title: Two new defective distributions based on the Marshall-Olkin extension.
abstract: The presence of immune elements (generating a fraction of cure) in survival data is common. These cases are usually modeled by the standard mixture model. Here, we use an alternative approach based on defective distributions. Defective distributions are characterized by having density functions that integrate to values less than 1, when the domain of their parameters is different from the usual one. We use the Marshall-Olkin class of distributions to generalize two existing defective distributions, therefore generating two new defective distributions. We illustrate the distributions using three real data sets.
authors.affiliation: Departamento de Estatística, Universidade Federal de São Carlos, São Carlos, SP, Brasil. ricardorocha23@hotmail.com.
authors.forename: Ricardo
authors.initials: R
authors.lastname: Rocha
authors.affiliation: School of Mathematics, University of Manchester, Manchester, UK.
authors.forename: Saralees
authors.initials: S
authors.lastname: Nadarajah
authors.affiliation: Departamento de Estatística, Universidade Federal de São Carlos, São Carlos, SP, Brasil.
authors.forename: Vera
authors.initials: V
authors.lastname: Tomazella
authors.affiliation: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, SP, Brasil.
authors.forename: Francisco
authors.initials: F
authors.lastname: Louzada
mesh_terms: D001499:Bayes Theorem; D003198:Computer Simulation; D003627:Data Interpretation, Statistical; D006801:Humans; D016013:Likelihood Functions; D015233:Models, Statistical; D016011:Normal Distribution; D013269:Stochastic Processes; D016019:Survival Analysis
keywords: Cure fraction; Defective models; Gompertz distribution; Inverse Gaussian distribution; Marshall–Olkin family; Survival analysis