mathematics Article Properties and Applications of a New Family of Skew Distributions Emilio Gómez-Déniz 1 , Barry C. Arnold 2 , José M. Sarabia 3 and Héctor W. Gómez 4, *   Citation: Gómez-Déniz, E.; Arnold, B.C.; Sarabia, J.M.; Gómez, H.W. Properties and Applications of a New Family of Skew Distributions. Mathematics 2021, 9, 87. https://doi. org/10.3390/math9010087 Received: 16 November 2020 Accepted: 29 December 2020 Published: 3 January 2021 Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional clai- ms in published maps and institutio- nal affiliations. Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and con- ditions of the Creative Commons At- tribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Quantitative Methods in Economics and TiDES Institute, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canarias, Spain; emilio.gomez-deniz@ulpgc.es 2 Statistics Department, University of California Riverside, Riverside, CA 92504, USA; barry.arnold@ucr.edu 3 Department of Quantitative Methods, CUNEF University, 28040 Madrid, Spain; josemaria.sarabia@cunef.edu 4 Departamento de Matemática, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile * Correspondence: hector.gomez@uantof.cl; Tel.: +56-55-2637-278 Abstract: We introduce two families of continuous distribution functions with not-necessarily symmetric densities, which contain a parent distribution as a special case. The two families proposed depend on two parameters and are presented as an alternative to the skew normal distribution and other proposals in the statistical literature. The density functions of these new families are given by a closed expression which allows us to easily compute probabilities, moments and related quantities. The second family can exhibit bimodality and its standardized fourth central moment (kurtosis) can be lower than that of the Azzalini skew normal distribution. Since the second proposed family can be bimodal we fit two well-known data set with this feature as applications. We concentrate attention on the case in which the normal distribution is the parent distribution but some consideration is given to other parent distributions, such as the logistic distribution. Keywords: logistic distribution; normal distribution; skew normal distribution; symmetric distribution 1. Introduction There are many situations in which empirical data show slight or marked asymmetry. This is frequently the case, for example, with actuarial and financial data which, in addition to this feature, have heavy tails reflecting the existence of extreme values. The se features mean that the data can not be adequately modeled by the Gauss (or normal) distribution. Furthermore, bimodal distributions appear naturally in many different scenarios. For ex- ample, in certain disease patterns, as well as in certain cancer incidence curves. Behind the bimodality (and multimodality as well) of some cancer incidence curves, and their study, clinicians can improve their understanding of cancer, the development process as well as the potential characteristics that identify cancer and that separate a particular type of cancer of all other types. This occurs, for example, in cases where there are two peaks of occurrence per age. The se cancers include Kaposi’s sarcoma and Hodgkin’s lymphoma. The latter type of cancer has two peaks of occurrence: in young people adults and middle-aged adults. On the other hand, the normal skew distribution appears naturally in stochastic frontier analysis when a normal distribution is assumed to represent the noise or idiosyncratic component and a half-normal distribution to represent the inefficiency term, in the event that the researcher imposes inefficient behavior on all firms in the sample of interest. See, for instance [1]. Recently, [2] introduces (using a finite mixture model) the zero inefficiency stochastic frontier model which can accommodate the presence of both efficient and inef- ficient firms in the sample by appearing various bimodal scenarios. The refore, it seems plausible to try to obtain families of distributions that incorporate bias to the normal distri- bution but that at the same time are more versatile in the sense of being able to adapt to the bimodal scenario that appears in different situations. Mathematics 2021, 9, 87. https://doi.org/10.3390/math9010087 https://www.mdpi.com/journal/mathematics