Some Results on the Non-Homogeneous Hofmann Process

Abstract

The classical counting processes (Poisson and negative binomial) are the most traditional discrete counting processes (DCPs); however, these are based on a set of rigid assumptions. We consider a non-homogeneous counting process (which we name non-homogeneous Hofmann process – NHP) that can generate the classical counting processes (CCPs) as special cases, and also allows modeling counting processes for event history data, which usually exhibit under- or over-dispersion. We present some results of this process that will allow us to use it in other areas and establish both the probability mass function (pmf) and the cumulative distribution function (cdf) using transition intensities. This counting process (CP) will allow other researchers to work on modelling the CP, where data dispersion exists in an efficient and more flexible way.

1. Introduction

In ref. [1], Hofmann introduced a new class of infinitely divisible mixed Poisson process (MPP), this broader class of CP allows obtaining other CCP by simply modifying or choosing its parameters, as well as Poisson, negative binomial, Poisson-Pascal among other distributions (see [2]). The family of distributions defined by Hofmann has been used in many types of applications of modelling and simulation studies that include topics such as accident models [3].

In this chapter, we analysed the event of number process $\left\{N\left(t\right),t\ge 0\right\}$ and used a broader CP, which is based on the Hofmann process. The appeal of this CP is that, analogous to the family of frequency distributions, it allows to generate several known CP. Through an NHP, we can generate the following as special cases: the Poisson counting process (PCP), the negative binomial counting process (NBCP) and the Poisson-Pascal process among other CCPs, and this allows us to obtain models for CP with under- or over-dispersion. The NHP was introduced by Hofmann [1] and has been used by other researchers [3, 4, 5]. Some properties of the NHP found by Walhin [2] are presented in this chapter, and we used the transition intensities to describe additional properties of the NHP.

The objective of this chapter is to present a unified view of related results on the NHP. The chapter is organised as follows: in Section 2, we present the NHP; in Section 3, we present some statistical properties, such as pmf and probability generating function (pgf), and formulas for the mean and variance are derived; in Section 4, we present various approaches for the NHP using CCP; in Section 5, we present other properties for NHP; finally, conclusions are presented.

2. Basic concepts of the NHP

Let us take $N\left(t\right)$ as the number of events that occurs in the time interval $\left(0,t\right]$ with $t>0$ and $N\left(0\right)=0$ . The probability of $n$ events occurring in this time interval is denoted by

$$P_n\left(t\right)=P\left[N\left(t\right)=n\right],n=\mathrm{0,1,2},\dots$$ E1

According to Dubourdieu [6], an MPP $\left\{N\left(t\right):t\ge 0\right\}$ is a PCP with rate $\mathrm{\Lambda }$ , where the non-negative random variable $\mathrm{\Lambda }$ is called a structure variable. The MPP has been studied by several authors [7, 8, 9].

When $\mathrm{\Lambda }$ is a continuous random variable with probability density function (pdf), $f\left(\lambda \right)$ , we can find probability by

$$\begin{array}{l}\underbrace{\mathbb{E}\left[P\left[N\left(t\right)=n,|,\mathrm{\Lambda }\right]\right]}=\underset{0}{\overset{\infty }{\int }}P\left[N\left(t\right)=n|,\mathrm{\Lambda }=\lambda \right]f\left(\lambda \right)\mathit{d\lambda }\\ P\left[N\left(t\right)=n\right]=\underset{0}{\overset{\infty }{\int }}{e}^{-\mathit{\lambda t}}\frac{{\left(\mathit{\lambda t}\right)}^n}{n!}f\left(\lambda \right)\mathit{d\lambda }.\end{array}$$ E2

For $n=0$ and $t>0$ we have

$$P_0\left(t\right)=\underset{0}{\overset{\infty }{\int }}{e}^{-\mathit{\lambda t}}f\left(\lambda \right)\mathit{d\lambda },$$ E3

The higher order derivatives of the last expression with respect to $t$ are

$$P_0^{\left(n\right)}\left(t\right)=\frac{d^n}{{\mathit{dt}}^n}{P}_0\left(t\right)={\left(-1\right)}^n\underset{0}{\overset{\infty }{\int }}{\lambda }^n{e}^{-\mathit{\lambda t}}f\left(\lambda \right)\mathit{d\lambda }.$$ E4

By substituting (4) into (2) we get

$$P_n\left(t\right)=\frac{t^n}{n!}\left[{\left(-1\right)}^n{P}_0^{\left(n\right)}\left(t\right)\right],n\ge 1$$ E5

The expressions (3) and (5) characterize an MPP with a continuous structure variable $\mathrm{\Lambda }$ . According to Hofmann [1], for the construction of examples, a special structure function is generally assumed, and from this the pmf is calculated by (3), (5). In most cases, this leads to formally complicated expressions. In ref. [1], Hofmann presents a CP called Hofmann process as an option to model the event number process given by (2) and whose general expression for (3) is as follows:

$$P_0\left(t\right)=exp\left\{-\theta \left(t\right)\right\}\theta \left(t\right)={\int }_0^t\lambda \left(\tau ;a\right)\mathit{d\tau }$$ E6

where $P_0\left(t\right)$ is a completely monotonic function¹. And $\lambda \left(\tau ;a\right)$ is a function of three parameters: $a\ge 0,$ $q>0$ and $\kappa \ge 0,$ which is a function infinitely divisible and given by

$$\lambda \left(\tau ;a\right)=\frac{q}{{\left(1+\kappa \tau \right)}^a}\forall \tau >0.$$ E7

Although $\lambda \left(\tau ;a\right)$ depends on three parameters, we use this notation given that the parameter $a$ provides various CCPs. We denote the NHP by $\mathcal{H}\left(a,q,\kappa \right)$ , if the pmf of $N\left(t\right)$ satisfies the expressions (5) and (6).

Using the expression (7), we get by integrating that

$$\theta \left(t\right)=\left\{\begin{array}{ll}ln\left[{\left(1+\mathit{\kappa t}\right)}^{q/\kappa }\right]& \text{if}a=1\\ \frac{q}{\kappa \left(1-a\right)}\left[{\left(1+\mathit{\kappa t}\right)}^{1-a}-1\right]& \text{if}a\ne 1\end{array}\right.$$ E8

By substituting (8) into (6)

$$P_0\left(t\right)=\left\{\begin{array}{ll}{\left(1+\mathit{\kappa t}\right)}^{-\frac{q}{\kappa }}& \text{if}a=1\\ exp\left\{-\frac{q}{\kappa \cdot \left(1-a\right)}\left[{\left(1+\mathit{\kappa t}\right)}^{1-a}-1\right]\right\}& \text{if}a\ne 1\end{array}\right.$$ E9

Remark 1.1: If in the expression (9) for $a=1$ we take the limit as $\kappa \to 0$ _, we have:

$$\underset{\kappa \to 0}{lim}{\left(1+\mathit{\kappa t}\right)}^{-\frac{q}{\kappa }}=e^{-\mathit{qt}},$$ E10

and the last expression agrees with the adequate $P_0\left(t\right)$ of a PCP with rate $\mathit{qt}$ .

3. Basic properties of the NHP

Theorem 1.2: Let $N\left(t\right)$ be an NHP then

1. The pgf of the process is given by

   $$G_N\left(z;t\right)=\left\{\begin{array}{ll}{\left(1+\kappa \left(1-z\right)t\right)}^{-q/\kappa }& \text{if}a=1\\ exp\left\{-\frac{q}{\kappa \left(1-a\right)}\left[{\left(1+\kappa \left(1-z\right)t\right)}^{1-a}-1\right]\right\}& \text{if}a\ne 1\end{array}\right.$$ E11

   Note that $G_N\left(z;t\right)=P_0\left(\left(1-z\right)t\right)$ with $0\le z<1$ .

2. The pmf of $N\left(t\right),$ for $t$ fixed, satisfies the following recursive formula:

   $$P_{n+1}\left(t\right)=\frac{\mathit{t\lambda }\left(t;a\right)}{n+1}\sum _{i=0}^n\left(\begin{array}{c}a+i-1\\ i\end{array}\right){\left(\frac{\mathit{\kappa t}}{1+\mathit{\kappa t}}\right)}^i{P}_{n-i}\left(t\right)$$ E12

   where $P_0\left(t\right)=G_N\left(0;t\right)$ is given by (9) and

   $$P_0^{\left(n+1\right)}\left(t\right)=\lambda \left(t;a\right)\sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right){\left(-1\right)}^{j+1}\frac{\mathrm{\Gamma }\left(a+j\right)}{\mathrm{\Gamma }\left(a\right)}{\left(\frac{\kappa }{1+\mathit{\kappa t}}\right)}^j{P}_0^{\left(n-j\right)}\left(t\right)$$

3. If $a=1$ the $P_n\left(t\right)$ satisfies the recurrence relation

   $$\frac{P_{n+1}\left(t\right)}{P_n\left(t\right)}=\frac{-t}{n+1}\frac{P_0^{\left(n+1\right)}\left(t\right)}{P_0^{\left(n\right)}\left(t\right)}=\frac{q+\mathit{\kappa n}}{1+\mathit{\kappa t}}\frac{t}{n+1}.$$ E13

4. The process $N\left(t\right)$ has a mean and variance given by

   $$\mathbb{E}\left[N\left(t\right)\right]=\mathit{qt}\text{and}\mathit{Var}\left[N\left(t\right)\right]=\left(1+\mathit{a\kappa t}\right)\mathbb{E}\left[N\left(t\right)\right]$$ E14

Proof:

See details in [2] or [10].

Note that from (14) we have that if $q\ne 0$ then:

$$\underset{t\to \infty }{lim}\frac{\mathbb{E}\left[N\left(t\right)\right]}{t}=q.$$ E15

It is possible from (14) to calculate the measure based on the variance-to-mean ratio (VMR) introduced by [11]:

$$\mathit{ID}\left(t\right)=\frac{\mathit{Var}\left[N\left(t\right)\right]}{\mathbb{E}\left[N\left(t\right)\right]}=1+\mathit{a\kappa t}.$$ E16

As $\mathit{ID}\left(t\right)>1$ , then using the criterion of the VMR, we have that the NHP is an over-dispersed CP and hence is an option for modelling over-dispersion in count data.

Using the expression (11), in Table 1 , we present the functions for $\mathit{qt}$ and $\mathit{\kappa t}$ that allow to obtain some CP. We consider the CCPs studied in [10], which are special cases of NHP when $a=1$ since this reduces to the Panjer counting process (see [12]). In addition, we consider other processes, such as the Neyman Type A process introduced by [13], the Poisson Pascal process introduced by [14] and the Pólya-Aeppli process introduced by [15].

	Counting process	$P_0\left[\left(1-z\right)t\right]$	Functions
			$\mathit{qt}$	$\mathit{\kappa t}$
Classical; (a = 1)	Poisson	$exp\left\{-\left(1-z\right)\mathit{\gamma t}\right\},\kappa \to 0$	$\mathit{\gamma t}$	$0$
	Negative binomial (or Pólya)	${\left[\frac{\delta }{\delta +\left(1-z\right)t}\right]}^{\gamma },\delta >0$	$\frac{\gamma }{\delta }t$	$\frac{t}{\delta }$
	Geometric	$\frac{\delta }{\delta +\left(1-z\right)t}$	$\frac{t}{\delta }$	$\frac{t}{\delta }$
Other; (a > 1)	Neyman Type A	$exp\left\{\gamma \left[exp\left\{\left(z-1\right)\mathit{\delta t}\right\}-1\right]\right\},a\to \infty$	$\mathit{\gamma \delta t}$	$\frac{\mathit{\delta t}}{a-1}$
	Poisson-Pascal	$exp\left\{\gamma \left[{\left(1+\left(1-z\right)\mathit{\delta t}\right)}^{-\left(a-1\right)}-1\right]\right\}$	$\left(a-1\right)\mathit{\gamma \delta t}$	$\mathit{\delta t}$
	Pólya-Aeppli	$exp\left\{\frac{-\left(1-z\right)\mathit{\gamma t}}{1-\left[1-{\left(1+\mathit{\delta t}\right)}^{-1}\right]z}\right\},a=2$	$\left(1+\mathit{\delta t}\right)\mathit{\gamma t}$	$\mathit{\delta t}$

Table 1.

Functions $\mathit{qt}$ and $\mathit{\kappa t}$ for some CCPs.

Source: own elaboration

3.1 NHP is infinitely divisible

The following relationships are identical to those of [16] which characterize infinitely divisible pmf:

Theorem 1.3: The pmf $\left\{P_n\left(t\right)\right\}$ with $P_0\left(t\right)>0$ is infinitely divisible if and only if satisfies that

$$\left(n+1\right)P_{n+1}\left(t\right)=\sum _{i=0}^n{r}_i\left(t\right)P_{n-i}\left(t\right)\text{for}t\text{fixed}.$$

where the quantities $r_n\left(t\right)$ with $n\in {\mathrm{\mathbb{Z}}}^{+}$ are nonnegative.

Proof: See details in [16].

Corollary 1.3.1: The pmf $\left\{P_n\left(t\right)\right\}$ of the NHP is infinitely divisible.

Proof:

By multiplying (12) by $\left(n+1\right)$ we get

$$\left(n+1\right)P_{n+1}\left(t\right)=\sum _{i=0}^n\mathit{t\lambda }\left(t;a\right)\left(\begin{array}{c}a+i-1\\ i\end{array}\right){\left(\frac{\mathit{\kappa t}}{1+\mathit{\kappa t}}\right)}^i{P}_{n-i}\left(t\right).$$

We denote

$$r_i\left(t;a\right)=\mathit{qt}\left(\begin{array}{c}a+i-1\\ i\end{array}\right)\frac{{\left(\mathit{\kappa t}\right)}^i}{{\left(1+\mathit{\kappa t}\right)}^{a+i}}i=0,1,\dots ,n.$$ E17

Note that $r_i\left(t;a\right)\ge 0$ , which allows to conclude that $P_n\left(t\right)$ is infinitely divisible.

The following relationship is given by [17]: all log-convex distributions are infinitely divisible but not all log-concave distributions are infinitely divisible.

Theorem 1.4: Let $N\left(t\right)$ be an infinitely divisible ${\mathrm{\mathbb{Z}}}^{+}$ -valued random variable with pmf $P_n\left(t\right)$ . Then

$$\mathbb{E}\left[N\left(t\right)\right]=\sum _{i=0}^{\infty }{r}_i\left(t;a\right)$$ E18

Proof:

We know that the expectation of $N\left(t\right)$ it is given by

$$\begin{array}{l}\mathbb{E}\left[N\left(t\right)\right]=\sum _{n=1}^{\infty }{\mathit{nP}}_n\left(t\right)=\sum _{m=0}^{\infty }\left(m+1\right)P_{m+1}\left(t\right)\\ =\sum _{m=0}^{\infty }\sum _{i=0}^m{r}_i\left(t;a\right)P_{m-i}\left(t\right)\end{array}$$

Now, by interchanging the order of summation, we get

$$\begin{array}{l}\mathbb{E}\left[N\left(t\right)\right]=\sum _{i=0}^{\infty }\sum _{m=i}^{\infty }{r}_i\left(t;a\right)P_{m-i}\left(t\right)=\sum _{i=0}^{\infty }{r}_i\left(t;a\right)\sum _{m=i}^{\infty }{P}_{m-i}\left(t\right)\\ \underset{j=m-i}{=}\sum _{i=0}^{\infty }{r}_i\left(t;a\right)\sum _{j=0}^{\infty }{P}_j\left(t\right)=\sum _{i=0}^{\infty }{r}_i\left(t;a\right).\end{array}$$

which completes the proof.

4. NHP in terms of CCPs

In this section, we present various approaches for the NHP using CCP.

4.1 NHP as a non-homogeneous pure birth process

We use logarithmic differentiation to find the derivative of (5) and we get

$$\frac{P_{n^{\prime }}\left(t\right)}{P_n\left(t\right)}=\frac{n}{t}+\frac{P_0^{\left(n+1\right)}\left(t\right)}{P_0^{\left(n\right)}\left(t\right)}$$

Then

$$P_{n^{\prime }}\left(t\right)=\frac{n}{t}{P}_n\left(t\right)+\frac{P_0^{\left(n+1\right)}\left(t\right)}{P_0^{\left(n\right)}\left(t\right)}{P}_n\left(t\right)$$ E19

From (5), we obtain

$$\begin{array}{c}\frac{n}{t}{P}_n\left(t\right)=-\frac{{\left(-1\right)}^{n-1}}{\left(n-1\right)!}{t}^{n-1}{P}_0^{\left(n\right)}\left(t\right)=\frac{{\left(-1\right)}^{n-1}}{\left(n-1\right)!}{t}^{n-1}{P}_0^{\left(n\right)}\left(t\right)\left(-\frac{P_0^{\left(n-1\right)}\left(t\right)}{P_0^{\left(n-1\right)}\left(t\right)}\right)\\ =-\frac{P_0^{\left(n\right)}\left(t\right)}{P_0^{\left(n-1\right)}\left(t\right)}{P}_{n-1}\left(t\right)\end{array}$$

By substituting in (19), we have

$$P_{n^{\prime }}\left(t\right)=\left(-\frac{P_0^{\left(n\right)}\left(t\right)}{P_0^{\left(n-1\right)}\left(t\right)}\right)P_{n-1}\left(t\right)-\left(-\frac{P_0^{\left(n+1\right)}\left(t\right)}{P_0^{\left(n\right)}\left(t\right)}\right)P_n\left(t\right).$$ E20

We denote

$${\lambda }_n\left(t;a\right)=-\frac{P_0^{\left(n+1\right)}\left(t\right)}{P_0^{\left(n\right)}\left(t\right)}=-\frac{d}{\mathit{dt}}ln\left[{\left(-1\right)}^n{P}_0^{\left(n\right)}\left(t\right)\right].$$ E21

In ref. [18], Lundberg shows that this corresponds to the transition intensities. Then from (20) and (21), we can derive the following system of Kolmogorov differential equations that must be satisfied by the NHP:

$$\begin{array}{l}{P}_0^{\prime }\left(t\right)=-{\lambda }_0\left(t;a\right)P_0\left(t\right)\\ P_n^{\prime }\left(t\right)={\lambda }_{n-1}\left(t;a\right)P_{n-1}\left(t\right)-{\lambda }_n\left(t;a\right)P_n\left(t\right)\text{for}n\ge 1.\end{array}$$ E22

By notation, we denote ${\lambda }_0\left(t;a\right)={\theta }^{\prime }\left(t\right)=\frac{q}{{\left(1+\mathit{\kappa t}\right)}^a}.$ With initial conditions

$$P_0\left(0\right)=1\text{and}{P}_n\left(0\right)=0\forall n\ge 1$$ E23

Using the method given in ref. [18], we find that the solution of (22) is given by

$$P_n\left(t\right)={\int }_0^t{\lambda }_{n-1}\left(\tau ;a\right)P_{n-1}\left(\tau \right)exp\left\{-{\int }_{\tau }^t{\lambda }_{n-1}\left(\nu ;a\right)\mathit{d\nu }\right\}\mathit{d\tau }\text{for}n\ge 1.$$

From the system of equations given in (22), we have that the NHP is a non-homogeneous pure birth process (NHPBP), which agrees with the definition given by Seal in ref. [19]. So, if $N\left(t\right)$ satisfies (6), then $N\left(t\right)$ is an NHPBP with transition intensities given by (21).

4.2 NHP as MPP

The list of equivalences provided by Lundberg in ref. [18] is satisfied by the NHP defined in (6), which is presented in the following theorem:

Theorem 1.5: Let $N\left(t\right)$ be an NHP with marginal pmf, given by (5) and transition intensities, given by (21). Then:

1. ${\lambda }_n\left(t;a\right)$ satisfy ${\lambda }_{n+1}\left(t;a\right)={\lambda }_n\left(t;a\right)-\frac{{\lambda }_n^{\prime }\left(t;a\right)}{{\lambda }_n\left(t;a\right)}$ for $n=0,1,\dots$

2. $P_n\left(t\right)$ and ${\lambda }_n\left(t;a\right)$ satisfy the relation

$$\frac{P_n\left(t\right)}{P_{n-1}\left(t\right)}=\frac{t}{n}{\lambda }_{n-1}\left(t;a\right)\text{for}n=1,2,\dots$$ E24

Proof:

1. By finding the derivative of function (21) with respect to $t$ , we obtain

   $$\begin{array}{c}{\lambda }_n^{\prime }\left(t;a\right)=-\left[\frac{P_0^{\left(n+2\right)}\left(t\right)P_0^{\left(n\right)}\left(t\right)-P_0^{\left(n+1\right)}\left(t\right)P_0^{\left(n+1\right)}\left(t\right)}{{\left(P_0^{\left(n\right)}\left(t\right)\right)}^2}\right]\\ =-\frac{P_0^{\left(n+2\right)}\left(t\right)}{P_0^{\left(n+1\right)}\left(t\right)}\frac{P_0^{\left(n+1\right)}\left(t\right)}{P_0^{\left(n\right)}\left(t\right)}+{\left(-\frac{P_0^{\left(n+1\right)}\left(t\right)}{P_0^{\left(n\right)}\left(t\right)}\right)}^2\\ =-{\lambda }_{n+1}\left(t;a\right){\lambda }_n\left(t;a\right)+{\left[{\lambda }_n\left(t;a\right)\right]}^2\end{array}$$

   By dividing by ${\lambda }_n\left(t;a\right)$ , we have

   $$\frac{{\lambda }_n^{\prime }\left(t;a\right)}{{\lambda }_n\left(t;a\right)}={\lambda }_n\left(t;a\right)-{\lambda }_{n+1}\left(t;a\right)$$ E25

2. By substituting (21) into (13), we get:

   $$\frac{P_n\left(t\right)}{P_{n-1}\left(t\right)}=\frac{\frac{{\left(-1\right)}^n}{n!}{t}^n{P}_0^{\left(n\right)}\left(t\right)}{\frac{{\left(-1\right)}^{n-1}}{\left(n-1\right)!}{t}^{n-1}{P}_0^{\left(n-1\right)}\left(t\right)}=-\frac{t}{n}\frac{P_0^{\left(n\right)}\left(t\right)}{P_0^{\left(n-1\right)}\left(t\right)}=\frac{t}{n}{\lambda }_{n-1}\left(t;a\right),$$

which completes the proof. □

In ref. [7], it is proved that the above three statements are equivalent.

Corollary 1.5.1: Let $N\left(t\right)$ be an NHP with transition intensities given by (21), then

$$\frac{P_n\left(t\right)}{P_0\left(t\right)}=\prod _{j=1}^n\frac{t{\lambda }_{j-1}\left(t;a\right)}{j}$$ E26

Proof:

Note that

$$\frac{P_n\left(t\right)}{P_0\left(t\right)}=\prod _{j=1}^n\frac{P_j\left(t\right)}{P_{j-1}\left(t\right)}.$$

Substituting (24) in the above expression completes the proof.

Corollary 1.5.2: Let $N\left(t\right)$ be an NHP with transition intensities given by (21), then

$$\prod _{j=0}^{n-1}{\lambda }_j\left(t;a\right)={\left(-1\right)}^n\frac{P_0^{\left(n\right)}\left(t\right)}{P_0\left(t\right)}n\ge 1.$$ E27

Proof:

From (21), we get

$$\prod _{j=0}^{n-1}{\lambda }_j\left(t;a\right)=\prod _{j=0}^{n-1}\left(-\frac{P_0^{\left(j+1\right)}\left(t\right)}{P_0^{\left(j\right)}\left(t\right)}\right)={\left(-1\right)}^n\frac{P_0^{\left(n\right)}\left(t\right)}{P_0\left(t\right)}.$$

This finishes the proof of Corollary.

The following additional properties set in ref. [9] are also satisfied by NHP:

Proposition 1.6: Let $\left\{N\left(t\right);t\ge 0\right\}$ be an NHP and $\mathrm{\Lambda }$ the continuous structure variable of the MPP. Then:

1. The transition intensities are such that

$$\mathbb{E}\left[\mathrm{\Lambda }|N\left(t\right)=n\right]={\lambda }_n\left(t;a\right).$$ E28

and

$$\mathit{Var}\left[\mathrm{\Lambda }|N\left(t\right)=n\right]=-{\lambda }_n^{\prime }\left(t;a\right).$$ E29

2. The mean of $N\left(t\right)$ is given by

$$\mathbb{E}\left[N\left(t\right)\right]=t\mathbb{E}\left[\mathrm{\Lambda }\right].$$ E30

3. The mean of $\mathrm{\Lambda }$ is given by

$$\mathbb{E}\left[\mathrm{\Lambda }\right]=-P_0^{\prime }\left(0\right).$$ E31

Proof:

1. From (2), taking the expected value of $\mathrm{\Lambda }$ , conditioning on $N\left(t\right)$ _, we get

$$\mathbb{E}\left[\mathrm{\Lambda }|N\left(t\right)=n\right]=\underset{0}{\overset{\infty }{\int }}\frac{\lambda e^{-\mathit{\lambda t}}{\left(\mathit{\lambda t}\right)}^{n}f\left(\lambda \right)}{n!P\left[N\left(t\right)=n\right]}\mathit{d\lambda }=\frac{n+1}{t}\frac{P_{n+1}\left(t\right)}{P_n\left(t\right)}.$$ E32

By substituting (24) into (32), we have

$$\mathbb{E}\left[\mathrm{\Lambda }|N\left(t\right)=n\right]={\lambda }_n\left(t;a\right).$$

Analogously, we can show that

$$\mathbb{E}\left[{\mathrm{\Lambda }}^2|N\left(t\right)=n\right]=\underset{0}{\overset{\infty }{\int }}\frac{{\lambda }^2{e}^{-\mathit{\lambda t}}{\left(\mathit{\lambda t}\right)}^{n}f\left(\lambda \right)}{n!P\left[N\left(t\right)=n\right]}\mathit{d\lambda }=\frac{\left(n+2\right)\left(n+1\right)}{t^2}\frac{P_{n+2}\left(t\right)}{P_n\left(t\right)}.$$ E33

By substituting (24) into (33), we have

$$\mathbb{E}\left[{\mathrm{\Lambda }}^2|N\left(t\right)=n\right]={\lambda }_{n+1}\left(t;a\right){\lambda }_n\left(t;a\right).$$

Then the conditional variance of $\mathrm{\Lambda }$ _, given that $N\left(t\right)=n$ _, is

$$\mathit{Var}\left[\mathrm{\Lambda }|N\left(t\right)=n\right]={\lambda }_{n+1}\left(t;a\right){\lambda }_n\left(t;a\right)-{\lambda }_n^2\left(t;a\right),$$

and substituting Eq. (25) into the above yields the result.

2. We use the law of total expectation to find the expected value

$$\begin{array}{c}\mathbb{E}\left[\mathrm{\Lambda }\right]=\mathbb{E}\left[\mathbb{E}(\mathrm{\Lambda }|N\left(t\right)=n)\right]=\sum _{n=0}^{\infty }\mathbb{E}\left(\mathrm{\Lambda }|N\left(t\right)=n\right)P\left[N\left(t\right)=n\right]\\ =\sum _{n=0}^{\infty }{\lambda }_n\left(t;a\right)P_n\left(t\right)\end{array}$$

By substituting (24) into the above expression, we get

$$\mathbb{E}\left[\mathrm{\Lambda }\right]=\sum _{n=0}^{\infty }\frac{n+1}{t}{P}_{n+1}\left(t\right)=\sum _{j=0}^{\infty }\frac{r_j\left(t;a\right)}{t}=\frac{1}{t}\mathbb{E}\left[N\left(t\right)\right].$$

And the proof is completed.

3. The pgf of $N\left(t\right)$ is defined as

$$\begin{array}{c}\underbrace{G_N\left(z;t\right)}=\sum _{n=0}^{\infty }{z}^n{P}_n\left(t\right)=\sum _{n=0}^{\infty }{z}^n\underset{0}{\overset{\infty }{\int }}\frac{{\left(\mathit{\lambda t}\right)}^n}{n!}{e}^{-\mathit{\lambda t}}f\left(\lambda \right)\mathit{d\lambda }\\ P_0\left[\left(1-z\right)t\right]=\underset{0}{\overset{\infty }{\int }}\left[\sum _{n=0}^{\infty }\frac{{\left(\mathit{z\lambda t}\right)}^n}{n!}\right]e^{-\mathit{\lambda t}}f\left(\lambda \right)\mathit{d\lambda }=\underset{0}{\overset{\infty }{\int }}{e}^{\lambda \left(z-1\right)t}f\left(\lambda \right)\mathit{d\lambda }\\ =M_{\mathrm{\Lambda }}\left[\left(z-1\right)t\right].\end{array}$$ E34

We make $z=0$ in the above expression and we have

$$P_0\left(t\right)=M_{\mathrm{\Lambda }}\left(-t\right)$$

Now, if we differentiate both sides with respect to $t$ , we obtain

$$P_0^{\prime }\left(t\right)=-M_{\mathrm{\Lambda }}^{\prime }\left(-t\right)$$

We complete the proof by substituting $t=0$ in the above expression. □

According to Walhin and Paris in ref. [20], the intensity of the stochastic process $N\left(t\right)$ in the period $\left[t,t+1\right]$ is

$$\mathbb{E}\left[N\left(t+1\right)-N\left(t\right)|N\left(t\right)=n\right]=\mathbb{E}\left[\mathrm{\Lambda }|N\left(t\right)=n\right].$$

The moment generating function of the process will uniquely determine the distribution of the process, on comparing expression (34) with $P_0\left[\left(1-z\right)t\right]$ given for $a=1$ and as shown in Table 1 , we find the particular cases: the PCP if $\mathrm{\Lambda }\sim {\delta }_{\gamma }\left(\lambda \right)$ (i.e. has a degenerate cdf at $\lambda =\gamma$ ), the NBCP if $\mathrm{\Lambda }\sim \mathrm{\Gamma }\left(\gamma ,\delta \right)$ and the Geometric Counting Process if $\mathrm{\Lambda }\sim exp\left(\delta \right)$ .

5. Additional properties

In this Section, we will introduce several other properties of the NHP.

5.1 Other expressions for ${\mathbf{P}}_{\mathbf{n}}\left(\mathbf{t}\right)$ in terms of ${\lambda }_{\mathbf{n}}\left(\mathbf{t};\mathbf{a}\right)$

Theorem 1.7: Let $N\left(t\right)$ be an NHP with transition intensities given by (21), then

$$P_n\left(t\right)=Q_n\left(t\right)-Q_{n+1}\left(t\right)\text{for}n\ge 1,$$

where $Q_0\left(t\right)$ is Heaviside’s step function and

$$Q_{n+1}\left(t\right)={\int }_0^t{\lambda }_n\left(v;a\right)P_n\left(v\right)\mathit{dv}.$$ E35

Proof:

We write the expression (22) as

$$\frac{d\left[P_n\left(\tau \right)\right]}{\mathit{d\tau }}={\lambda }_{n-1}\left(\tau ;a\right)P_{n-1}\left(\tau \right)-{\lambda }_n\left(\tau ;a\right)P_n\left(\tau \right)\text{for}n\ge 1.$$

By integration of the above expression with respect to $\tau$ between $0$ and $t$ , we get

$$\begin{array}{c}{\int }_0^{t}d\left[P_n\left(\tau \right)\right]={\int }_0^t{\lambda }_{n-1}\left(\tau ;a\right)P_{n-1}\left(\tau \right)\mathit{d\tau }-{\int }_0^t{\lambda }_n\left(\tau ;a\right)P_n\left(\tau \right)\mathit{d\tau }\\ {\left.P_n\left(\tau \right)\right|}_0^t=Q_n\left(t\right)-Q_{n+1}\left(t\right)\text{for}n\ge 1.\end{array}$$ E36

Since $P_n\left(0\right)=0,\forall n\ge 1,$ so the proof is completed.

Corollary 1.7.1: Let $N\left(t\right)$ be an NHP with transition intensities given by (21), then

$$P\left[N\left(t\right)>n\right]=Q_{n+1}\left(t\right)\text{for}n\ge 0$$ E37

Proof: The proof consists of a direct calculation

$$\begin{array}{l}P\left[N\left(t\right)>n\right]=1-P\left[N\left(t\right)\le n\right]\\ =1-\sum _{j=0}^n{P}_j\left(t\right)=1-P_0\left(t\right)-\sum _{j=1}^n{P}_j\left(t\right)\end{array}$$

Using the previous result:

$$\begin{array}{l}P\left[N\left(t\right)>n\right]=1-P_0\left(t\right)-\sum _{j=1}^n\left[Q_j\left(t\right)-Q_{j+1}\left(t\right)\right]\\ =1-P_0\left(t\right)-\left[Q_1\left(t\right)-Q_{n+1}\left(t\right)\right]\end{array}$$ E38

Note that

$$Q_1\left(t\right)={\int }_0^t{\lambda }_0\left(v;a\right)P_0\left(v\right)\mathit{dv}=-{\int }_0^t{P}_0^{\prime }\left(v\right)\mathit{dv}=-{\left.P_0\left(v\right)\right|}_0^t=1-P_0\left(t\right)$$

Replacing $Q_1\left(t\right)$ in (38) the proof is completed.

The expression (37) allows to calculate the cdf of an NHP.

Corollary 1.7.2: The function $Q_{n+1}\left(t\right)$ satisfies the following condition:

$$\underset{t\to \infty }{lim}{Q}_{n+1}\left(t\right)=1\text{for}n\ge 0.$$ E39

Proof:

From (37), we get

$$\underset{t\to \infty }{lim}{Q}_{n+1}\left(t\right)=\underset{t\to \infty }{lim}\left[1-\sum _{j=0}^n{P}_j\left(t\right)\right].$$

As we have for $n\ge 1:P_n\left(\infty \right)=0$ , and using the above relationship

$$\underset{t\to \infty }{lim}{Q}_{n+1}\left(t\right)=1-\underset{t\to \infty }{lim}{P}_0\left(t\right).$$

For example, from expression (9) when $a=1$ _, we have:

$$P_0\left(t\right)={\left(1+\mathit{\kappa t}\right)}^{-\frac{q}{\kappa }}\text{for}\frac{q}{\kappa }>0$$ E40

and we take the limit as $t\to \infty$ _, we get:

$\underset{t\to \infty }{lim}{Q}_{n+1}\left(t\right)=1-\underset{t\to \infty }{lim}{\left(1+\mathit{\kappa t}\right)}^{-\frac{q}{\kappa }}=1.$ □

Proposition 1.8: Let $N\left(t\right)$ be an NHP with transition intensities given by (21), then

$$exp\left\{-\underset{t}{\overset{t+h}{\int }}{\lambda }_n\left(v;a\right)\mathit{dv}\right\}=\frac{P_0^{\left(n\right)}\left(t+h\right)}{P_0^{\left(n\right)}\left(t\right)}\text{for}h\ge 0.$$ E41

Proof:

By substituting (28) into (40), we have

$\begin{array}{c}exp\left\{-\underset{t}{\overset{t+h}{\int }}{\lambda }_n\left(v;a\right)\mathit{dv}\right\}=exp\left\{\underset{t}{\overset{t+h}{\int }}\frac{P_0^{\left(n+1\right)}\left(v\right)}{P_0^{\left(n\right)}\left(v\right)}\mathit{dv}\right\}\\ =exp\left\{\underset{t}{\overset{t+h}{\int }}d\left[ln\left(P_0^{\left(n\right)}\left(v\right)\right)\right]\right\}\\ =exp\left\{{\left..ln\left[P_0^{\left(n\right)}\left(v\right)\right]\right|}_t^{t+h}\right\}=\frac{P_0^{\left(n\right)}\left(t+h\right)}{P_0^{\left(n\right)}\left(t\right)}.\end{array}$

Corollary 1.8.1: Let $N\left(t\right)$ be an NHP. If the probability that no event occurs in a small interval of length $h$ is denoted by $P_0\left(t,t+h\right)$ , that is $P_0\left(t,t+h\right)=P\left(N\left(t+h\right)-N\left(t\right)=0\right)$ _, then

$$P_0\left(t+h\right)=P_0\left(t\right)\cdot P_0\left(t,t+h\right)\text{for}t,h\ge 0.$$ E42

Proof:

According to Lundberg in [18]:

$$P\left(N\left(t+h\right)=0|N\left(t\right)=0\right)=exp\left\{-\underset{t}{\overset{t+h}{\int }}{\lambda }_0\left(u\right)\mathit{du}\right\}$$ E43

where ${\lambda }_0\left(t\right)$ denotes the intensity function associated with the time-dependent (or nonstationary) PCP. If we make $n=0$ in (40), then we obtain

$$P_0\left(t,t+h\right)=exp\left\{-\underset{t}{\overset{t+h}{\int }}{\lambda }_0\left(v;a\right)\mathit{dv}\right\}=\frac{P_0\left(t+h\right)}{P_0\left(t\right)}$$ E44

Thus,

$$P_0\left(t+h\right)=P_0\left(t\right)\cdot P_0\left(t,t+h\right)\text{for}t,h\ge 0.$$ I134

The expression obtained in (41) may be interpreted as if no event occurred, then the NHP has independent increments.

Lemma 1.9: Let $N\left(t\right)$ be an NHP with transition intensities given by (21). Then this CP satisfies

$$\sum _{j=0}^m\frac{{\lambda }_j^{\prime }\left(t;a\right)}{{\lambda }_j\left(t;a\right)}={\lambda }_0\left(t;a\right)-{\lambda }_{m+1}\left(t;a\right)\text{for}\mathrm{all}m\ge 0.$$ E45

Proof:

From (25), we have

$$\frac{{\lambda }_j^{\prime }\left(t;a\right)}{{\lambda }_j\left(t;a\right)}={\lambda }_j\left(t;a\right)-{\lambda }_{j+1}\left(t;a\right)\text{for}\mathrm{all}j\ge 0.$$ E46

Thus, (44) turns out the $m$ th partial sum of a telescoping series and from here

$$\sum _{j=0}^m\frac{{\lambda }_j^{\prime }\left(t;a\right)}{{\lambda }_j\left(t;a\right)}={\lambda }_0\left(t;a\right)-{\lambda }_{m+1}\left(t;a\right)\text{for}\mathrm{all}m\ge 0.$$

Now, using the above lemma, we will prove the following proposition:

Proposition 1.10: Let $N\left(t\right)$ be an NHP with marginal pmf given by (5), then $P_n\left(t\right)$ satisfies that

1. Process with time-dependent increments

   $$\underset{h\to 0}{lim}\frac{P_{n,n+1}\left(t,t+h\right)}{h}={\lambda }_n\left(t;a\right)$$

2. The probability that no event occurs in $\left(t,t+h\right]$ is

   $$P_0\left(t,t+h\right)=1-h{\lambda }_0\left(t;a\right)+o\left(h\right)$$ E47

3. The probability that one event occurs in $\left(t,t+h\right]$ is

   $$P_1\left(t,t+h\right)=h{\lambda }_0\left(t;a\right)-o\left(h\right)$$ E48

4. Faddy’s conjecture ²: If the transition intensities be an increasing sequence with $n$ , i.e,

   $${\lambda }_0\left(t;a\right)<{\lambda }_1\left(t;a\right)<\dots <{\lambda }_n\left(t;a\right),\text{for}\mathrm{any}\text{fixed}t$$ E49

then $\mathit{Var}\left[N\left(t\right)\right]>\mathbb{E}\left[N\left(t\right)\right]$ , this last inequality is reversed for a decreasing sequence.

Proof:

1. As the NHP is an MPP then, according to Lundberg in [18], for $0\le u<v,i\le j$ , $N\left(t\right)$ satisfies:

   $$\underset{P_{i,j}\left(u,v\right)}{\underbrace{P\left(N\left(v\right)=j|N\left(u\right)=i\right)}}=\left(\begin{array}{c}j\\ i\end{array}\right){\left(\frac{u}{v}\right)}^i{\left(1-\frac{u}{v}\right)}^{j-i}\frac{P_j\left(v\right)}{P_i\left(u\right)}$$ E50

   Replacing the expression $P_n\left(t\right)$ given in (12), when $\kappa \ne 0$ , we obtain in (49) that the transition probabilities for the NHP are:

   $$\begin{array}{l}{P}_{i,j}\left(u,v\right)=\left(\begin{array}{c}j\\ i\end{array}\right){\left(\frac{u}{v}\right)}^i{\left(1-\frac{u}{v}\right)}^{j-i}\frac{P_j\left(v\right)}{P_i\left(u\right)}\\ =\left(\begin{array}{c}j\\ i\end{array}\right){\left(\frac{u}{v}\right)}^i{\left(\frac{v-u}{v}\right)}^{j-i}\left[\frac{\frac{{\left(-1\right)}^j{v}^j{P}_0^{\left(j\right)}\left(v\right)}{j!}}{\frac{{\left(-1\right)}^i{u}^i{P}_0^{\left(i\right)}\left(u\right)}{i!}}\right]\\ =\frac{{\left(u-v\right)}^{j-i}}{\left(j-i\right)!}\frac{P_0^{\left(j\right)}\left(v\right)}{P_0^{\left(i\right)}\left(u\right)}\\ =\prod _{m=1}^{j-i}\left[\frac{v-u}{m}{\lambda }_{m+i-1}\left(u;a\right)\right]exp\left\{-\underset{u}{\overset{v}{\int }}{\lambda }_j\left(w;a\right)\mathit{dw}\right\}.\end{array}$$ E51

   We complete the proof of the theorem by the following steps: Rewrite the product in (50) by replacing all instances of $i=n$ , $j=n+1$ , $u=t$ and $v=t+h$ _, and we make the limit as $h$ approaches zero. Then the transition intensities given by (21) represent the instantaneous transitions probabilities of the NHP.

2. Certainly, the function given by (9) is continuous for $t\ge 0$ and also analytic, due to $P_0^{\left(n\right)}\left(t\right)$ _, exists for all $n\ge 1$ . Then it is possible to express $P_0\left(t+h\right)$ through a Taylor series as follows:

   $$P_0\left(t+h\right)=\sum _{m=0}^{\infty }\frac{h^m}{m!}{P}_0^{\left(m\right)}\left(t\right).$$ E52

   By substituting the expression for the $m$ th derivative of $P_0\left(t\right)$ obtained given by (27) in (51), we have:

   $$P_0\left(t+h\right)=P_0\left(t\right)+\sum _{m=1}^{\infty }\frac{h^m}{m!}\left[{\left(-1\right)}^m\left(\prod _{j=0}^{m-1}{\lambda }_j\left(t;a\right)\right)P_0\left(t\right)\right].$$ E53

   Notice that $P_0\left(t+h\right)$ satisfies (41), then (52) is similar to:4

   $$P_0\left(t\right)\cdot P_0\left(t,t+h\right)=P_0\left(t\right)\left[1+\sum _{m=1}^{\infty }{\left(-1\right)}^m\frac{h^m}{m!}\left(\prod _{j=0}^{m-1}{\lambda }_j\left(t;a\right)\right)\right]$$ E54

   Let $n=m-1$ then:

   $$\begin{array}{c}{P}_0\left(t,t+h\right)=1+\sum _{n=0}^{\infty }{\left(-1\right)}^{n+1}\frac{h^{n+1}}{\left(n+1\right)!}\left(\prod _{j=0}^n{\lambda }_j\left(t;a\right)\right)\\ =1-h\sum _{n=0}^{\infty }\frac{{\left(-h\right)}^n}{\left(n+1\right)!}\left(\prod _{j=0}^n{\lambda }_j\left(t;a\right)\right)\end{array}$$ E55

   From the expansion of the first terms of (54), we get:

   $$P_0\left(t,t+h\right)=1-h{\lambda }_0\left(t;a\right)+o\left(h\right)$$ E56

   where

   $$o\left(h\right)=\sum _{n=1}^{\infty }\frac{{\left(-h\right)}^{n+1}}{\left(n+1\right)!}\prod _{j=0}^n{\lambda }_j\left(t;a\right).$$

   The last function satisfies that $\underset{h\to 0}{lim}o\left(h\right)/h=0$ ([21, 22]).

3. From (55) and the fact $P_0\left(t,t+h\right)=P\left(N\left(t+h\right)-N\left(t\right)=0\right)$ , we obtain

   $$P\left(N\left(t+h\right)-N\left(t\right)>0\right)=1-P_0\left(t,t+h\right).$$ E57

   Given that the NHP $N\left(t\right)$ is an NHPBP and assuming that we have in a small time interval, then there will be only two cases: there is a birth or not in that period. Thus,

   $$P\left(N\left(t+h\right)-N\left(t\right)>0\right)=P\left(N\left(t+h\right)-N\left(t\right)=1\right)=P_1\left(t,t+h\right).$$

   Then, from (56), we obtain:

   $$P_1\left(t,t+h\right)=h{\lambda }_0\left(t;a\right)-o\left(h\right),$$ E58

   provided that $h$ is infinitesimal.

4. According to Steutel et al. in ref. [16], a non-degenerate distribution $\left\{P_n\left(t\right)\right\}$ is log-convex if and only if $P_n\left(t\right)>0$ for all $n\ge 0$ and $\left\{\frac{P_{n+1}\left(t\right)}{P_n\left(t\right)}\right\}$ is a nondecreasing sequence. By assumption

   $$\frac{P_n\left(t\right)}{P_{n-1}\left(t\right)}<\frac{P_{n+1}\left(t\right)}{P_n\left(t\right)}\text{for}\text{some}n\ge 1$$ E59

   By substituting (5) into (58)

   $$\begin{array}{l}\frac{\frac{t^n}{n!}\left[{\left(-1\right)}^n{P}_0^{\left(n\right)}\left(t\right)\right]}{\frac{t^{n-1}}{\left(n-1\right)!}\left[{\left(-1\right)}^{n-1}{P}_0^{\left(n-1\right)}\left(t\right)\right]}<\frac{\frac{t^{n+1}}{\left(n+1\right)!}\left[{\left(-1\right)}^{n+1}{P}_0^{\left(n+1\right)}\left(t\right)\right]}{\frac{t^n}{n!}\left[{\left(-1\right)}^n{P}_0^{\left(n\right)}\left(t\right)\right]}\\ \frac{1}{n}\left(-\frac{P_0^{\left(n\right)}\left(t\right)}{P_0^{\left(n-1\right)}\left(t\right)}\right)<\frac{1}{n+1}\left(-\frac{P_0^{\left(n+1\right)}\left(t\right)}{P_0^{\left(n\right)}\left(t\right)}\right)\\ \frac{1}{n}{\lambda }_{n-1}\left(t;a\right)<\frac{1}{n+1}{\lambda }_n\left(t;a\right)\end{array},$$

   we know $1<\frac{n+1}{n}$ for all $n$ . Hence, we have the following:

   $${\lambda }_{n-1}\left(t;a\right)<\frac{n+1}{n}{\lambda }_{n-1}\left(t;a\right)<{\lambda }_n\left(t;a\right).$$ E60

   Thus, we obtain that (48) is satisfied and, therefore, the conjecture holds.

The expression (48) allows to identify under- or over-dispersion of a CP, then we can classify the process according to the fixed criteria given in (16).

Corollary 1.10.1: If $a\ne 0$ and $N\left(t\right)$ is an NHP, then it does not have independent increments.

Proof:

From theorem 1.5, we know that an NHP is an MPP. According to McFadden in ref. [9], if $\left\{N\left(t\right),t\ge 0\right\}$ is a CP with independent increments, then its transition intensities satisfy that ${\lambda }_0\left(t;a\right)={\lambda }_1\left(t;a\right),$ but by expression (48), we get

$${\lambda }_0\left(t;a\right)=\frac{q}{{\left(1+\mathit{\kappa t}\right)}^a}\ne \frac{\mathit{a\kappa }}{1+\mathit{\kappa t}}+\frac{q}{{\left(1+\mathit{\kappa t}\right)}^a}={\lambda }_1\left(t;a\right)\text{if}a\ne 0$$ E61

And therefore, $N\left(t\right)$ is a CP that does not have independent increments.

This was to be expected since that MPP has stationary increments but does not meet the condition of independent increments (see [23]).

6. Conclusions

In this chapter, we studied the NHP presenting some of its properties indicating that it is a good option for modelling CP regardless of the fact that it presents under- or over-dispersion.

Using transition intensities, we found some properties of the NHP and provided explicit analytic expressions for its pmf and cdf.