Characterizing PH distributions

Figure 2.9: Set ${\cal S}^{(2)}$ and set ${\cal S}^{(2)^*}$. Here, Exp denotes the exponential distribution, and E$_2$ denotes the Erlang-2 distribution.

All prior work on characterizing ${\cal S}^{(n)}$ has focused on characterizing ${\cal S}^{(2)^*}$, where ${\cal S}^{(2)^*}$ is the set of distributions that are well-represented by a two-phase Coxian$^+$ PH distribution. Observe ${\cal S}^{(2)^*} \subset {\cal S}^{(2)}$. Altiok [6] showed a sufficient condition for a distribution to be in ${\cal S}^{(2)^*}$. More recently, Telek and Heindl [190] proved the necessary and sufficient condition for a distribution to be in ${\cal S}^{(2)^*}$. While neither Altiok nor Telek and Heindl expressed these conditions in terms of normalized moments, the results can be expressed more simply with our normalized moments:

Theorem 2   [144,190] $G\in {\cal S}^{(2)^*}$ iff $G$ satisfies exactly one of the following three conditions:

$$\begin{eqnarray*} & & \mbox{(i) } \frac{9m_2^G-12+3\sqrt{2}(2-m_2^G)^{\frac{3}{2... ...iii) } \frac{3}{2} m_2^G < m_3^G \quad\mbox{and}\quad 2 < m_2^G. \end{eqnarray*}$$

Figure 2.9 shows an exact characterization of set ${\cal S}^{(2)}$ and set ${\cal S}^{(2)^*}$ via normalized moments. We provide an exact characterization of set ${\cal S}^{(2)}$ in [147]. In this chapter, we will characterize ${\cal S}^{(n)}$, for all integers $n \geq 2$.