Properties of PH distributions

Below, we summarize some of the basic properties of the PH distribution. First, the set of PH distributions is quite broad and, in theory, any nonnegative distribution can be approximated arbitrarily closely by a PH distribution.

Proposition 2   [132] The set of PH distributions is dense in the set of nonnegative distributions (distributions with support on $[0,\infty)$).

Observe that Proposition 1 follows immediately from Proposition 2.

Second, the set of PH distributions is closed under some operations. In particular, a mixture of independent PH distributions is a PH distribution, and the convolution of independent PH distributions is a PH distribution.

Proposition 3   [111] Consider two PH distributions: PH( $\Vec{\tau_1},\mathbf{T_1}$) with distribution function $F_1(\cdot)$ and PH( $\Vec{\tau_2},\mathbf{T_2}$) with distribution function $F_2(\cdot)$. A mixture of the two PH distribution, which has distribution function $pF_1(\cdot)+(1-p)F_2(\cdot)$, is a PH distribution, PH( $\Vec\tau,\mathbf{T}$), where

$$\Vec\tau = (p\Vec{\tau_1}, (1-p)\Vec{\tau_2}) \quad\mbox{and... ... & \mathbf{0}\\ \mathbf{0} & \mathbf{T_2} \end{array}\right).$$

Here, $\mathbf{0}$ denotes a zero matrix.

The convolution of the two PH distributions, PH( $\Vec{\tau_1},\mathbf{T_1}$) and PH( $\Vec{\tau_2},\mathbf{T_2}$), is a PH distribution, PH( $\Vec\tau,\mathbf{T}$), where

$$\Vec\tau = (\Vec{\tau_1}, \tau_{10}\Vec{\tau_2}) \quad\mbox{... ...}\Vec{\tau_2}\\ \mathbf{0} & \mathbf{T_2} \end{array}\right).$$

Here, $\Vec{t_1}=-\mathbf{T_1}\Vec{1}$ and $\tau_{10} = 1 - \Vec{\tau_1}\Vec{1}$, where $\Vec{1}$ is a column vector of 1's.

To shed light on the expression $F(\cdot) = pF_1(\cdot) + (1-p)F_2(\cdot)$, consider a random variable $V_1$ whose distribution function is $F_1(\cdot)$ and a random variable $V_2$ whose distribution function is $F_2(\cdot)$. Then, random variable

$$V = \left\{\begin{array}{ll} V_1 & \mbox{with probability } p\\ V_2 & \mbox{with probability } 1-p \end{array}\right.$$

has distribution function $F(\cdot)$. Below, unless otherwise stated, we denote the (cumulative) distribution function of a distribution, $X$, by $X(\cdot)$.

Definition 9   Let $V_X$ be a random variable having a distribution $X$. We denote the cumulative distribution function by $X(\cdot)$, namely

$$X(x) \equiv \mbox{Pr}(V_X \leq x).$$

Finally, the distribution function, the density function, the moments, and the Laplace transform of a PH distribution have simple mathematical expressions.

Proposition 4   [111] The distribution function of PH( $\Vec{\tau},\mathbf{T}$) is given by

$$F(x) = 1 - \Vec{\tau} \exp(\mathbf{T}x) \Vec{1}$$

for $x\geq 0$, where the matrix exponential is defined by $\exp(\mathbf{X}) = \sum_{i=0}^\infty \frac{1}{i!} \mathbf{X}^i$. The density function of PH( $\Vec{\tau},\mathbf{T}$) is given by

$$f(x) = \Vec{\tau} \exp(\mathbf{T}x) \Vec{t}$$

for $x\geq 0$, where $\Vec{t}=-\mathbf{T}\Vec{1}$.

Let $X$ be a random variable with the PH( $\Vec{\tau},\mathbf{T}$) distribution. Then,

$$\mbox{{\bf\sf E}}\left[ X^i \right] = i! \Vec{\tau} (-\mathbf{T}^{-1})^i \Vec{1}$$

for $i\geq 1$. The Laplace transform of PH( $\Vec{\tau},\mathbf{T}$) is given by

$$\widetilde X(s) = \tau_0 + \Vec\tau (s\mathbf{I}-\mathbf{T})^{-1} \Vec{t},$$

where $\tau_0 = 1 - \Vec\tau\Vec{1}$ and $\mathbf{I}$ is an identity matrix.