New defective models based on the Kumaraswamy family of distributions with application to cancer data sets

An alternative to the standard mixture model is proposed for modeling data containing cured elements or a cure fraction. This approach is based on the use of defective distributions to estimate the cure fraction as a function of the estimated parameters. In the literature there are just two of these distributions: the Gompertz and the inverse Gaussian. Here, we propose two new defective distributions: the Kumaraswamy Gompertz and Kumaraswamy inverse Gaussian distributions, extensions of the Gompertz and inverse Gaussian distributions under the Kumaraswamy family of distributions. We show in fact that if a distribution is defective, then its extension under the Kumaraswamy family is defective too. We consider maximum likelihood estimation of the extensions and check its finite sample performance. We use three real cancer data sets to show that the new defective distributions offer better fits than baseline distributions.