Finite mixtures of unimodal beta and gamma densities and the <img src="/fulltext-image.asp?format=htmlnonpaginated&src=Y50G758153125885_html\180_2012_367_Article_IEq1.gif" border="0" alt="$$k$$" />-bumps algorithm

This paper addresses the problem of estimating a density, with either a compact support or a support bounded at only one end, exploiting a general and natural form of a finite mixture of distributions. Due to the importance of the concept of multimodality in the mixture framework, unimodal beta and gamma densities are used as mixture components, leading to a flexible modeling approach. Accordingly, a mode-based parameterization of the components is provided. A partitional clustering method, named $$k$$ -bumps, is also proposed; it is used as an ad hoc initialization strategy in the EM algorithm to obtain the maximum likelihood estimation of the mixture parameters. The performance of the $$k$$ -bumps algorithm as an initialization tool, in comparison to other common initialization strategies, is evaluated through some simulation experiments. Finally, two real applications are presented.