Alizadeh et al. Journal of Statistical Distributions and Applications (2015) 2:4 DOI 10.1186/s40488-015-0027-7 METHODOLOGY Open Access The beta Marshall-Olkin family of distributions Morad Alizadeh1, Gauss M. Cordeiro2, Edleide de Brito3* and Clarice Garcia B. Demétrio4 *Correspondence:
[email protected] Abstract 3 Departamento de Estatística, We study general mathematical properties of a new generator of continuous Universidade Federal da Bahia, Salvador, Bahia, Brazil distributions with three extra shape parameters called the beta Marshall-Olkin family. Full list of author information is We present some special models and investigate the asymptotes and shapes. The new available at the end of the article density function can be expressed as a mixture of exponentiated densities based on the same baseline distribution. We derive a power series for its quantile function. Explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon and Rényi entropies and order statistics, which hold for any baseline model, are determined. We discuss the estimation of the model parameters by maximum likelihood and illustrate the flexibility of the family by means of two applications to real data. PACS: 02.50.Ng, 02.50.Cw, 02.50.-r Mathematics Subject Classification (2010): 62E10, 60E05, 62P99 Keywords: Generated family; Marshall-Olkin family; Maximum likelihood; Moment; Order statistic; Quantile function; Rényi entropy 1 Introduction Recently, some attempts have been made to define new families to extend well-known distributions and at the same time provide great flexibility in modelling data in practice. So, several classes by adding one or more parameters to generate new distributions have been proposed in the statistical literature.