Key idea

While our goal is to characterize the set ${\cal S}^{(n)}$, this characterization turns out to be ugly. One of the key ideas is that there is a set ${\cal T}^{(n)}\supset {\cal S}^{(n)}$ which is very close to ${\cal S}^{(n)}$ in size, such that ${\cal T}^{(n)}$ has a simple specification. To provide a simple specification of ${\cal T}^{(n)}$, we define an alternative to the standard moments, which we refer to as normalized moments:

Definition 6   Let $\mu_k^F$ be the $k$-th moment of a distribution $F$ for $k=1,2,3$. The normalized k-th moment $m_k^F$ of $F$ for $k=2,3$ is defined to be

$$m_2^F = \frac{\mu_2^F}{(\mu_1^F)^2} \quad\mbox{and}\quad m_3^F = \frac{\mu_3^F}{\mu_1^F\mu_2^F}.$$

Notice the relationship between the normalized moments and the coefficient of variability $C_F$ and the skewness $\gamma_F$ of $F$:

$$m_2^F = C_F^2 + 1 \quad\mbox{and}\quad m_3^F = \nu_F\sqrt{m_2^F},$$

where $\nu_F=\mu_3^F/(\mu_2^F)^{3/2}$. ($\nu_F$ and $\gamma_F$ are closely related, since $\gamma_F=\bar{\mu}_3^F/(\bar{\mu}_2^F)^{3/2}$, where $\bar{\mu}_k^F$ is the centralized $k$-th moment of $F$ for $k=2,3$.)

Figure 2.4: Set ${\cal T}^{(n)}$ is depicted as a function of the normalized moments. Sets ${\cal T}^{(n)}$ are delineated by solid lines, which include the border, and dashed lines, which do not include the border ($n=2,3,32$). Here, Exp denotes the exponential distribution, and E$_n$ denotes the Erlang-$n$ distribution. Observe that all possible nonnegative distributions lie within the region delineated by the two dotted lines: $m_2\geq 1$ and $m_3\geq m_2$ [96]. Also, a distribution $G$ is in ${\cal PH}_3$ iff its normalized moments satisfy $m_3^G > m_2^G > 1$ [88].

Now, ${\cal T}^{(n)}$ can be defined via normalized moments (see Figure 2.4).

Definition 7   For integers $n \geq 2$, let ${\cal T}^{(n)}$ denote the set of distributions, $F$, that satisfy exactly one of the following two conditions:

$$\begin{eqnarray*} & & \mbox{(i) } m_2^F > \frac{n+1}{n} \quad\mbox{and}\quad m_3... ..._2^F = \frac{n+1}{n} \quad\mbox{and}\quad m_3^F = \frac{n+2}{n}. \end{eqnarray*}$$