, Cordeiro, Ortega, & Sarabia, 2012), the exponencialized exponential-Poisson (Ristić & Nadarajah, 2014), the logistic- G (Tahir, Cordeiro, Alzaatreh, Mansoor, & Zubair, 2016a), the new Weibull-G (Tahir et al., 2016b), the new gamma-G (de Brito, Rêgo, de Oliveira, & Gomes-Silva, 2017) and the normal-G (Silveira et al., 2019). Probability mixture models are often used in data modeling with more than one mode. Bimodal distributions are quite useful tools because they model relevant variables in nature. For instance, the sizes of the weaver ant workers are bimodally distributed (Nichols & Padgett, 2006) as well as the number of cases per year of Hodgkin’s lymphoma (Mauch, Armitage, Diehl, Hoppe, & Weiss, 1999). In inferential terms, this modeling can be difficult since mixture models can admit a reasonable amount of parameters. It is well-known that probability mixture models can lead to identifiability problems making parametric inferences becomes a hard task. The two- component normal mixture model is a classic example of this problem that can be found (Teicher, 1961). In this work we employ a method to generate classes of probability distributions (de Brito, Rêgo, de Oliveira, & Gomes-Silva, 2019) to build a class whose cumulative distribution function (cdf) is written as a composition of the standard normal cdf and two baselines, namely, G 1 and G 2 . One of the new features of this method is working with multiple baselines. Since this method considers the Lebesgue integral instead of the Riemann integral, one can choose either continuous or discrete distributions to be baselines. The proposed class is called Normal-tangent-logarithm- (G 1 ,G 2 ) , NTL- (G 1 ,G 2 ) for short, and besides furnish a more parsimonious model, it holds interesting properties, like the parametric identifiability (under certain conditions) and the bimodality of some special cases. The name of the class alludes to the monotonic functions that are used in its definition.