Estimation of Exponential Mixtures

Exponential mixtures are distributions of the form 8.1$$ \sum\limits_{i = 0}^k {{p_i}} \varepsilon xp\left( {{\lambda _i}} \right) $$With$$ {p_o} + \ldots + {p_k} = 1 $$and$$ {\lambda _i} > 0\left( {0 \leqslant i \leqslant k} \right) $$ Considering the huge literature on normal mixtures (see §3.4), the treatment of exponential mixtures is rather limited. A possible reason, as illustrated in this chapter, is that the components of (8.1) are much more difficult to distinguish than in the normal case of §3.4. Exponential mixtures with a small number of components are nonetheless used in the modeling of phenomena with positive output and long asymmetric tails, mainly in survival and duration setups, like the applications mentioned in Titterington, Smith and Makov (1985, p.17-21). We also illustrate this modeling in the case of hospitalization durations for which a two or three component exponential mixture is appropriate.