Panjer class revisited: one formula for the distributions of the Panjer (a,b,n) class

Abstract

The loss count distributions whose probabilities ultimately satisfy Panjer’s recursion were classified between 1981 and 2002; they split into six types, which look quite diverse. Yet, the distributions are closely related – we show that their probabilities emerge out of one formula: the binomial series. We propose a parameter change that leads to a unified, practical and intuitive, representation of the Panjer distributions and their parameter space. We determine the subsets of the parameter space where the probabilities are continuous functions of the parameters. Finally, we give an inventory of parameterisations used for Panjer distributions.

1. Introduction

1.1 Motivation

The loss count distributions of the Panjer class are popular in insurance and beyond, in particular the classical models Poisson, binomial and negative binomial (NB). Apart from being realistic and intuitive, they are easy to handle computationally as their probabilities satisfy a recursive formula. For more flexibility, it was successfully tried to extend the class of distributions, while preserving most of their convenient properties. The above three models are the only nontrivial ones fulfilling Panjer’s recursion thoroughly (Sundt & Jewell, 1981), but two models satisfy it for all probabilities but the lowest step ( $p_{0}$ to $p_{1}$ ), namely the logarithmic distribution and the so-called Engen or extended truncated negative binomial distribution (ETNB) (Willmot, 1988). Finally, Hess et al. (2002) classified the discrete loss distributions fulfilling the recursion for all but a finite number of initial probabilities, by extending the NB distribution further and adding an extension of the logarithmic distribution. Overall, the resulting class embraces six types of distributions which, however, look quite diverse. So, the picture is complete, but a bit heterogeneous.

1.2 Research context

In the actuarial literature, extensions of the Panjer class beyond the three classical models have been investigated since the 1980s, when Panjer’s famous recursive algorithm to calculate the aggregate loss distribution in the collective model emerged (Panjer, 1981). It was successfully tried to generalise the recursive formula for the probabilities in a number of ways, which leads to a huge variety of distributions, see, for example, Sundt & Jewell (1981), Willmot and Panjer (1987), Schröter (1990), Gerhold et al. (2010), the survey paper Sundt (2002), and the textbooks Panjer & Willmot (1992), Klugman et al. (2008). However, as for example, Albrecher et al. (2017) note in a compact review of the Panjer class, some of the findings are older and date back at least to Johnson & Kotz (1969).

We focus on the particular extension of the Panjer class that keeps the original recursive formula for the probabilities but does not require it for some initial probabilities. We call this the general Panjer class; it combines a large flexibility for the initial probabilities with a distribution tail geometry somewhat extending that of the three classical distributions. The papers being most relevant for this class were already mentioned: Willmot (1988) and Hess et al. (2002) extended the classical three Panjer models to the final total of six distribution types, and Gerhold et al. (2010) developed a generalised and numerically superior recursive Panjer algorithm.

1.3 Objective

In this paper, we complement the theory by showing how the probabilities of the general Panjer class emerge out of one formula, namely the binomial series. This enhances and extends unified views on parts of the Panjer class as given by Panjer & Willmot (1992) and Fackler (2011). Beyond being instructive, the resulting representation of the general Panjer class of distributions can ease implementation and use of the models in practice, providing all-in-one formulae for probabilities and moments, expressed in terms of the parameters of Panjer’s recursion. A slight transform of the traditional recursion parameters simplifies both the formulae (slightly) and the geometry of the parameter space (greatly).

1.4 Outline

In section 2, we define the general Panjer class of distributions, reconciling diverse definitions appearing in the literature, and discuss useful parameterisations. In section 3, we rearrange the binomial series such that it yields the probabilities of half of the general Panjer class, based on a common parameter space. In section 4, we study some limiting cases, which yield the other half of the class, and determine where the probabilities are continuous functions of their parameters, and where not. Section 5 wraps up the classification; section 6 comments on parameter inference. The appendix provides some technical details and an inventory of parameterisations being used for members of the Panjer class.

2. Preliminaries

For the sake of precision, we must go through some technicalities; however, intuition will be provided on the way. Let us first collect and reconcile definitions of the Panjer class(es) found in the literature, adding an alternative parameter s.

2.1 Definitions

Definition 2.1. For a positive integer k and real $a,b,s=a+b$ , Panjer’s recursion is as follows:

(1) \begin{equation}p_{k}=\left(a+\frac{b}{k}\right)p_{k-1}=\left\{ a\left(1-\frac{1}{k}\right)+s\frac{1}{k}\right\} p_{k-1}\end{equation}

For any $n\in\mathbb{N}_{0}$ , the wide Panjer $\left(a,b,n\right)$ class is the class of nontrivial counting distributions whose probabilities $p_{k}$ satisfy Panjer’s recursion for some a, b and all integers $k>n$ .

The proper Panjer $\left(a,b,n\right)$ class is the subclass thereof where in addition $p_{k}=0$ for all integers $k<n$ .

The narrow Panjer $\left(a,b,n\right)$ class is the subclass thereof whose distributions cannot be written as a left-truncation of some other distribution of some $\left(a,b,m\right)$ class.

Nontrivial means not (almost surely) constant, such that at least two $p_{k}$ are positive. Left-truncation (of discrete loss distributions) means setting some initial probabilities to 0 and rescaling the remaining probabilities accordingly, which yields again a distribution. Truncation to the left of $p_{m}$ formally means left-truncation at $m-1$ ; following Hess et al. (2002) we call this m-truncation:

(2) \begin{equation}p_{k}(m)\,:\!=\,\begin{cases}0, & k<m\\[4pt]\frac{p_{k}}{1-\sum_{j=0}^{m-1}p_{j}}, & k\geq m\end{cases}\end{equation}

Instead of the $\left(a,b,n\right)$ class one also speaks of the Panjer distributions of order n. The general Panjer class is the union of the Panjer classes of any order, and likewise has a wide, a proper and a narrow variant.

For the general concept of truncation see Klugman et al. (2008), who define the wide Panjer distributions (of order $n=0,1$ ). The wide classes are totally ordered: for $n<m$ the $\left(a,b,n\right)$ class is a subset of the $\left(a,b,m\right)$ class.

Panjer & Willmot (1992) (for $n=0,1$ ) and Hess et al. (2002) define the $\left(a,b,n\right)$ class in the proper sense. Proper Panjer classes of different order are disjoint. If we m-truncate a wide $\left(a,b,m\right)$ model, we get the (unique) proper model having the same parameters a, b, m. The original wide model is a mixture of this corresponding proper model with a loss count model having support in $\left\{ 0,...,m-1\right\} $ . In other words, in a wide $\left(a,b,m\right)$ model the initial probabilities $p_{0},...,p_{m-1}$ can be chosen (thus are parameters) with the only restriction that their sum must be less than 1. The recursion determines the tail starting from $p_{m}$ . When Panjer class distributions are analysed, it is sufficient (and common) to look at the proper models; the wider ones are embraced by choosing the initial probabilities as described here.

The distributions of the general narrow Panjer class are specified by Hess et al. (2002), who call them basic claim number distributions and distinguish six types: Poisson, binomial, NB, logarithmic, extended NB and extended logarithmic, see Table 1 with integers $n_{0}$ , $m_{0}$ and further parameters being positive real unless specified otherwise.

Table 1. Narrow Panjer class.

If for $n\leq m$ one starts with a narrow $\left(a,b,n\right)$ model and m-truncates it, one gets a proper $\left(a,b,m\right)$ distribution. All proper distributions can be generated this way, so it is ultimately sufficient to classify narrow Panjer models and generate proper and wide models having the same parameters a, b by changing the first probabilities and rescaling the tail.

Finally, note that the three variants of the Panjer $\left(a,b,0\right)$ class coincide.

2.2 Practical use

Wide Panjer models are useful in situations where one considers the tails of Panjer distributions as realistic but needs much more flexibility for the initial probabilities. In particular, flexibility about $p_{0}$ often helps fitting real insurance data, for a discussion see section 6.7 of Klugman et al. (2008), who speak of zero-modification and zero-truncation (which latter is notably 1-truncation in our terminology). We will take a closer look at this combination of “free” initial probabilities and subsequent Panjer tail in section 6.

Proper and narrow models can be adequate for situations where the lowest loss counts are impossible, for example, when modelling the number of risks contributing to an accumulation loss. Reinsurance treaties protecting such losses usually have a n risks warranty, that is, losses are only covered when at least n risks are affected. $n=2$ is most common, but in life or personal accident accumulation covers one finds, for example, $n=6$ .

Panjer’s classical recursive algorithm to calculate the aggregate loss distribution in the collective model can be extended to all proper Panjer loss count distributions (Hess et al., 2002), so they all share this important ease of calculation that made the classical Panjer distributions so popular in insurance. Gerhold et al. (2010) found that Panjer’s algorithm can be numerically unstable with the two “Extended” distribution types and developed (see their section 5) a slower but stable and far more general algorithm, which embraces further loss count distributions, including in particular the wide Panjer class (see their Corollary 4.7).

However, for the latter class one can often alternatively work with the classical algorithm. To see this, let us call the loss count N, the loss severity X and the aggregate loss S. In a wide model of order m, the initial probabilities $p_{0},...,p_{m-1}$ sum up to $q<1$ and we can write the cdf of S as

\begin{equation*}F_{S}\left(x\right)=\sum_{k=0}^{\infty}p_{k}F_{X}^{k*}\left(x\right)=\sum_{k=0}^{m-1}p_{k}F_{X}^{k*}\left(x\right)+\left(1-q\right)F_{\bar{S}}\left(x\right)\end{equation*}

where $\bar{S}$ is the aggregate loss of the collective model with the same severity X but a thinned loss count $\bar{N}$ , namely the m-truncation of N with probabilities according to Formula 2. So, $F_{S}$ can be calculated from the convolutions of $F_{X}$ up to order $m-1$ , plus $F_{\bar{S}}$ resulting from the Panjer algorithm for proper models.

2.3 Consolidation

We will see below that in a narrow $\left(a,b,n\right)$ model the integer n is uniquely determined by a and b and is thus not a parameter. It will further turn out that many (but not all) formulae become simpler if we use the parameter $s=a+b$ instead of b. Both parameterisations are equivalent and easy to convert into each other, so one can switch between them according to convenience.

Lemma 2.2. For any real a,b (or equivalently a, s) the sequence

(3) \begin{equation}r_{k}\,:\!=\,\frac{1}{k!}\prod_{i=1}^{k}\left(b+ai\right)=\frac{1}{k!}\prod_{i=0}^{k-1}\left(s+ai\right),\quad k\in\mathbb{N}_{0}\end{equation}

(where $r_{0}=1$ ) satisfies Panjer’s recursion.

Proof. For $k \gt 0$ , we have

\begin{equation*}\left(a+\frac{b}{k}\right)r_{k-1}=\frac{b+ak}{k}\frac{1}{\left(k-1\right)!}\prod_{i=1}^{k-1}\left(b+ai\right)=\frac{1}{k!}\prod_{i=1}^{k}\left(b+ai\right)=r_{k}\end{equation*}

Note that we did not require the $r_{k}$ to be probabilities, not even up to a factor. They may partly equal 0 or have varying sign. Yet, Panjer’s recursion holds for all $k>0$ . So, if the $r_{k}$ have ultimately the same sign, one can possibly use the (rescaled) tail of the sequence as loss count probabilities, which would yield a proper Panjer distribution. This is indeed the key idea of this paper and will lead to a common representation of the key figures of the proper Panjer distributions.

Definition 2.3. We call $\left(r_{k}\right)$ as in Formula 3 the Panjer sequence. Where we want to emphasise that $r_{k}$ is a function of a and b, we write $r_{k}\left(a,b\right)$ ; with alternative parameters we write analogously $r_{k}\left(a, s\right)$ , etc.

Before using the Panjer sequence in a general setting, we relate it to the classical Panjer distributions, whose probabilities and recursion parameters can be written in all-in-one formulae (Fackler, 2011):

(4) \begin{equation}p_{k}=\left(1+\frac{\lambda}{\alpha}\right)^{-\alpha}\frac{\lambda^{k}}{k!}\prod_{i=0}^{k-1}\frac{\alpha+i}{\alpha+\lambda},\qquad a=\frac{\lambda}{\alpha+\lambda},\quad s=\frac{\alpha\lambda}{\alpha+\lambda}\end{equation}

Here $\lambda>0$ is the expectation. $\alpha>0$ yields the NB distribution, while $\alpha\in-\mathbb{N}$ satisfying $-\alpha>\lambda$ yields the binomial distribution. The limits for $\alpha\rightarrow\pm\infty$ are well defined and both yield the Poisson distribution. To avoid infinite parameters, one can equivalently work with $\lambda$ and $c=\frac{1}{\alpha}$ ; however, here the parameter space is not so easy to write down (see Appendix B for this and alternative parameterisations). It gets even more intricate if we rewrite the probabilities in terms of a and s, but this is the variant that we need for the general treatment of the Panjer class. We have $\alpha=\frac{s}{a}$ and $\lambda=\frac{s}{1-a}$ . Some algebra yields

(5) \begin{equation}p_{k}=\left(1-a\right)^{\frac{s}{a}}\frac{1}{k!}\prod_{i=0}^{k-1}\left(s+ai\right)=\left(1-a\right)^{\frac{s}{a}}r_{k}\left(a, s\right)\end{equation}

Thus, for appropriate a, s (which we will specify below), the Panjer sequence generates the probability function of the Panjer $\left(a,b,0\right)$ class. Poisson corresponds to the (well-defined) limit for $a\rightarrow0$ , where $s=\lambda$ .

Let us introduce a third parameterisation, which only embraces binomial and NB but will be useful below. Using parameters $a\neq0$ and $\alpha=\frac{s}{a}$ , it is a hybrid of the two preceding variants:

(6) \begin{equation}p_{k}=\left(1-a\right)^{\alpha}r_{k}\left(a,\alpha\right),\qquad r_{k}\left(a,\alpha\right)=\frac{a^{k}}{k!}\prod_{i=0}^{k-1}\left(\alpha+i\right)=a^{k}\binom{\alpha+k-1}{k}\end{equation}

3. Rearranging the Binomial Series

It is well known that for certain real $x,\gamma$ the Taylor series

(7) \begin{equation}\left(1+x\right)^{\gamma}=\sum_{k=0}^{\infty}\binom{\gamma}{k}x^{k}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\prod_{i=0}^{k-1}\left(\gamma-i\right)\end{equation}

converges (absolutely), in particular in the following situations:

• $\gamma\in\mathbb{N}$ ,

• $\left|x\right|<1$ ,

• $\left|x\right|=1$ and $\gamma>0$ .

Formula 7 is called the binomial series or generalised binomial formula.

Now choose real $a\neq0$ and s; set $x=-a$ , $\gamma=-\frac{s}{a}$ . If the latter two figures are in the appropriate range, we can rewrite the binomial series in terms of a and s as

(8) \begin{equation}\left(1-a\right)^{-\frac{s}{a}}=\sum_{k=0}^{\infty}\frac{\left(-a\right)^{k}}{k!}\prod_{i=0}^{k-1}\left(-\frac{s}{a}-i\right)=\sum_{k=0}^{\infty}\frac{1}{k!}\prod_{i=0}^{k-1}\left(s+ai\right)=\sum_{k=0}^{\infty}r_{k}\left(a, s\right)\end{equation}

and have found the (well-defined and finite) sum of the Panjer sequence. In particular, this applies to the following cases, which are special cases of the above three situations, but disjoint and grouped in a different way:

Figure 1 Admissible parameters a, s and $a,\alpha.$

Definition 3.1. We call real a, s strongly admissible if

\begin{equation*}\left(a, s\right)\in\mathcal{S}\,:\!=\,\left\{ \left(a,-an_{0}\right)\left|\:a\in\left(-\infty,0\right),n_{0}\in\mathbb{N}\right.\right\} \cup\left(0,1\right)\!\times\!\left(0,\infty\right)\cup\left(0,1\right]\!\times\!\left(\left(-\infty,0\right)\!\setminus\!-\mathbb{N}\right)\subset\mathbb{R}^{2}\end{equation*}

that is, if they lie in one of the following three disjoint areas (see also Figure 1, left side):

The delay of $\left(a, s\right)\in\mathcal{S}$ is the integer

\begin{equation*}m_{0}\left(a, s\right)\,:\!=\,\begin{cases}0, & s>0\\[4pt]1+\left[-\frac{s}{a}\right], & s<0\end{cases}\end{equation*}

where $\left[x\right]$ is the largest integer not exceeding x, such that for $s<0$ we have $0\leq m_{0}-1<-\frac{s}{a}<m_{0}$ .

Finally, for $n\in\mathbb{N}_{0}$ we define the subsets

\begin{equation*}\mathcal{S}_{n}\,:\!=\,\left\{ \left(a, s\right)\in\mathcal{S}\left|\,m_{0}\left(a, s\right)\leq n\right.\right\} ,\quad\mathcal{S}_{-1}\,:\!=\,\emptyset,\qquad\mathcal{S}_{\left(n\right)}\,:\!=\,\left\{ \left(a, s\right)\in\mathcal{S}\left|\,m_{0}\left(a, s\right)=n\right.\right\} =\mathcal{S}_{n}\!\setminus\!\mathcal{S}_{n-1}\end{equation*}

As we will see, these subsets structure a large parameter space. For orientation, in Figure 1, left side, the $\mathcal{S}_{\left(n\right)}$ , $n\geq1$ , are the triangles below the a-axis; they include their respective right border and exclude the other borders. The latter borders are where the step function $m_{0}\left(a, s\right)$ changes value. $\mathcal{S}_{0}=\mathcal{S}_{\left(0\right)}$ is the union of Areas 1 and 2 above the a-axis, while $\mathcal{S}_{n}$ , $n\geq1$ , unites these areas with the first n triangles below the a-axis.

Lemma 3.2. For strongly admissible a, s, the RHS of the formula with corresponding delay $m_{0}=m_{0}\left(a, s\right)$ ,

(9) \begin{equation}\left(1-a\right)^{-\frac{s}{a}}-\sum_{j=0}^{m_{0}-1}r_{j}\left(a, s\right)=\sum_{k=m_{0}}^{\infty}r_{k}\left(a, s\right)\end{equation}

converges and has either positive or negative summands, except that in Area 1 we ultimately, namely for all $k>n_{0}$ , have $r_{k}\left(a, s\right)=0$ .

Proof. Formula 9 is the rearranged (and convergent) binomial series as given in Formula 8. The summands $r_{k}$ , $k\geq0$ , are, up to a positive factor, products of the factors $\left(s+ai\right)$ , where $i=0,...,k-1$ .

In Area 1, these factors are initially positive, but $\left(s+an_{0}\right)=0$ . Thus, $r_{0},...,r_{n_{0}}$ are positive, while the further $r_{k}$ equal 0.

In Area 2, both a and s are positive, such that all $\left(s+ai\right)$ and all $r_{k}$ are positive.

In Area 3, we have $s+a\left(m_{0}-1\right) \lt 0 \lt s+am_{0}$ . So, the $\left(s+ai\right)$ are negative for $i<m_{0}$ and positive for $i\geq m_{0}$ . Thus, all $r_{k}$ with $k\geq m_{0}$ have the same sign, either positive or negative.

Definition 3.3. For any $k\in\mathbb{N}_{0}$ and strongly admissible a, s with delay $m_{0}=m_{0}\left(a, s\right)$ , we set

(10) \begin{equation}p_{k}\left(a, s\right)\,:\!=\,\begin{cases}0, & k \lt m_{0}\\[4pt]\frac{r_{k}\left(a, s\right)}{\left(1-a\right)^{-s/a}-\sum_{j=0}^{m_{0}-1}r_{j}\left(a, s\right)}, & k\geq m_{0}\end{cases}\end{equation}

For any $m\in\mathbb{N}_{0}$ , where $m<n_{0}$ in Area 1, we set $M\left(a, s;\,m\right)\,:\!=\,\max\left(m,m_{0}\left(a, s\right)\right)$ and

(11) \begin{equation}p_{k}\left(a, s;\,m\right)\,:\!=\,\begin{cases}0, & k \lt M\\[4pt]\frac{r_{k}\left(a, s\right)}{\left(1-a\right)^{-s/a}-\sum_{j=0}^{M-1}r_{j}\left(a, s\right)}, & k\geq M\end{cases}\end{equation}

Proposition 3.4. The $\left(p_{k}\left(a, s\right)\right)$ and the $\left(p_{k}\left(a, s;\,m\right)\right)$ of Definition 3.3 constitute nontrivial loss count distribution models having (for fixed m) parameter space $\mathcal{S}$ . The latter model is the m-truncation of the former. The delay $m_{0}\left(a, s\right)$ indicates the first non-zero probability of $\left(p_{k}\left(a, s\right)\right)$ , while $M\left(a, s;\, m\right)$ is the corresponding index for $\left(p_{k}\left(a, s;\,m\right)\right)$ . Both distributions satisfy Panjer’s recursion with parameters a, s for all but the initial zero probabilities, that is, for $k>m_{0}$ and $k>M$ , respectively.

Proof. According to the preceding lemma, all $p_{k}\left(a, s\right)$ , $k\geq m_{0}$ , have the same sign (apart from possibly ultimately equaling 0) and sum up to 1, so they constitute a discrete loss distribution, which is nontrivial as at least two $p_{k}\left(a, s\right)$ are positive. Obviously, $\left(p_{k}\left(a, s;\, m\right)\right)$ is the m-truncation of $\left(p_{k}\left(a, s\right)\right)$ and in particular a distribution. The constraint $m<n_{0}$ for Area 1 ensures that it is not the trivial distribution being concentrated at $n_{0}\in\mathbb{N}$ . Both distributions inherit Panjer’s recursion from the $r_{k}\left(a, s\right)$ .

Corollary 3.5. For $m\leq m_{0}\left(a, s\right)$ , that is, $M=m_{0}$ , m-truncations coincide with the original model. For $m=M>m_{0}$ , $\left(p_{k}\left(a, s;\, m\right)\right)$ is different from $\left(p_{k}\left(a, s\right)\right)$ and thus a proper Panjer distribution that is not narrow. Thus, $\left(p_{k}\left(a, s\right)\right)$ is a narrow Panjer distribution of order $m_{0}$ , while $\left(p_{k}\left(a, s;\, m\right)\right)$ , for $m>m_{0}$ , is a proper (but not narrow) distribution of order m.

The parameters of narrow distributions $\left(p_{k}\left(a, s\right)\right)$ with delay $m_{0}\in\mathbb{N}_{0}$ lie in $\mathcal{S}_{\left(m_{0}\right)}$ , while the parameters a, s of proper distributions $\left(p_{k}\left(a, s;\, m\right)\right)$ with $m\geq m_{0}\left(a, s\right)$ lie in $\mathcal{S}_{m}$ .

Proof. The assertions follow immediately from the construction of the $p_{k}\left(a, s\right)$ and $p_{k}\left(a, s;\, m\right)$ , and from the definition of the parameter sets $\mathcal{S}_{n}$ and $\mathcal{S}_{\left(n\right)}$ .

Example 3.6. For $s>0$ (and $m_{0}=0$ ), Formula 10 defines the combined binomial (Area 1)/NB (Area 2) distribution as parameterised in Formula 5. If $m>0$ , Formula 11 yields respective m-truncations of the binomial and NB distributions that are proper, but not narrow, $\left(a,b,m\right)$ models.

Let us now rewrite Formula 11, which for $m=m_{0}$ embraces Formula 10, in terms of the parameters a and $\alpha=\frac{s}{a}$ used in Formula 6. As $a\neq0$ for all strongly admissible a, s, this is possible and an equivalent parameterisation.

(12) \begin{equation}p_{k}\left(a,\alpha;\, m\right)\,:\!=\,\frac{r_{k}\left(a,\alpha\right)}{\left(1-a\right)^{-\alpha}-\sum_{j=0}^{M-1}r_{k}\left(a,\alpha\right)}=\frac{a^{k}\binom{\alpha+k-1}{k}}{\left(1-a\right)^{-\alpha}-\sum_{j=0}^{M-1}a^{j}\binom{\alpha+j-1}{j}},\quad k\geq M\end{equation}

Area 3 in this parameterisation means $0<a\leq1$ , $\alpha<0$ , $\alpha\notin-\mathbb{N}$ , which is a union of squares, see Figure 1, right side. In particular, we have $0\leq m_{0}-1<-\alpha<m_{0}$ .

Example 3.7. If $m=m_{0}$ , Formula 12 with parameters $\left(a,\alpha\right)$ in Area 3 parameterises the extended negative binomial (ENB) distribution as defined by Hess et al. (2002), a further of their six types of narrow $\left(a,b,n\right)$ models. $M=m>m_{0}$ yields m-truncations of ENB that are proper, but not narrow, $\left(a,b,m\right)$ models.

The special case $m=m_{0}=1$ , $-1<\alpha<0$ (first square below the a-axis) has been known for long and was termed ETNB distribution (Willmot 1988). However, like Hess et al. (2002), we prefer the name ENB and confirm their reasoning from our perspective: ETNB has $p_{0}=0$ , but it is no 1-truncation of any Panjer distribution – it is a narrow $\left(a,b,1\right)$ model. While one can indeed interpret ETNB as an extension of 1-truncated NB (which we will generalise below), an alternative perspective fits better to the unified view we are developing here: via Formula 10, ETNB and the rest of ENB ( $\alpha<-1$ , that is, the further squares below the a-axis, or equivalently triangles in terms of a, s) are an extension of the $\left(a,b,0\right)$ class; focusing on $a>0$ (Areas 2, 3), one can see ENB as an extension of the non-truncated NB model.

4. Embracing Limiting Cases

Definition 3.3 yielded three of the six narrow $\left(a,b,n\right)$ distributions, plus the corresponding proper distributions emerging via m-truncation. Now we show that it yields the other three types too, as limiting cases at (part of) the border of the parameter space. To this end, let us complement Definition 3.1.

Definition 4.1. We call real a, s admissible if

\begin{equation*}\left(a, s\right)\in\mathcal{A}\,:\!=\,\left\{ \left(a,-an_{0}\right)\left|\:a\in\left(-\infty,0\right),n_{0}\in\mathbb{N}\right.\right\} \cup\left\{ 0\right\} \!\times\!\left(0,\infty\right)\cup\left(0,1\right)\!\times\!\mathbb{R}\cup\left\{ 1\right\} \!\times\!\left(-\infty,0\right)\subset\mathbb{R}^{2}\end{equation*}

that is, if they lie in one of the following six disjoint areas (see also Figure 1, left side):

The delay of $\left(a, s\right)\in\mathcal{A}$ is the integer

(13) \begin{equation}m_{0}\left(a, s\right)\,:\!=\,\begin{cases}0, & s>0\\[4pt]1+\left[-\frac{s}{a}\right], & s\leq0\end{cases}\end{equation}

such that for $s\leq0$ we have $0\leq m_{0}-1\leq-\frac{s}{a}<m_{0}$ .

Finally, for $n\in\mathbb{N}_{0}$ we define the subsets

\begin{equation*}\mathcal{A}_{n}\,:\!=\,\left\{ \left(a, s\right)\in\mathcal{A}\left|\,m_{0}\left(a, s\right)\leq n\right.\right\} ,\ \ \ \mathcal{A}_{-1}\,:\!=\,\emptyset,\ \ \ \mathcal{A}_{\left(n\right)}\,:\!=\,\left\{ \left(a, s\right)\in\mathcal{A}\left|\,m_{0}\left(a, s\right)=n\right.\right\} =\mathcal{A}_{n}\!\setminus\!\mathcal{A}_{n-1}\end{equation*}

One sees at a glance that strongly admissible parameters are admissible and that the two definitions of the delay are consistent – as are the definitions of the respective subsets of $\mathcal{A}$ and $\mathcal{S}$ :

Lemma 4.2. We have $\mathcal{A}_{0}=\mathcal{A}_{\left(0\right)}=\left\{ \left(a, s\right)\in\mathcal{A}\left|\,s>0\right.\right\} $ and, for $n\in\mathbb{N}$ ,

\begin{equation*}\mathcal{A}_{n}=\left\{ \left(a, s\right)\in\mathcal{A}\left|\,s \gt 0\vee s \gt -na\right.\right\} ,\qquad\mathcal{A}_{\left(n\right)}=\left\{ \left(a, s\right)\in\mathcal{A}\left|\,-na \lt s\leq\left(1-n\right)a\right.\right\}\end{equation*}

For the subsets of $\mathcal{S}$ the analogous formulae hold and, for $n\in\mathbb{N}_{0}$ , $\mathcal{S}_{n}=\mathcal{S}\cap\mathcal{A}_{n}$ , $\mathcal{S}_{\left(n\right)}=\mathcal{S}\cap\mathcal{A}_{\left(n\right)}$ .

Proof. Straightforward algebra.

Formulae 10 and 11, which generate narrow/proper Panjer distributions, hold for Areas 1, 2, 3. They extend in a straightforward way to Area 4, which is adjacent (in topological sense) to Areas 1, 2.

Proposition 4.3. The $p_{k}\left(a, s\right)$ and $p_{k}\left(a, s;\, m\right)$ from Definition 3.3 are continuous on $\mathcal{S}$ and can be continuously extended to Area 4.

Proof. As all $p_{k}\left(a, s;\, m\right)$ are algebraic functions of the $p_{k}\left(a, s\right)$ , $k\in\mathbb{N}_{0}$ , it is sufficient to show continuity on $\mathcal{S}$ for the latter, which is clear from Formulae 10 and 3.

For $s>0$ , we can rearrange Formula 9 as

(14) \begin{equation}\left(1-a\right)^{-\frac{s}{a}}-\sum_{j=0}^{M-1}r_{j}\left(a, s\right)=\sum_{k=M}^{\infty}r_{k}\left(a, s\right)\end{equation}

where all summands but the first are continuous on $\mathbb{R}^{2}$ . Now recall from the binomial series that for $-1<a<1$ and $\gamma\in\mathbb{R}$ , $\left(1-a\right)^{\gamma}$ is well defined, finite and continuous in both variables. Thus, $\left(1-a\right)^{-\frac{s}{a}}$ is well defined and continuous for $\left(a, s\right)\in\left(\left(-1,1\right)\!\setminus\!\left\{ 0\right\} \right)\!\times\!\left(0,\infty\right)$ . The continuous extension to $\left\{ 0\right\} \!\times\!\left(0,\infty\right)$ , that is, Area 4, is straightforward via $\lim_{a\rightarrow0}\left(1-a\right)^{-\frac{s}{a}}=e^{s}$ . So, both sides of Equation (14) are continuous functions on $\mathcal{S}$ plus Area 4, and further non-zero, such that $r_{k}\left(a, s\right)$ divided by the LHS yields $p_{k}\left(a, s;\, m\right)$ being a continuous function on that domain.

Example 4.4. With $r_{k}\left(a, s\right)=\frac{s^{k}}{k!}$ on Area 4, Definition 3.3 here yields the Poisson distribution and its m-truncations, which are limiting cases of (m-truncated) binomial and NB distributions having the same parameter s.

The continuity of the $p_{k}\left(a, s\right)$ and $p_{k}\left(a, s;\, m\right)$ on the “upper” $\mathcal{A}_{0}$ (Areas 1, 2, 4) is remarkable as $\mathcal{A}_{0}$ has a weird geometry, see Figure 1, left side. Instead, continuity on Area 3 does not mean too much as this area is separated (in topological sense) from $\mathcal{A}_{0}$ and splits into small separated pieces: the triangular $\mathcal{S}_{\left(n\right)}$ , $n>0$ . However, things become interesting once we involve the remaining areas: Area 6 connects the pieces of Area 3, while Area 5 connects Area 3 and Area 2, see again Figure 1, left side.

Our aim is to define the $p_{k}\left(a, s\right)$ and $p_{k}\left(a, s;\, m\right)$ on Areas 5 and 6 in a reasonable way, which would mean having them on the whole parameter set $\mathcal{A}$ . With them being defined on $\mathcal{S}$ in a discrete manner via the step function $m_{0}\left(a, s\right)$ , we cannot expect them to be continuous on $\mathcal{A}$ . Yet, it will turn out that we get surprisingly close.

We can in the following treat Areas 5 and 6 largely in parallel, but the setting is a bit intricate. It is more convenient to work with the parameters $a,\alpha$ , which for $a\neq0$ ( $\mathcal{A}$ without Area 4) are equivalent to a, s and can be converted via smooth functions ( $\alpha=\frac{s}{a}$ , $s=\alpha a$ ), such that properties like continuity are not affected by this parameter change. For orientation see the right side of Figure 1 emphasising that the geometry of $\mathcal{A}$ is much simpler (utterly rectangular) in terms of $a,\alpha$ – as is the delay formula:

Definition 4.5. We call real $a,\alpha$ (strongly) admissible if $a\neq0,s=\alpha a$ are (strongly) admissible. We call the set of such parameters $\mathcal{A}^{*}$ ( $\mathcal{S}^{*}$ ) and write, equivalently to earlier formulae,

\begin{align*}m_{0}\left(a,\alpha\right) & \,:\!=\,\begin{cases}0, & \alpha \gt 0\\[4pt]1+\left[-\alpha\right], & \alpha\leq0\end{cases}\\[7pt]p_{k}\left(a,\alpha;\, m\right) & \,:\!=\,\begin{cases}0, & k \lt M=\max\left(m,m_{0}\left(a,\alpha\right)\right)\\[4pt]\dfrac{r_{k}\left(a,\alpha\right)}{\left(1-a\right)^{-\alpha}-\sum_{j=0}^{M-1}r_{j}\left(a,\alpha\right)}, & k\geq M\end{cases}\end{align*}

which for $m=m_{0}$ (and $M=m_{0}$ ) embraces $p_{k}\left(a,\alpha\right)$ . We write finally, for $n\in\mathbb{N}_{0}$ ,

\begin{align*}\mathcal{A}_{n}^{*} & \,:\!=\,\left\{ \left(a,\alpha\right)\in\mathcal{A}^{*}\left|\,m_{0}\left(a,\alpha\right)\leq n\right.\right\} ,\quad\mathcal{A}_{-1}^{*}\,:\!=\,\emptyset,\qquad\mathcal{A}_{\left(n\right)}^{*}\,:\!=\,\left\{ \left(a,\alpha\right)\in\mathcal{A}^{*}\left|\,m_{0}\left(a,\alpha\right)=n\right.\right\}\\& \qquad\qquad\qquad\quad\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad =\mathcal{A}_{n}^{*}\!\setminus\!\mathcal{A}_{n-1}^{*}\end{align*}

and analogously for the subsets of $\mathcal{S}^{*}$ .

Lemma 4.6. For $n\in\mathbb{N}$ we have

\begin{align*}\mathcal{A}_{0}^{*} & =\mathcal{A}_{\left(0\right)}^{*}=\left\{ \left(a,\alpha\right)\in\mathcal{A}^{*}\left|\,s>0\right.\right\} ,\;\mathcal{A}_{n}^{*}\left\{ \left(a,\alpha\right)\in\mathcal{A}^{*}\left|\,s>0\vee\alpha>-n\right.\right\} ,\\& \qquad\qquad\qquad \mathcal{A}_{\left(n\right)}^{*}\left\{ \left(a,\alpha\right)\in\mathcal{A}^{*}\left|\,-n<\alpha\leq1-n\right.\right\}\end{align*}

For the subsets of $\mathcal{S}^{*}$ the analogous formulae hold and, for $n\in\mathbb{N}_{0}$ , $\mathcal{S}_{n}^{*}=\mathcal{S}^{*}\cap\mathcal{A}_{n}^{*}$ , $\mathcal{S}_{\left(n\right)}^{*}=\mathcal{S}^{*}\cap\mathcal{A}_{\left(n\right)}^{*}$ .

Proof. Straightforward translation from the $\left(a, s\right)$ world.

Definition 4.7. For all $\left(a,-n\right)$ in Areas 5 and 6 of $\mathcal{A}^{*}$ (where $a>0$ , $n\in\mathbb{N}_{0}$ , and $m_{0}=n+1$ ), we set

(15) \begin{equation}p_{k}\left(a,\alpha=-n\right)\,:\!=\,\left.p_{k}\left(a,\alpha\right)\right|_{\alpha=-n}\,:\!=\,\lim_{\alpha\nearrow-n}p_{k}\left(a,\alpha\right)\end{equation}

and set, for $m\in\mathbb{N}_{0}$ , the m-truncations $p_{k}\left(a,\alpha=-n;\, m\right)$ thereof as usual, according to Formula 2.

Proposition 4.8. The $p_{k}\left(a,\alpha=-n\right)$ and the $p_{k}\left(a,\alpha=-n;\, m\right)$ of the preceding definition constitute well-defined discrete loss distributions on Areas 5 and 6. Together with the earlier analogous definitions on $\mathcal{S^{*}}$ , they define $\left(p_{k}\left(a,\alpha\right)\right)$ and its m-truncations on $\mathcal{A}^{*}$ .

$p_{k}\left(a,\alpha\right)$ is continuous on $\mathcal{S^{*}}$ , while on $\mathcal{A^{*}}$ it is continuous in a (for fixed $\alpha$ ) and left-continuous in $\alpha$ with discontinuities at Areas 5 and 6 (for $k>n=-\alpha$ ). Instead, the m-truncation $p_{k}\left(a,\alpha;\, m\right)$ is continuous on $\mathcal{A}_{m}^{*}$ .

For $\left(a,\alpha\right)\in\left(0,1\right)\!\times\!\left\{ 0\right\} $ (Area 5) or $\left(a,\alpha\right)\in\left(0,1\right]\!\times\!-\mathbb{N}$ (Area 6), we have with $m_{0}\left(a,\alpha\right)=1-\alpha$ that $p_{k}\left(a,\alpha\right)=0$ for $k<m_{0}$ , while for $k\geq m_{0}$

(16) \begin{align}p_{k}\left(a,\alpha=-n\right)& =\frac{a^{k}/\binom{k}{m_{0}}}{\sum_{j=m_{0}}^{\infty}a^{j}/\binom{j}{m_{0}}}\nonumber\\& =\frac{a^{k}/\binom{k}{m_{0}}}{\left(-1\right)^{m_{0}}m_{0}\left\{ \left(\ln\left(1-a\right)\right)\left(1-a\right)^{m_{0}-1}-\sum_{j=1}^{m_{0}-1}\left(\left(-a\right)^{j}\binom{m_{0}-1}{j}\sum_{i=m_{0}-j}^{m_{0}-1}\frac{1}{i}\right)\right\} }\end{align}

with $m_{0}$ notably equaling $1+n$ .

For $m\in\mathbb{N}_{0}$ and $M=\max\left(m,m_{0}\right)$ , we have analogously $p_{k}\left(a,\alpha\right)=0$ for $k<M$ , while for $k\geq M$

(17) \begin{align}& p_{k}\left(a,\alpha=-n;\, m\right)=\frac{a^{k}/\binom{k}{m_{0}}}{\sum_{j=M}^{\infty}a^{j}/\binom{j}{m_{0}}}\nonumber\\& =\frac{a^{k}/\binom{k}{m_{0}}}{\left(-1\right)^{m_{0}}m_{0}\left\{ \left(\ln\left(1-a\right)\right)\left(1-a\right)^{m_{0}-1}-\sum_{j=1}^{m_{0}-1}\left(\left(-a\right)^{j}\binom{m_{0}-1}{j}\sum_{i=m_{0}-j}^{m_{0}-1}\frac{1}{i}\right)\right\} +\sum_{j=m_{0}}^{M-1}a^{j}/\binom{j}{m_{0}}}\end{align}

Proof. The proof and more details are set out in Appendix A.

Note that Formulae 16 and 17 contain the parameter $\alpha$ only indirectly, via $m_{0}$ , which will below make conversion to the parameters a, s very easy. Both formulae represent the respective probabilities in two ways, a compact and a complex one. The former contains an infinite sum, which may complicate its use in practice, so it is good to have both variants.

One sees quickly that the parameter spaces can be allocated just as in Corollary 3.5.

Corollary 4.9. The parameters of narrow distributions $\left(p_{k}\left(a,\alpha\right)\right)$ with delay $m_{0}\in\mathbb{N}_{0}$ lie in $\mathcal{A}_{\left(m_{0}\right)}^{*}$ , where $-m_{0}<\alpha\leq1-m_{0}$ . The parameters $a,\alpha$ of proper distributions $\left(p_{k}\left(a,\alpha;\, m\right)\right)$ with $m\geq m_{0}\left(a,\alpha\right)$ lie in $\mathcal{A}_{m}^{*}$ , where $\alpha>-m$ .

Example 4.10. The compact part of Formula 16 (with the infinite sum in the denominator) is given, for $\alpha=1-m_{0}\in-\mathbb{N},0<a\leq1$ , by Hess et al. (2002), who term the resulting narrow distribution type extended logarithmic distribution. It corresponds to Area 6 of the parameter space and Formula 17 yields the corresponding m-truncations and proper models.

The given name becomes clear if we at last look at Area 5, where $\alpha=1-m_{0}=0,0<a<1$ . Here Formula 16 simplifies to

\begin{equation*}p_{k}\left(a,\alpha=0\right)=\frac{a^{k}/k}{-\ln\left(1-a\right)},\qquad k\geq1\end{equation*}

which is the logarithmic distribution, one of the two well-known narrow Panjer models of order 1 and the sixth narrow model type as specified by Hess et al. (2002). Again, Formula 17 yields the corresponding m-truncations and proper distributions.

For ease of orientation, we restate the parameter space of the narrow Panjer class in formulae and in Figure 2, in terms of both $\left(a, s\right)$ and $\left(a,\alpha\right)$ . We add the geometric distribution, which is the special NB model having $\frac{s}{a}=\alpha=1$ .

\begin{equation*}\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}l}(1) & \textrm{B} & a<0, & s>0, & \alpha\in-\mathbb{N};\\[2pt](2) & \textrm{NB} & 0<a<1, & s>0, & \alpha>0;\\[2pt] & \textrm{Geo} & 0<a<1, & s=a, & \alpha=1;\\[2pt] (3) & \textrm{ENB} & 0<a\leq1, & s<0, & \alpha\in\left(-\infty,0\right)\!\setminus\!-\mathbb{N};\\[2pt] & \textrm{ETNB} & 0<a\leq1, & -1<s<0, & \alpha\in\left(-1,0\right);\\[2pt] (4) & \textrm{P} & a=0, & s>0;\\[2pt] (5) & \textrm{Log} & 0<a<1, & s=0, & \alpha=0;\\[2pt] (6) & \textrm{ELog} & 0<a\leq1, & s<0, & \alpha\in-\mathbb{N}\end{array}\end{equation*}

The left side of Figure 2 is a redesign of Figure 7.2.1 of Panjer & Willmot (1992), which represents the narrow $\left(a,b,0\right)$ and $\left(a,b,1\right)$ classes in terms of $\left(a,b\right)$ . We have added ELog and the ENB parts beyond ETNB (below the $s=-a$ line), which constitute the narrow models of higher order. To the right we show the same parameter space apart from Poisson ( $a=0$ ), translated to $\left(a,\alpha\right)$ .

Figure 2 Narrow Panjer class.

As noted earlier, the overall ENB parameter space splits into separate pieces, with ELog models in between. The delay $m_{0}$ is constant on each ENB piece and the (E)Log model bordering above. The logarithmic model separates ENB from NB, which has a topologically connected parameter space, but the intermediate geometric distribution can be seen as separating the very skewed NB models between Log and Geo from the “higher” NB models being closer to the limiting Poisson model.

As for the subsets of the parameter space, in both charts of Figure 2, $\mathcal{A}_{0}^{\left(*\right)}=\mathcal{A}_{\left(0\right)}^{\left(*\right)}$ corresponds to the part of the parameter space above (and excluding) the a-axis, plus the area left of the $\alpha$ -axis. $\mathcal{A}_{\left(1\right)}^{\left(*\right)}$ is the log line segment plus the first ENB piece (ETNB) immediately below, which on the left side is triangular and on the right side square. $\mathcal{A}_{\left(2\right)}^{\left(*\right)}$ is the second ENB triangle/square including its upper border, which is the first part of ELog; and so on.

The continuity of $p_{k}\left(a,\alpha;\, m\right)$ on $\mathcal{A}_{m}^{*}$ gives another insight if we for the moment leave the (topologically separated) binomial area to the left of the $\alpha$ -axis aside. The lowest part of $\mathcal{A}_{m}^{*}$ (to the right of the $\alpha$ -axis) is $\mathcal{A}_{(m)}^{*}$ , the parameter space of the narrow Panjer model of order m having probabilities $p_{k}\left(a,\alpha\right)=p_{k}\left(a,\alpha;\, m\right)$ . Instead, the highest part of $\mathcal{A}_{m}^{*}$ is Area 2, where the $p_{k}\left(a,\alpha;\, m\right)$ represent the m-truncated NB distribution. By this perspective, $p_{k}\left(a,\alpha;\, m\right)$ can be seen as representing a continuous extension of m-truncated NB from Area 2 to $\mathcal{A}_{m}^{*}$ , or say: from $\alpha>0$ down to $\alpha>-n$ . In particular for $m=1$ , this means extending 1-truncated NB to the first ENB piece. This justifies the traditional name ETNB; however, it shall be noted that this extension also embraces the logarithmic model in between, which allows for a continuous passage from 1-truncated NB to ETNB.

Remark 4.11. While the NB model is often interpreted as being opposed to binomial ( $\alpha=\pm1,\pm2,\pm3,...$ , starting with geometric vs Bernoulli), Figure 2 reveals another opposition, on the right side: binomial vs extended logarithmic. For each $\alpha=-1,-2,-3,...$ there is a pair of models, which are indeed based on the same binomial series with exponent $-\alpha\in\mathbb{N}$ :

to the left of the $\alpha$ -axis $\left(1-a\right)^{-\alpha}$ , $a<0$ , generating a binomial model of “size” $n_{0}=-\alpha$ , where the $r_{k}$ (and $p_{k}$ ) are initially positive, but for $k\geq n_{0}+1=1-\alpha$ equal 0, which leads to a model being concentrated on $\left\{ 0,1,...,-\alpha\right\} $ ,

to the right of the $\alpha$ -axis $\left(1-a\right)^{-\alpha}$ , $0<a\leq1$ , generating an ELog model with positive probabilities starting at $m_{0}$ , where likewise the $r_{k}$ , for $k\geq m_{0}=1-\alpha$ , equal 0, but can be “rescaled” to $p_{k}>0$ via L’Hôpital’s rule.

If one, for the sake of symmetry with ELog, restricts binomial to $-1\leq a<0$ , one gets the binomial models having probability $0<p\leq\frac{1}{2}$ . The conversion of parameters for this and other Panjer distributions can be deduced most easily from Table 3 in Appendix B.

5. Wrap Up

Let us collect and complement the main results, formulating them in terms of the parameters a, s, which work for the whole Panjer class including Area 4.

Theorem 5.1. The general narrow Panjer class $\textrm{Pan}\left(a, s\right)$ and the general proper Panjer class $\textrm{Pan}\left(a, s;\, m\right)$ of distribution models can be described by the parameters

\begin{equation*}\left(a, s\right)\in\mathcal{A}=\left\{ \left(a,-an_{0}\right)\left|\:a\in\left(-\infty,0\right),n_{0}\in\mathbb{N}\right.\right\} \cup\left\{ 0\right\} \!\times\!\left(0,\infty\right)\cup\left(0,1\right)\!\times\!\mathbb{R}\cup\left\{ 1\right\} \!\times\!\left(-\infty,0\right)\subset\mathbb{R}^{2}\end{equation*}

and (for the latter)

\begin{equation*}\mathbb{N}_{0}\ni m\geq m_{0}\left(a, s\right)=\begin{cases}0, & s>0\\[4pt]1+\left[-\frac{s}{a}\right], & s\leq0\end{cases}\end{equation*}

with the additional constraint $m<n_{0}$ for $\left(a, s\right)\in\left\{ \left(a,-an_{0}\right)\left|\:a\in\left(-\infty,0\right),n_{0}\in\mathbb{N}\right.\right\} $ . Technically, one can drop the constraint $m\geq m_{0}$ but needs it if one wants uniqueness of parameters.

For $n\in\mathbb{N}_{0}$ , the narrow Panjer class of order n corresponds to the parameter subspace

\begin{equation*}\mathcal{A}_{\left(n\right)}=\left\{ \left(a, s\right)\in\mathcal{A}\left|\,-na \lt s\leq\left(1-n\right)a\right.\right\} ;\qquad\mathcal{A}_{\left(0\right)}=\left\{ \left(a, s\right)\in\mathcal{A}\left|\,s \gt 0\right.\right\}\end{equation*}

while the proper $\left(a,b,n\right)$ class corresponds to parameters in

\begin{equation*}\mathcal{A}_{n}\!\times\!\left\{ n\right\} ,\qquad\mathcal{A}_{n}=\left\{ \left(a, s\right)\in\mathcal{A}\left|\,s \gt -n\,a^{+}\right.\right\}\end{equation*}

$\mathcal{A}$ splits into six areas corresponding to six types of distributions:

\begin{equation*}\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}l}(1) & \textrm{B} & a<0, & s>0, & -\frac{s}{a}=n_{0}\in\mathbb{N};\\[1pt](2) & \textrm{NB} & 0<a<1, & s>0;\\[1pt](3) & \textrm{ENB} & 0<a\leq1, & s<0, & -\frac{s}{a}\notin\mathbb{N};\\[1pt](4) & \textrm{P} & a=0, & s>0;\\[1pt](5) & \textrm{Log} & 0<a<1, & s=0;\\[1pt](6) & \textrm{ELog} & 0<a\leq1, & s<0, & -\frac{s}{a}=n\in\mathbb{N};\end{array}\end{equation*}

The probabilities of the general narrow/proper Panjer distribution are given, with $p_{k}\left(a, s\right)=p_{k}\left(a, s;\, m_{0}\right)$ , by the following formula:

(18) \begin{equation}p_{k}\left(a, s;\, m\right)=\begin{cases}0, & k \lt M\\[4pt]\dfrac{r_{k}\left(a, s\right)}{\left(1-a\right)^{-s/a}-\sum_{j=0}^{M-1}r_{j}\left(a, s\right)}, & k\geq M\end{cases}\end{equation}

which emerges from the binomial series and where

\begin{equation*}r_{k}\left(a, s\right)=\frac{1}{k!}\prod_{i=0}^{k-1}\left(s+ai\right),\;k\in\mathbb{N}_{0};\qquad M\left(a, s;\, m\right)=\max\left(m,m_{0}\left(a, s\right)\right)\end{equation*}

Formula 18 holds in a strict sense on the parameter subspace (types B, NB, ENB)

\begin{equation*}\left(a, s\right)\in\mathcal{S}=\left\{ \left(a,-an_{0}\right)\left|\:a\in\left(-\infty,0\right),n_{0}\in\mathbb{N}\right.\right\} \cup\left(0,1\right)\!\times\!\left(0,\infty\right)\cup\left(0,1\right]\!\times\!\left(\left(-\infty,0\right)\!\setminus\!-\mathbb{N}\right)\subset\mathcal{A}\end{equation*}

and extends to $\mathcal{A}\!\setminus\!\mathcal{S}$ as follows:

type P: for $\left(a, s\right)\in\left\{ 0\right\} \!\times\!\left(0,\infty\right)$ by simply replacing $\left(1-a\right)^{-s/a}$ by $e^{s}$ , which formally means setting

\begin{equation*}p_{k}\left(a=0,s;\, m\right)\,:\!=\,\left.p_{k}\left(a, s;\, m\right)\right|_{a=0}\,:\!=\,\lim_{a\rightarrow0}p_{k}\left(a, s;\, m\right)\end{equation*}

types Log, ELog: for $0<a\leq1,-\frac{s}{a}=n\in\mathbb{N}_{0}$ , but without $\left(a, s\right)=\left(1,0\right)$ , by setting

\begin{equation*}p_{k}\left(a, s=-na;\, m\right)\,:\!=\,\left.p_{k}\left(a, s;\, m\right)\right|_{s=-na}\,:\!=\,\lim_{s\nearrow-na}p_{k}\left(a, s;\, m\right)\end{equation*}

Here $m_{0}=n+1$ and for $k\geq M$ we have

(19) \begin{align}& p_{k}\left(a, s=-na;\, m\right)=\frac{a^{k}/\binom{k}{m_{0}}}{\sum_{j=M}^{\infty}a^{j}/\binom{j}{m_{0}}}\nonumber\\& =\frac{a^{k}/\binom{k}{m_{0}}}{\left(-1\right)^{m_{0}}m_{0}\left\{ \left(\ln\left(1-a\right)\right)\left(1-a\right)^{m_{0}-1}-\sum_{j=1}^{m_{0}-1}\left(\left(-a\right)^{j}\binom{m_{0}-1}{j}\sum_{i=m_{0}-j}^{m_{0}-1}\frac{1}{i}\right)\right\} +\sum_{j=m_{0}}^{M-1}a^{j}/\binom{j}{m_{0}}}\nonumber\\\end{align}

With these limiting cases, $p_{k}\left(a, s\right)$ is continuous on $\mathcal{A}$ apart from Areas 5 and 6 ( $s=-na$ , $n\in\mathbb{N}_{0}$ ), where we have discontinuity for $k>n$ . Instead, for fixed $m\geq m_{0}$ , $p_{k}\left(a, s;\, m\right)$ is a continuous function on $\mathcal{A}_{m}$ .

The general formula for the probability generating function of the proper general Panjer class (which embraces the narrow class for $m=m_{0}$ ) is

(20) \begin{equation}P_{\left(a, s;\, m\right)}\left(z\right)=\frac{\left(1-az\right)^{-s/a}-\sum_{j=0}^{M-1}r_{j}\left(az,s\right)}{\left(1-a\right)^{-s/a}-\sum_{j=0}^{M-1}r_{j}\left(a, s\right)}\end{equation}

which likewise holds in a strict sense for B, NB, ENB, and extends

to P via $a\rightarrow0$ , which simply means replacing $\left(1-a\right)^{-s/a}$ by $e^{s}$ and $\left(1-az\right)^{-s/a}$ by $e^{sz}$ ;

to (E)Log via $s\nearrow-na$ . Here, we have

(21) \begin{align}& P_{\left(a, s=-na;\, m\right)}\left(z\right)\,:\!=\,\left.P_{\left(a, s;\, m\right)}\left(z\right)\right|_{s=-na}=\frac{\sum_{j=M}^{\infty}\left(az\right)^{j}/\binom{j}{m_{0}}}{\sum_{j=M}^{\infty}a^{j}/\binom{j}{m_{0}}}\nonumber\\& =\frac{\left(-1\right)^{m_{0}}m_{0}\left\{ \left(\ln\left(1-az\right)\right)\left(1-az\right)^{m_{0}-1}-\sum_{j=1}^{m_{0}-1}\left(\left(-az\right)^{j}\binom{m_{0}-1}{j}\sum_{i=m_{0}-j}^{m_{0}-1}\frac{1}{i}\right)\right\} +\sum_{j=m_{0}}^{M-1}\left(az\right)^{j}/\binom{j}{m_{0}}}{\left(-1\right)^{m_{0}}m_{0}\left\{ \left(\ln\left(1-a\right)\right)\left(1-a\right)^{m_{0}-1}-\sum_{j=1}^{m_{0}-1}\left(\left(-a\right)^{j}\binom{m_{0}-1}{j}\sum_{i=m_{0}-j}^{m_{0}-1}\frac{1}{i}\right)\right\} +\sum_{j=m_{0}}^{M-1}a^{j}/\binom{j}{m_{0}}}\end{align}

With the notation $x_{\left(n\right)}=\binom{x}{n}n!=x\left(x-1\right)...\left(x-n+1\right)$ , $x\in\mathbb{R}$ , $n\in\mathbb{N}_{0}$ (with $x_{\left(0\right)}=1$ ), we have, for $a<1$ , the following recursion for the factorial moments of $N\sim\textrm{Pan}\left(a, s;\, m\right)$ :

(22) \begin{equation}\textrm{E}\left(N_{\left(n\right)}\right)=\frac{1}{1-a}\left\{ \left(\left(n-1\right)a+s\right)\textrm{E}\left(N_{\left(n-1\right)}\right)+M_{\left(n\right)}p_{M}\right\} ,\qquad n\in\mathbb{N}\end{equation}

The respective first and second moments, written compactly in terms of a, s, M, and the first non-zero probability $p_{M}$ , are

(23) \begin{equation}\textrm{E}\left(N\right)=\textrm{E}\left(N_{\left(1\right)}\right)=\frac{s+Mp_{M}}{1-a}\qquad \qquad \qquad \qquad \end{equation}

(24) \begin{equation}\textrm{E}\left(N_{\left(2\right)}\right)=\frac{s\left(a+s\right)+\left\{ \left(1-a\right)M+2a+s-1\right\} Mp_{M}}{\left(1-a\right)^{2}}\end{equation}

(25) \begin{equation}\textrm{Var}\left(N\right)=\frac{s+\left\{ \left(1-a\right)M+a-s\right\} Mp_{M}-M^{2}p_{M}^{2}}{\left(1-a\right)^{2}}\quad \end{equation}

Geometrically, all narrow and proper Panjer distributions have unimodal probability functions with an overall maximum probability and no further local maxima. For $s\leq0$ (Log, ENB, ELog) the mode is the first non-zero probability.

Proof. Recall that Hess et al. (2002) defined the six types of narrow and proper Panjer distributions (using diverse parameterisations), proving in particular that no more types exist. Our system describes the same distributions, rewritten uniformly in terms of the parameters a, s, m, and thus also covers the whole Panjer class. The optional constraint $m\geq m_{0}$ (which implies $M=m$ ) for the proper models ensures unique parameters: for all $m\leq m_{0}$ one has $p_{k}\left(a, s;\, m\right)=p_{k}\left(a, s;\, m_{0}\right)=p_{k}\left(a, s\right)$ .

The correspondence between the classes of order n and the subsets $\mathcal{A}_{n}$ , $\mathcal{A}_{\left(n\right)}$ of $\mathcal{A}$ results quickly from Corollaries 3.5 and 4.9.

The formulae for the probabilities and the assertions on continuity are earlier results of this paper (in particular from Proposition 4.8), translated from the $\left(a,\alpha\right)$ parameterisation where necessary. The latter does not cover Area 4, but this is topologically separated from Areas 5 and 6 and does not affect their continuity properties.

The general pgf formula is given in Table 1 of Hess et al. (2002) for the (E)NB distributions, in terms of $a,\alpha$ (which they call $\vartheta,\beta$ ). Converting to a, s, embracing B and P and generalising to the respective m-truncations is straightforward. The compact variant of the ELog formula is given there too; generalising to m-truncations is straightforward. The second (complex) pgf representation results from Formula 27 in Appendix A; both variants embrace Log ( $n=0$ , $0<a<1$ , $m_{0}=1$ ).

The recursion for the factorial moments results from the corresponding differential equation for the pgf, as given in Theorem 2.1(b) of Hess et al. (2002), after translation to our notation. The expectation follows immediately, the second moment after some algebra.

As for modes, if a probability function, on some interval, is positive and first rises then decreases, constituting a (possibly local) maximum, this implies that some $\frac{p_{k}}{p_{k-1}}$ are greater than 1, being followed by values smaller than 1. For the non-zero probabilities of proper Panjer distributions we have $\frac{p_{k}}{p_{k-1}}=a+\frac{b}{k}$ , which is a monotonic function in k for all admissible a, b and can shift from being $>1$ to $<1$ at most once. So, proper Panjer distributions cannot have a local maximum probability other than the global one, thus are unimodal.

As is well known, for $s>0$ the mode of the narrow distributions (B, NB, P) may be positive and may consist of two subsequent probabilities, which is, however, still interpreted as unimodal. This property is inherited by their left-truncations, the corresponding proper Panjer distributions.

For a proper Panjer distribution with $s\leq0$ we have $a>0$ , $b=s-a<0$ , such that overall for the non-zero probabilities we have $\frac{p_{k}}{p_{k-1}}=a+\frac{b}{k}$ with a factor $0<a+\frac{b}{k}<a\leq1$ . Thus, the probabilities strictly decrease from the first non-zero probability onwards.

Note that the recursion for the moments does not embrace the case $a=1$ , where at least the higher moments are infinite (Hess et al., 2002). We have included this border case in our theory for the sake of completeness; it is, however, not of much practical interest.

Example 5.2. For the Panjer $\left(a,b,0\right)$ class, the formulae for expectation and variance are simple and well known:

\begin{equation*}\textrm{E}\left(N\right)=\frac{s}{1-a},\qquad\textrm{Var}\left(N\right)=\frac{s}{\left(1-a\right)^{2}}\end{equation*}

The corresponding formulae for the proper $\left(a,b,1\right)$ class are

\begin{equation*}\textrm{E}\left(N\right)=\frac{s+p_{1}}{1-a},\qquad\textrm{Var}\left(N\right)=\frac{\left(s+p_{1}\right)\left(1-p_{1}\right)}{\left(1-a\right)^{2}}\end{equation*}

They are very compact; however, some complexity is hidden in $p_{1}$ . Note that they hold for both the narrow distributions of order 1 (Log and ETNB) and the 1-truncations of Poisson, binomial and NB. Instead, the better-known alternative formulae (see Appendix B.3.1 of Klugman et al. (2008))

\begin{equation*}\textrm{E}\left(N\right)=\frac{s}{\left(1-a\right)\left(1-p_{0}^{*}\right)},\qquad\textrm{Var}\left(N\right)=\frac{s\left(1-\left(1+s\right)p_{0}^{*}\right)}{\left(1-a\right)^{2}\left(1-p_{0}^{*}\right)^{2}}\end{equation*}

are more complex and hold only for the 1-truncations of the models of order 0, from which they take the no-loss probability $p_{0}^{*}$ .

6. Parameter Inference

With the multitude of situations where Panjer distributions could be applied, from abundant empirical data (wide models) to cases where low loss counts are impossible (proper models), a full exploration of parameter inference is beyond the scope of this paper; this would probably yield enough content for a separate paper (available data for examples permitting). Yet, some ideas for potentially successful procedures can be derived from generalising a graphical approach explained in section 6.5 of Klugman et al. (2008) for the classical distributions of order 0.

If the probabilities of a loss count distribution ultimately fulfil Panjer’s recursion, for large enough k, the empirical loss counts $n_{k}$ fulfil approximately, apart from random fluctuation, the equivalent formulae

\begin{equation*}\frac{n_{k}}{n_{k-1}}\approx a+\frac{b}{k},\qquad k\frac{n_{k}}{n_{k-1}}\approx ka+b\end{equation*}

For high k randomness and low likelihood will let most $n_{k}$ equal zero, for the lowest k the recursion may not hold; however, if we observe an intermediate interval of values k with non-zero $n_{k}$ , we can plot the $k\frac{n_{k}}{n_{k-1}}$ against k and check whether they approximate an affine linear function $ka+b=\left(k-1\right)a+s$ . If so, the distribution is a candidate for a general-Panjer fit and we can get initial (rough) estimates for a, b, s and $m_{0}$ , the latter being an approximate lower bound for m.

The assessment of m is crucial: if we can narrow down m to one or a few integers, we can do the final estimation of the further parameters for fixed m, which greatly reduces complexity of the Maximum Likelihood (or other) estimation algorithm.

• If we want to fit a proper model, m is often determined by the setting of the problem. For example, in the case of the six risks warranty (Cat reinsurance) mentioned earlier, it is clear that m most probably equals 6.

• Wide models have many parameters and need a lot of data. In such situations the plot should make clear where the Panjer recursion starts to hold. The initial probabilities before the Panjer tail are parameters and simply estimated by the respective sample frequencies.

Having fixed m, a look at the initial estimates for a and s will in many cases narrow down the parameter space to consider: to a rather small subset of $\mathcal{A}_{m}$ . The most benign cases are clear indications for either (possibly truncated) binomial or NB. ENB is slightly more complicated with the ELog stripes in between; however, as the $p_{k}\left(a, s;\, m\right)$ are continuous on $\mathcal{A}_{m}$ , complexity is lower than the above ELog formulae indicate. Bordering cases require some more work, just as in the classical case: when one is not sure whether binomial or NB fits better, one has to calculate both, plus the intermediate Poisson.

In such cases inference may technically work better when done separately per distribution type, possibly using parameters other than the unified ones. For example, for NB the parameterisation NB2 as given in Appendix B, which uses shape parameter $\alpha$ and expectation $\lambda$ , is preferable as the respective MLE estimators are asymptotically independent, see section 9.8 of Panjer & Willmot (1992).

Table 2. Loss count example.

Example 6.1. For illustration we revisit Example 6.2 of Klugman et al. (2008) treating an MTPL loss count data set, see Table 2. The $k\frac{n_{k}}{n_{k-1}}$ show a slow but clear upward trend, albeit one must acknowledge that the last points are based on very scarce data. However, the data seem to indicate that if modelled with a Panjer distribution, the resulting parameter a should be positive.

So, among the models of order 0, the first choice would be NB. Here one would expect to get a rather small $a\in\left(0,1\right)$ and an even smaller $s>0$ , which would lead to a likewise small $\alpha=\frac{s}{a}>0$ and thus to a rather skewed NB distribution.

Alternatively, one can try wide $\left(a,b,1\right)$ models, which are called zero-modified by Klugman et al. (2008) and have a free parameter $p_{0}$ followed by a Panjer tail. Willmot (1988) derived (in his section 6) common MLE formulae for zero-modified NB/ETNB, embracing the intermediate Log as limiting case. This means inference based on the parameter space $\mathcal{A}_{1}$ , or more exactly $\mathcal{A}_{1}^{*}$ as the parameters $a,\alpha$ were used. The MLE estimators were applied to a number of MTPL data sets, including (as Data Set 5) the one discussed here. The resulting MLE estimates are the sample frequency $\widehat{p_{0}}=\frac{7840}{9461}=0.829$ and $\widehat{\alpha}=-0.103$ , $\widehat{a}=0.380$ , which means $\widehat{s}=\widehat{a}\widehat{\alpha}=-0.391$ . The resulting fit turns out to be statistically reasonable; the negative, but quite small, $\widehat{\alpha}$ means that the estimated tail model is ETNB and quite similar to Log ( $\alpha=0$ ).

However, noting that the slope of the function $k\frac{n_{k}}{n_{k-1}}$ rises after the first steps, one could alternatively try wide models of a bit higher order. After the second step the function $k\frac{n_{k}}{n_{k-1}}$ looks rather steep; an affine linear approximation in this area would lead to a larger a than with the above models, possibly greater than $0.5$ , and to a negative s maybe close to $-1$ . The resulting $\alpha$ would be in or around the interval $\left[-2,-1\right]$ . So, plausible values for $m_{0}$ are 1, 2 and possibly 3. Having already tried $m=0,1$ , the next (and due to data scarcity probably last reasonable) option is $m=2$ . In a wide model of order 2, $p_{0}$ and $p_{1}$ are estimated by the empirical frequencies. The further probabilities constitute the Panjer tail; the respective parameter space for a, s is $\mathcal{A}_{2}$ ; tail inference is based on the observed $n_{k}$ with $k\geq2$ . Here we have 304 such observations, which is not abundant but could suffice for reliable inference of the two tail parameters.

7. Conclusion

We have seen that the representation of the general Panjer class of distributions in terms of a, s yields (mostly) simpler formulae and more intuitive insight than the traditional representation in terms of a, b. Other parameterisations, for example, $a,\alpha$ , can yield still simpler formulae and additional insight, but they usually do not work for the whole class, see the overview in Appendix B and there in particular Table 3. It seems that some of the most common parameterisations, while being very practical for a specific model type, obscure the links to other parts of the Panjer class.

Beyond being instructive, the unified view of the general Panjer class of distributions via the parameters a, s, m can ease implementation and use of the models in practice, providing (mostly compact) all-in one formulae for their probabilities and moments.