Finite mixtures of unimodal beta and gamma densities and the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$k$$</EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$k$$</EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$k$$</EquationSource> </InlineEquation>-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. Copyright Springer-Verlag Berlin Heidelberg 2013