Abstract
Abstract We study general mathematical properties of a new generator of continuous distributions with three extra shape parameters called the beta Marshall-Olkin family. We present some special models and investigate the asymptotes and shapes. The new 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
Highlights
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
For the beta Marshall-Olkin (BMO)-N and Beta Marshall-Olkin gamma (BMO-Ga) distributions introduced in Sections 3.1 and 3.3, the quantities ωr,k can be expressed in terms of the Lauricella functions of type A defined by FA(n)(a; b1, . . . , bn; c1, . . . , cn; x1, . . . , xn) =
14 Concluding remarks We define a new class of models, named the beta Marshall-Olkin-G (BMO-G) family of distributions by adding three shape parameters, which generalizes some well-known distributions in the statistical literature such as the normal, Weibull and beta distributions
Summary
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. We propose a new wider class of continuous distributions called the beta Marshall-Olkin (BMO) family by taking W [ G(x)] =. The BMO family is simulated by inverting (3) as follows: if V has a beta distribution with positive parameters a and b, the solution of the nonlinear equation xq = G−1. For the BMO-N and BMO-Ga distributions introduced in Sections 3.1 and 3.3, the quantities ωr,k can be expressed in terms of the Lauricella functions of type A (see Exton 1978; Trott 2006) defined by FA(n)(a; b1, . Equations (14) and (15) are the main results of this section
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