Organization of this chapter

The rest of this chapter is organized as follows. Section 2.2 provides a brief tutorial on the PH distribution. For the purpose of understanding the rest of the chapter, one only needs to understand the definitions of the PH distribution and its subclasses. In Section 2.3, we review the state of the art in the moment matching algorithm and characterization of ${\cal S}^{(n)}$.

Sections 2.4-2.8 are devoted to the characterization of ${\cal S}^{(n)}$ and three variants of moment matching algorithms. The characterization of ${\cal S}^{(n)}$ is covered primarily in Section 2.4. In Section 2.5, we study properties of the EC distribution in depth. In particular, we will find that for the purpose of moment matching it suffices to narrow down the set of EC distributions further from six free parameters to five free parameters, by optimally fixing one of the parameters. Section 2.6 provides the Simple solution, Section 2.7 provides the Complete solution, and Section 2.8 provides the Positive solution.