5.1.1 Main Properties of the Claim Number Process

The distribution of N(t), the number of claims up to time t, can be given in explicit form

It can also be introduced via the (probability) generating function

(5.1.1)

which is at least defined for |z| ≤ 1, but often the radious of convergence is larger. Apart from the expression p_n(t) we will introduce a (c.d.f) of N(t), that is, the probability of at most r claims up to time t:

(5.1.2)

Among the main characteristics of a claim number process we should mention a variety of possible moments and derived quantities. The function Q_t(z) can be used to evaluate these consecutive moments of N(t). Here are a few of the most important qualitative indices.

1. For |z| < 1, the rth order derivative of Q_t with respect to z is given by
   
   In terms of expectations this reads as
   (5.1.3)
   The factorial moments of N(t) can be obtained from this by letting z ↑ 1:
   (5.1.4)
   In many cases Q_t(z) will be analytic at z = R > 1. Then for 0 ≤ u < R − 1
   (5.1.5)
2. From (5.1.4) one easily finds the successive moments of N(t). Most importantly we begin with the mean number of claims, which is obtained from the first derivative of Q_t(z) at 1:
   (5.1.6)
3. The variance of the number of claims is obtained from (5.1.4) with r = 1 and r = 2 and gives
   (5.1.7)
4. The index of dispersion is defined by
   (5.1.8)
   This quantity is popular among actuaries. As we will see soon, it is equal to 1 for the Poisson process. As such, the value of the index of dispersion makes it possible to call a claim number process overdispersed (with respect to the Poisson case) if its index is greater than 1. The concept underdispersed is defined similarly. The index of dispersion has been introduced to indicate that claim numbers in a portfolio show a greater volatility than one could expect from a Poisson process; in particular overdispersion appears regularly in actuarial portfolios.

In the following we collect a broad set of models. Some of them – like the Poisson and the negative binomial model – are extremely popular while others are modifications intended to allow better data fitting. Several examples are based on an underlying probabilistic structure that has a natural actuarial context.

5.2.1 The Homogeneous Poisson Process

A homogeneous Poisson process with intensity (or rate) λ > 0 is a counting process that satisfies the following properties:

1. Start at 0: almost surely.
2. Independent increments: for any , , the increments are mutually independent for i = 1, …, n.
3. Poisson increments: for any
   

Without any doubt the homogeneous Poisson process Ñ(t) is the most popular among all claim number processes in the actuarial literature. Because of its benchmark character we deal with it first. We then offer a number of generalizations in which the homogeneous Poisson process acts as the main building block.

From the definition it follows that the increments are independent and also stationary, that is

with

Consequently

while also so that

It further follows that

Properties of the homogeneous Poisson process Ñ(t) and jump arrival times

are as follows:

1. The inter‐arrival times where T₀ := 0, are i.i.d. exponential random variables with mean 1/λ:
   
2. Order statistics property: the conditional distribution of (T₁, …, T_n) given for some equals the distribution of the order statistics U_{1, n} ≤ U_{2, n} ≤ … ≤ U_{n, n} of independent uniform (0, t) distributed random variables (e.g., see Sato [666]).
3. Memoryless property: it follows from the first property that the distribution of the time until the next arrival is independent of the time t we have already been waiting for that arrival:
   
   This property gives the homogeneous Poisson process a special role among all claim number processes and may be seen as one of the main reasons for its popularity from a modelling perspective.
4. Jump sizes: as t ↓ 0
   
   where the notation f(t) = o(t) means . Hence, at any point in time, no more than one claim can occur with positive probability.

Finally, note that the homogeneous Poisson process can be constructed based on the sequence of i.i.d. exponential random variables (with rate λ > 0) {W_j, j ≥ 1}, that is,

5.2.2 Inhomogeneous Poisson Processes

A more general definition of the Poisson process goes as follows:

A general Poisson process is a stochastic process N = N_μ = {N_μ(t);t ≥ 0} that satisfies the following properties:

1. Càdlàg paths: the paths of N are almost surely càdlàg functions, that is right‐continuous with existing left limits.
2. Start at 0:N_μ(0) = 0 almost surely.
3. Independent increments: for any , , the increments are mutually independent for i = 1, …, n.
4. Poisson increments: for a càdlàg function with for allt ≥ 0, the increments have the following distribution:
   
   The non‐decreasing function μ is called the mean‐value function ofN_μ.

It is natural to call μ the mean‐value function as it describes the expectation of the process increments:

The homogeneous Poisson process described in the preceding subsection is the special case of a linear mean‐value function,

for some intensity λ > 0.

An inhomogeneous Poisson process can be defined through a deterministic time‐change of a homogeneous process.

1. Let Ñ₁(t) be a homogeneous Poisson process with intensity λ = 1 and let μ(.) be a valid mean‐value function, then the process defined by is an inhomogeneous Poisson process with mean‐value function μ.
2. Conversely, every inhomogeneous Poisson process N has a representation as a time‐changed Poisson process, that is, {N_μ(μ⁻¹(t)), t ≥ 0} is a standard homogeneous Poisson process.

In other words, the intensity changing over time can equivalently be interpreted as going through time with constant intensity but varying speed. For this reason μ is also referred to as operational time: whereas time runs linearly for a homogeneous Poisson process, it can be seen to speed up or slow down according to μ for an inhomogeneous Poisson process.

If μ is continuous and strictly increasing with , then the inverse function μ⁻¹ exists and the inhomogeneous Poisson process can be converted back to a homogeneous one with intensity 1 by a time change using μ⁻¹.

In many applications it is even assumed that μ is absolutely continuous, that there exists a non‐negative function λ(.) such that

where the function λ(.) is called the intensity function. Note that for a homogeneous Poisson process with intensity λ > 0 we have λ(t) ≡ λ, t ≥ 0.

5.2.3 Mixed Poisson Processes

A far‐reaching generalization of the ordinary Poisson process is obtained when the parameter λ is replaced by a random variable Λ with mixing or structure distribution function F_Λ. This increases the flexibility of the model due to additional parameters while keeping some of the main properties of the Poisson case. A common interpretation of this model extension is given in terms of a counting process which consists of two or more different sub‐processes that individually behave as a Poisson process with a specific intensity value (e.g., with a heterogeneous group of policyholders each producing claims according to a simple Poisson process but with different intensities). Note, however, that here for each sample path of N(t) the realization of Λ is chosen once at the beginning (i.e., in the above interpretation all counts then come from one of these sub‐processes).

A mixed Poisson process can be represented as

where Ñ(t) is a homogeneous Poisson process and Λ is a positive random variable, and

(5.2.9)

while the generating function is given by

(5.2.10)

We evaluate the characteristics of the mixed Poisson process. It follows from the above equations that

The latter relations immediately yield a linearly increasing expected growth

if exists. Similarly,

and more generally

(again given that the respective moments of Λ exist). The expression for the variance shows that among all mixed Poisson processes the Poisson process has the smallest variance. Actually the variance will be linear if and only if Λ is degenerate and this happens exactly for the Poisson case. The index of dispersion here equals

which is constant in time for the homogeneous Poisson process. Since this index is always greater than 1, mixed Poisson processes are natural candidates to model overdispersed claim number processes. Another way of highlighting the role of the random variable Λ is the observation that for s ≥ 0,

which for tends to the Laplace transform of of F_Λ. This in turn implies that

(5.2.11)

Again, conditional on N_Λ(t) = n, the occurrence times of the n claim events are uniformly distributed on (0, t), that is, mixed Poisson processes also satisfy the order statistics property. It also essentially characterizes the mixed Poisson processes (see [492], page 160). However, for a non‐degenerate mixing variable, a mixed Poisson process no longer has independent increments (but conditionally independent ones).

The mixed Poisson process N_Λ = {N_Λ(t) t ≥ 0} was introduced to actuaries by Dubourdieu [310]. For a very thorough treatment of mixed Poisson processes and their stochastic properties, see Grandell [406]. We give a few special cases that have found their way into the actuarial literature.

5.2.4 Doubly Stochastic Poisson Processes

Whereas in mixed Poisson processes the constant (static) intensity of a homogeneous Poisson process is randomized, in a doubly stochastic Poisson process the entire mean‐value function of an inhomogeneous Poisson process is randomized.

Generalities on doubly stochastic Poisson processes.

Let denote the homogeneous Poisson process with intensity λ. Independently, let M = {M(t);t ≥ 0} be a stochastic process that has almost surely non‐decreasing càdlàg paths with M(0) = 0 and for . Then the process is called a doubly stochastic Poisson process directed by M. Doubly stochastic Poisson processes are also called Cox processes based on the paper [229] where this concept appeared first in a general form (the first treatment of a specific form of such a process for actuarial purposes goes back even further, to Ammeter [36], see the Notes at the end of the chapter).

Hereafter, unless otherwise stated, we assume the intensity λ of the underlying homogeneous Poisson process to be 1.

If there exists a non‐negative process Λ = {Λ(t);t ≥ 0} such that M has the representation

then Λ is called the intensity process.

If M(t) = Λ(0) ⋅ t for some non‐negative random variable Λ(0), then the transformed process is again a mixed Poisson process.

Given a sample path μ(t) of M(t), it holds that for s < t

Note that we do not need information concerning the full path of Λ between s and t, but rather the values at s and t. This is not the case if we write this in terms of the intensity process {Λ(t);t ≥ 0}:

From the conditional distribution of N(t) given {Λ(u), 0 ≤ u ≤ t} we obtain

For the first two moments of doubly stochastic Poisson processes, note that from the conditional distribution of N(t) − N(s) given {Λ(u), s ≤ u ≤ t} we find

from which

while from

we obtain

from which the overdispersion for doubly stochastic Poisson processes follows.

A probabilistic construction of the claim arrival process of a doubly stochastic Poisson process can be based on the one‐to‐one correspondence between the claim arrival times and claim number process

(5.2.13)

If W_j () denote independent standard exponentially distributed random variables, then

(5.2.14)

For a fixed time horizon T > 0, a sample path of N(t) can now be generated using the following steps for a given stochastic model for M(t):

1. Simulate a path from M(t) according to some suitably fine discretization grid.
2. Repeatedly draw independent standard exponentially distributed random variables until their sum exceeds the level , and compute the claim arrival times according to (5.2.14).
3. Determine the sample path from the sampled claim arrival times using (5.2.13).

While this algorithm is constructed directly on the very nature of a doubly stochastic Poisson process, more efficient algorithms are available. See Korn et al. [497] for an application of the acceptance/rejection method in this context.

Doubly stochastic Poisson processes directed by Lévy subordinators.

In [695], Selch introduced Cox processes Ñ_λ(M(t)) directed by a Lévy subordinator M. Lévy processes are characterized by their independent and stationary increments. In comparison with linear functions which have constant increments, Lévy processes have i.i.d. increments. Homogeneous Poisson processes hence are an example of Lévy processes. In the following we only state some basic facts and properties that we need for later purposes. For more mathematical details on Lévy processes, see, for example, Sato [666], Schoutens [683], and Applebaum [44].

A stochastic process X = {X(t);t ≥ 0} is called a Lévy process if it has the following properties:

1. Start at 0:X(0) = 0 almost surely.
2. Independent increments: for any , , the increments are mutually independent fori = 1, …, n.
3. Stationary increments: for any 0 < s < tthe increments satisfyX(t) − X(s) = _dX(t − s).
4. Stochastic continuity: for allt ≥ 0 andɛ > 0 one has .

For every Lévy process X one can construct a version that has càdlàg paths almost surely, and we can hence assume the càdlàg property (cf. [666]). A Lévy process with almost surely non‐decreasing paths is called a Lévy subordinator, denoted here by M.

Due to the stochastic continuity of the process, the jumps of a Lévy process cannot appear at any fixed point in time with a positive probability. Furthermore, due to the càdlàg property of the paths, only countably many jumps can occur in any finite time interval and the number of jumps with size larger than some arbitrary ɛ > 0 is finite.

One then considers the jump measure

where , and the so‐called Lévy measure

One typically imposes an integrability condition such as for the small jumps to be well behaved. The famous Lévy–Khintchine representation completely characterizes the distribution of the Lévy process in terms of the Lévy measure and an extra drift parameter b_X under the above integrability condition:

Let M be a Lévy subordinator. The Laplace transform of M(t) can be expressed in terms of the so‐called Laplace exponent by

where the Laplace exponent is a Bernstein function derived from the characteristics of the subordinator as

We give here some classical examples of Lévy subordinators.

1. The homogeneous Poisson process with intensity λ > 0 with Laplace exponent and Lévy characteristics b_M = 0 and .
2. The gamma (α, β) subordinator: with M(t) being gamma distributed with density
   
   The Laplace exponent is
   
   while , b_M = 0 and
   
   Moreover
   
   from which
   
3. The inverse Gaussian (β, η) subordinator: here M(t) is assumed to follow the classical inverse Gaussian distribution as given in example (c) of a mixed Poisson process with μ = (β/η)t and b = η⁻² (cf. Section 5.2.3):
   
   The Laplace exponent is
   
   Then , b_M = 0 and _. Moreover
   
   from which
   

Due to the jumps of the subordinator, the basic Poisson model is here extended to allow for simultaneous claim arrivals. The Lévy subordinator M serves as the (random) operational time (also referred to as stochastic clock).

The discontinuity of paths of M entails that M is not differentiable. It follows that no random intensity Λ exists for these Cox processes. Also, the process cannot be converted back to the underlying independent Poisson process by a time change with the inverse of the directing process.

Moreover, Selch [695] showed that Ñ_λ(M(t)) is a Poisson cluster process, which means that the cluster sizes are i.i.d. and their arrival times are determined by an independent Poisson process:

where {L(t) : t ≥ 0} denotes a Poisson process with intensity λ_L = Ψ_M(λ), and where, independent of L, Y₁, Y₂, … are i.i.d. copies of a cluster size random variable Y with Laplace transform . Hence the waiting times between clusters should be exponentially distributed for this model to be applicable.

It may be desirable that the operational time, although randomly running slower and faster, behaves on average like real time, that is,

which by the properties of Lévy processes is equivalent to

(5.2.15)

Condition (5.2.15) is referred to as time normalization.

Note that the marginal distributions of N(t) at a given time point t equal that of a mixed Poisson process. For the finite‐dimensional distribution of N with time points 0 = t₀ < t₁ < … < t_n and integers 0 = k₀ ≤ k₁ ≤ … ≤ k_m, Selch [695] provides the expression

(5.2.16)

with Δk_j = k_j − k_j−1 and Δt_j = t_j − t_j−1 (j = 1, …, n).

For the marginal distribution of N(t) and a gamma (α, β) subordinator, this formula must reduce to a negative binomial distribution. Indeed, then the derivatives of the Laplace transform are

so that

which corresponds to (5.2.12).

We summarize the four models based on Poisson processes (PP) in Table 5.1.

Homogeneous PP	Inhomogeneous PP
Ñ λ(t)
μ(t) = λt, λ(u) ≡ λ
Mixed PP	Doubly stochastic PP
M(t) = Λt

To illustrate the differences between the inhomogeneous Poisson and doubly stochastic Poisson counting process models, Figure 5.2 depicts a simulated path from each as well as a sample path of a homogeneous Poisson process with the same number of expected claims. We also give the expected value curve together with bands that are four standard deviations wide in each case. While the inhomogeneous Poisson process follows a non‐linear trend but shows Poisson jumps, the doubly stochastic Poisson example allows for larger jump sizes, but maintains the same expected claim number per time unit throughout the realization.

We emphasize that a main difference between a Pólya process (i.e., mixed Poisson process with gamma‐distributed Λ) and a doubly stochastic Poisson process directed by a gamma subordinator is that in the latter case for each new time interval the realization of Λ is sampled anew, whereas for the Pólya process the same value is used throughout. From this point of view, the interpretation of heterogeneous policies to motivate this model is more intuitive for the doubly stochastic Poisson process, as at each time instant the next claim(s) may come from a different group of policies with another claim intensity. In particular, when considering model fitting to discrete claim counts in Section 5.4 under an i.i.d. assumption for the available realizations of claim numbers, the doubly stochastic Poisson process is arguably the more natural continuous‐time version of this model.

Multivariate extensions.

The general framework of Cox processes allows several multivariate extensions. Multivariate Cox processes driven by shot‐noise processes are treated in Scherer et al. [670]. Another natural way to introduce dependence between claim numbers of different lines of business or categories is to start with a priori independent Poisson processes for each, but then direct them all by the same Lévy subordinator. This causes simultaneous claim occurrences in the different business lines in an intuitive way (cf. Selch [695]). To this end let N_j = {N_j(t); t ≥ 0} count the claims arriving in each portfolio j = 1, …, d up to time t ≥ 0. In a traditional modelling approach claim arrivals in each portfolio j are assumed to follow independent homogeneous Poisson processes with individual intensities λ_j > 0. Let be the corresponding multivariate standard Poisson process with independent marginals with intensity vector λ = (λ₁, …, λ_d). The multivariate Cox process N = (N₁, …, N_d) is then defined through

(5.2.17)

This process naturally generates dependence between the d portfolios, as the stochastic factor M similarly affects all components. Furthermore, while the underlying Poisson process yields only jumps of size one, the subordinator and therefore time can now jump, leading to simultaneous claim arrivals within and between the individual portfolios in the time‐changed process.

Expression (5.2.16) for the finite‐dimensional distributions of N with time points 0 = t₀ < t₁ < … < t_n and integer vectors 0 = k₀ ≤ k₁ ≤ … ≤ k_n with now generalizes to (cf. [695]):

(5.2.18)

with , Δt_i = t_i − t_i−1, i = 1, …, n, where we use the notation |x| = |(x₁, …, x_d)| = |x₁| + … + |x_d|, and k! = k₁!…k_d!.

5.3.1 The Nearly Mixed Poisson Model

A further generalization of the mixed Poisson model, hence of the Poisson process, is obtained by the assumption that the counting process satisfies the following condition (cf. (5.2.11)): there exists a non‐negative random variable Λ such that as ,

A counting process that satisfies this condition will be called nearly mixed Poisson. We mention that this definition lacks a dynamic background, while the mixed Poisson process can be constructed using precise dependence rules between the successive variables {T_i, i ≥ 1}. Nevertheless, the nearly mixed Poisson process encompasses quite a number of other models.

The convergence in distribution is equivalent to pointwise convergence of the corresponding Laplace transforms. Hence for all θ ≥ 0 one has that

(5.3.19)

where is the Laplace transform of a distribution F_Λ that we keep calling the mixing distribution. By the continuity of the limit we see that (5.3.19) is equivalent to the condition that for all values of θ ≥ 0,

By the definition of the generating function

for all w ≥ 0. The main difference to the mixed Poisson model is that for the latter there is equality in the above relation while for the nearly mixed Poisson case, the generating function attains the quantity only in the limit. Clearly, conditioning on the random variable Λ we see that, for any ,

By the central limit theorem for the Poisson process one arrives at

where Φ is the standard normal distribution function. Also the infinitely divisible processes and renewal processes discussed below are examples of nearly mixed Poisson processes.

5.3.2 Infinitely Divisible Processes

The Poisson process directed by a Lévy subordinator discussed in Section 5.2.4 is itself a Lévy process (in fact, another Lévy subordinator), and as such has i.i.d. increments. The general class of counting processes with i.i.d. increments is of interest. For this to hold the distribution at each time point t has to be infinitely divisible (i.e., is the distribution of a sum of n i.i.d. random variables for any ), but from Sato [666, Cor. 27.5] it follows that the class of discrete infinitely divisible distributions coincides with the one of compound Poisson distributions with integer‐valued jump size distribution. The probability generating function of an infinitely divisible counting process hence can be expressed as

where is the probability generating function of a discrete random variable G on the strictly positive integers. In the special case where g(z) = z one gets back to the classical Poisson process. For other choices of g(z), the counting process should be called a (discrete) compound Poisson process. Note that instead of “stretching” the time axis as in the subordination approach one here specifies explicitly a discrete distribution for the number of additional claims at each Poisson instant. In fact, all processes that can be constructed this way correspond to a Lévy subordinator (e.g., see [666, Thm. 24.11]), but the explicit alternative construction in terms of the random variable G is quite intuitive.

The probabilities are given in the form

where is the k‐fold convolution of the distribution {g_n} with probability generating function g(z). Needless to say the explicit evaluation of the above probabilities is mostly impossible because of the complex form of the convolutions.

A general form for can be given by using Faa di Bruno’s formula

where the inside sum runs over all integers a_j ≥ 0 for which a₁ + a₂ + ⋯ + a_r = i and simultaneously a₁ + 2a₂ + ⋯ + ra_r = r. From the first two r values one obtains the expressions

Kupper [518] gives some nice applications of such a construction of infinitely divisible processes to model claim counts, where the Poisson process refers to the number of accidents and the number of casualties in accident i is given by the random variable G_i.

Any infinitely divisible process provides an example of a nearly mixed Poisson process as long as . Here so that the limiting mixing distribution is again degenerate but now at the point .

Here are a few more explicit cases that have appeared in the actuarial literature.

1. As a first particular case one finds a generalized Poisson–Pascal processintroduced by Kestemont and Paris [486], where g(z) = (1 − β(z − 1))^−α is the generating function of a negative binomial random variable (this is not to be confounded with the gamma‐subordinated Poisson process, where the overall number of claims up to time t is negative binomially distributed).
2. If g(z) = (e^θz − 1)(e^θ − 1)⁻¹ then G is a truncated Poisson and the corresponding claim process is of Neyman‐type A.
3. If g(z) = ((1 − θ)z)(1 − θz)⁻¹ with 0 < θ < 1, then G is a truncated geometric. The generating function is
   
   The probabilities can be evaluated explicitly in terms of Laguerre polynomials
   
   The latter counting process is called the Pólya–Aeppli process.

5.3.3 The Renewal Model

If the inter‐claim times (with T₀ = 0) are i.i.d. (non‐negative) random variables with cumulative distribution function (c.d.f.) F_W, then the corresponding claim number process N(t) is called a renewal process. This process was introduced in risk theory by Sparre Andersen [707], so that often this model is referred to as the Sparre Andersen model. Since

one has that

In general, successive convolutions of a distribution cannot be evaluated explicitly. The only simple exception is provided by the exponential case F_W(t) = 1 − e^−λt, for which N(t) is a Poisson process. The Poisson process is actually the only mixed Poisson process that is simultaneously a renewal process.

The generating function is also cumbersome, since

(5.3.20)

From (5.3.20), (5.1.4) and it follows that

where U(t) is the classical renewal function generated by W. The evaluation of the variance is already somewhat tedious but leads to

where σ² = Var(W).

Any renewal process generated by a distribution with a finite mean is a nearly mixed Poisson. This follows from the so‐called weak law of large numbers from renewal theory: if the mean μ of the interclaim time distribution is finite, then where Λ is degenerate at the point 1/μ.

The renewal model is an extension of the Poisson process that allows for an elegant mathematical treatment. We refer to Feller [345] and Rolski et al. [652] for detailed accounts. At the same time, its practical importance is limited, as the implicitly introduced particular memory pattern between claim times is typically hard to interpret or justify in the insurance context.

5.3.4 Markov Models

Another model for the claim number process can be provided by a pure (non‐homogeneous) birth process. Here the probabilities {p_n(t);n ≥ 0, t ≥ 0} satisfy a system of Kolmogorov difference‐differential equations, well‐known from the theory of continuous‐time Markov chains. Then

(5.3.21)

where we take λ₋₁(t) = 0. The transition probabilities can sometimes be explicitly evaluated.

1. The first such example is the Poisson process where λ_n(t) = λ.
2. The mixed Poisson process also has a representation of the above form, with
   
3. Another one is provided by the Yule process where λ_n(t) = nλ. It is then easy to show that for n ≥ 0,
   
   so that the generating function is given by
   

For details and more examples, see [492, Ch. 7].

5.5.1 Modelling Yearly Claim Counts

Statistical analysis of claim count data is typically performed by generalizing the normal theory using the exponential family of distributions, which comprises popular models for claim counts such as the Poisson, binomial and other models, and which can be extended to important cases such as the negative binomial distribution.

The exponential family of distributions is defined through the probability density functions f for which

for parametrization (θ, ϕ) with θ the parameter of interest and ϕ a nuisance parameter, and some functions b only depending on θ, and a and c only depending on ϕ.

Typical examples are

1. the Poisson distribution with
   
   for which , a(ϕ) = ϕ = 1, , and !
2. the binomial model with
   
   for which , a(ϕ) = ϕ = 1, , and
3. the negative binomial model as introduced in (5.2.12) with t = 1 satisfies
   
   so that , a(ϕ) = ϕ = 1, , and . Hence the negative binomial example only belongs to the exponential family for known α.

This family can be treated using classical likelihood theory stating that with

one has

from which we obtain that

since U = (Y − b(θ))/a(ϕ) and U^′ = −b^″(θ)/a(ϕ). Hence ϕ is a dispersion parameter, while b^′(θ) gives the expected value of the variable Y:

1. in the Poisson case we obtain the well‐known results
   
2. in the binomial setting
   
3. in the negative binomial case
   

These results confirm the underdispersion in case of the binomial model and the overdispersion in the negative binomial case as it was already found for all mixed Poisson models.

Now the expected value can be modelled as a function of a covariate (vector), either in a parametric or non‐parametric way. Here we consider the application with accident year or claim arrival year t as a covariate. While in practice the non‐parametric approach often gives a start, we first discuss here the parametric modelling approach since the non‐parametric solution uses the theory behind the parametric analysis.

• Parametric modelling happens through generalized linear models (GLM). A standard reference here is McCullagh and Nelder [566]. A GLM allows the transformation of the systematic part of a model without changing the distribution of the variable under consideration. The link function g connects the mean μ_i of the observed variables Y_i, i = 1, …, n, with the covariate. Assuming a simple linear predictor we then propose
  
  where g is a smooth monotonic function and t₁, …, t_n are the values of the covariate. For instance, when using the Poisson or negative binomial model for Y, is a popular link function. The linear predictors
  

  are used in maximizing the log‐likelihood based on the vector y = (y₁, …, y_n) of observations

  

  where Θ = (θ₁, …, θ_n), and θ_i = (g∘b^′)⁻¹(η_i) denotes the inverse function of the composition of g and b^′. The functions a, b and c may vary with i, for instance to allow different binomial parameters n = n_i for each observation of a binomial response or to allow for different α in the negative binomial model. Also, for practical work, it suffices to consider a_i(ϕ) = ϕ/ω_i where the weights ω_i are known constants. It will often be convenient to consider Var(Y) as a function of , and then we can define a function V such that V (μ) = b^″(θ)/ω so that Var(Y) = V (μ)ϕ.

  Likelihood maximization then proceeds by partial differentiation of ℓ with respect to each β parameter:

  

  Moreover

  

  where the last step follows from ∂μ/∂θ = b^″(θ). Hence

  

  and the equations to solve are

  

  These equations are in fact the solving equations of non‐linear least squares regression analysis with objective function , where μ_i depends non‐linearly on the β parameters. This set of equations is then solved iteratively. In many cases the nature of the response distribution is not known precisely, and it is only possible to specify the relationship between the variance and the mean of the responses, that is, the function V can be specified, but little more. Then, quasi‐likelihood using the log quasi‐likelihood

  

  leads to exactly the same equations as the full likelihood approach.

  As a goodness‐of‐fit criterion the scaled deviance

  

  is proposed, which constitutes twice the difference between the maximum achievable log‐likelihood obtained by setting , and that achieved by the model under investigation. Moreover it can be shown that for large enough sample size n

  (5.5.27)

  where dim(β) denotes the number of regression parameters β. In our example dim(β₀, β₁) = 2.

  The scaled deviance based on the fitted model can be written as

  (5.5.28)

  with denoting the estimates of the parameters θ_i based on the full model with n parameters with maximum achievable likelihood ℓ(y;y). Here is known as the deviance for the current model.

  Hence based on (5.5.27) and (5.5.28) we find an estimator for the parameter ϕ:

  (5.5.29)

  Another important goodness‐of‐fit measure for a given regression model is the explained deviance

  

  where D(no regression) denotes the deviance for the model with only an intercept η_i = β₀ (i = 1, …, n). This generalizes the adjustment coefficient in classical linear regression and expresses which percentage of the total deviance before regression has been explained by using a regression model, which is the complement of how much unexplained deviance remains after using the postulated regression model.

  Finally, when ϕ is unknown, nested GLMs with p₀ and p₁ (p₀ < p₁) parameters (and corresponding deviances denoted by D₀ and D₁) can be compared with the generalized likelihood ratio test, rejecting [H₀: model with p₀ parameters holds] for large values of

  

  with under H₀.

  • Poisson model: The deviance of the Poisson model is given by
    
    If a constant term is included in the model it can be shown that , and hence then
    

    which for large μ can be approximated by

    

    which is the so‐called Pearson χ² statistic.

    Overdispersion can be deduced from fitting a Poisson model with which the ϕ parameter using (5.5.29) is larger than 1.

  • Negative binomial model: here the log‐likelihood is given by
    
    from which a maximum likelihood estimator of α can be traced back through a non‐linear equation involving the digamma function. The deviance is then given by
    
• Non‐parametric modelling can be performed using generalized additive modelling (GAM) where the expected counts are modelled through a smooth non‐parametric function not depending on specific parameters. Here the linear predictor predicts through some smooth monotonic function of the expected value of the response, and the response may follow any exponential family distribution, or simply have a mean‐variance relationship permitting to use the quasi‐likelihood approach. The resulting model is then of the kind
  
  Given knot locations s₁ < s₂ < … < s_m, one defines a basis b₁(t) = 1, b₂(t) = t, b_i+2(t) (i = 1, …, m − 2) where b_i+2(t) denote cubic splines linked to the knots. Then, the function h at t_i is estimated through
  
  which yields a generalized linear model in the unknown parameters β₁, …, β_m. This then leads to minimizing the penalized objective function using a smoothing parameter λ which controls the weight to be given to the smoothing:
  
  with β = (β₁, …, β_m)^t and S a symmetric m × m matrix only depending on the knots. Such analysis allows to validate a parametric proposal for the shape of a parametric regression function of λ or μ. We refer to Wood [795] for more details.

5.5.2 Modelling the Claim Arrival Process

If one has more refined information on the times of claim occurrences, and hence the waiting times between these events, one can analyze the Poisson properties for these data based on the order statistics property and the independence and exponentiality of the waiting times W_i, i = 1, …, n. To verify the order statistics property one plots the time points T_j against j:

(5.5.30)

This plot should closely follow the line , where denotes the average waiting time, as an estimate of 1/λ. This line hence represents how the arrival times increase with the claim number if the intensity were constant. Linearity corresponds to the uniform distribution of the arrival times over the interval [0, T] where T represents the end of the inspected time interval. Deviation from this linear structure is then a first indication that the counting process is not a homogeneous Poisson process. A formal statistical test on uniformity can, for instance, be performed using the Kolmogorov–Smirnov test for uniformity comparing the empirical distribution function against the c.d.f. t/T of the uniform [0, T] distribution, rejecting uniformity for too large values of .

The exponentiality of the waiting times W_i can also be verified through the exponential QQ‐plot of these inter‐arrival times:

(5.5.31)

A formal test for exponentiality based on the exponential QQ‐plot is given by the Shapiro–Wilk test based on the correlation coefficient r_Q of this QQ‐plot and rejecting exponentiality for too low values of r_Q. Of course graphically one can also inspect for a constant derivative of mean excess plot based on the observed W values.

In order to see how the intensity λ varies over time one can use a moving average approach, plotting

(5.5.32)

against i (i = 1, …, n). Of course under a simple Poisson counting process no trends should be visible in such plots. If deviations from a homogeneous Poisson behavior are detected in the plots, one can start evaluating the inhomogenous Poisson behavior and try to estimate the mean value function through (5.5.32). Then after a time change using the inverse function μ⁻¹(t) one can validate the homogeneous Poisson behavior of the time‐changed process in order to inspect possible inhomogeneous Poisson behavior.

When large clusters of arrivals appear that cannot be predicted using an inhomogeneous Poisson behavior, a next option is to consider Cox process models, for instance directed by a Lévy subordinator. In this setting Zhang and Kou [806] proposed estimating using a kernel estimator based on the observed arrival times in the observed time window [0, T]

with bandwidth h.

For the estimation of a Cox process directed by a Lévy subordinator model such as the gamma or inverse Gaussian subordinators, we assume that the process N is observed up to a time horizon T > 0 on a discrete time grid 0 := t₀ < t₁ < … < t_m := T with m monitoring points. Moreover we assume an equidistant grid with step size h = T/m, that is, t_i = hi for i = 0, 1, …, m. For instance the grid size could be a day or 1 week. We denote the observations of N, or the observed frequencies, at times t_i as

while . Following the Lévy property of such process N the increments (i = 1, …, m) are i.i.d. observations of the infinitely divisible distribution N(h). Using (5.2.16) we find that the log likelihood of the Cox process with the intensity parameter λ of the underlying Poisson process Ñ and the parameter vector Θ of the Lévy subordinator (i.e., Θ = (α, β) in the gamma case, and Θ = (β, η) in the inverse Gaussian model) is given by

(5.5.33)

This results in optimizing

with respect to (λ, Θ).

Note that the intensity parameter λ can be selected upfront assuming time normalization as the sample mean of the process distribution

To explore the seasonality effect further, a seasonal ARIMA model is fitted to the observed intensities (e.g., see Shumway [699] or Box and Jenkins [158]). Given that many days did not have a claim at all, the data were divided into one‐week intervals. The average waiting time per week is then used as an estimate of 1/λ in that week.

A seasonal ARIMA process, or SARIMA(p, d, q)(P, D, Q)_m process X_t, is an ARIMA (p, d, q) process where the residuals ɛ_t are ARIMA(P, D, Q) with respect to lag m:

where L_mX_t = X_t−m, L = L₁, while ϕ_i, Φ_i are constants that determine the autocorrelation of the process with past values, and θ_i, Θ_i modulate how much past shocks have impacted on the current value. All ɛ_t are assumed to be i.i.d. normally distributed with mean 0 and constant variance σ². The seasonality of the data can be inspected using an auto‐correlation function (ACF), which is defined for any process with mean μ_t and variance as

In a seasonal process one expects significant auto‐correlation with the data from the previous corresponding season. The partial auto‐correlation function (PACF) is defined as the auto‐correlation conditional on all values of the process between s and t, which eliminates the correlation originating from the values of X between s and t. Using the function auto.arima in R, an ARIMA(2, 1, 1)(0, 0, 1)₅₂ model with mean 0 is selected given by

with . When looking for a seasonal ARIMA with lag 26 the resulting model ARIMA(2, 1, 1)(0, 0, 1)₂₆ is

(5.5.34)

with . In Figure 5.7 we contrast the observed intensities with a simulated process from model (5.5.34), and add the ACF and PACF plots as well as a normal QQ‐plot of the residuals of the fitted model. The observed data seem to show larger clustering. Process (5.5.34) also generates negative intensity values, which of course is not admissible. The normality of the residuals is also strongly rejected, with a correlation of 0.75 on the normal QQ‐plot of the residuals. Several power transformations were therefore used, leading to an ARIMA(2, 1, 1)(0, 0, 2)₂₆ model based on the log‐transformed intensities:

(5.5.35)

with . Now the correlation coefficient of the normal QQ‐plot of the residuals is 0.95, and the clusters are better approximated using (5.5.35), but still the clustering is underestimated.

Applying the likelihood method based on (5.5.33) using a daily observation grid over the entire observation period using a gamma and an inverse Gaussian model leads to the following parameters:

• gamma: , and
• inverse Gaussian: , and .

We simulated 100 sample paths from the fitted Cox models in order to construct confidence intervals at each time point. Both models do fit inside these bands, while the log‐likelihood values in the fitted parameters are − 3633.67 for the gamma model and − 3634.07 for the inverse Gaussian model. In order to evaluate the fit of these models, the Cox models directed by the gamma model were applied locally using a moving window of 30 days. The recorded mean values of the estimated local processes are also given in Figure 5.8, in which no stochastic structure was detected. The exponentiality of the waiting times between clusters also appears to be satisfied, and simulated processes based on the inverse Gaussian fit show similar cluster sizes, as illustrated in Figure 1.9.

Applying the likelihood method based on (5.5.33) to using a daily observation grid over the entire observation period using a gamma and an inverse Gaussian model leads to the following parameters:

• gamma: , and
• inverse Gaussian: , and .

Again we simulated 100 sample paths from the fitted Cox models in order to construct confidence intervals at each time point. Both models do fit inside these bands, while the log‐likelihood values in the fitted parameters are almost equal: − 2462.77 for the gamma model and − 2462.73 for the inverse Gaussian model. In order to evaluate the fit of these models, the Cox models directed by the gamma model were applied locally using a moving window of 30 days. The recorded mean values of the estimated local processes are also given in Figure 5.10, in which no stochastic structure was detected. Also the exponentiality of the waiting times between clusters appears to be satisfied, and simulated processes based on the inverse Gaussian fit (see Figure 5.10) show similar cluster sizes as in Figure 1.8.

For the estimation of a multivariate Cox process directed by a Lévy subordinator we again assume that the multivariate process N is observed up to a time horizon T > 0 on a discrete time grid 0 := t₀ < t₁ < … < t_m := T with m monitoring equidistant grid points with step size h = T/m, that is, t_i = hi for i = 0, 1, …, m. We denote the observations of N, or the observed frequencies, at times t_i as

while . Following the Lévy property of such a process N, the increments (j = 1, …, m) are i.i.d. observations of the infinitely divisible distribution N(h). Using (5.2.18) optimizing the log likelihood of the Cox process with the intensity parameter vector λ of the underlying Poisson process Ñ_λ and the parameter vector Θ of the Lévy subordinator (i.e., Θ = (α, β) in the gamma case, and Θ = (β, η) in the inverse Gaussian model) results in optimizing

(5.5.36)

with respect to λ, Θ.

Here, the use of time normalization means

	λ1	λ2	λ3
inverse Gaussian	180.9	152.6	56.0
gamma	180.9	152.6	56.0

5.6.1 Number of Claims under Excess‐loss Reinsurance

Under an XL contract, the first‐line insurer has to pay the part of each claim that is below the retention, so that N_D(t) = N(t). On the other hand, the reinsurer only has to pay for claims which touch the reinsured layer. Let be the probability that a claim exceeds the retention level u. Then

with

(5.6.37)

Here denotes the probability that there are exactly k claims occurring in the time interval from 0 to t.

An alternative expression can be obtained if we look at the generating function . It is a straightforward calculation to relate it to the generating function of the original claim number:

(5.6.38)

Written in this fashion, the variable N_R(t) can be considered as a thinned version of the original N(t).

From the above, one can quickly derive information on the moments of N_R(t). For any non‐negative integer k

In particular

For the dispersion one has

which indicates that the smaller s_u, the closer the dispersion gets to the value 1, which constitutes the Poisson case (this, by the way, is a neat way to intuitively see why the Poisson distribution is a natural model for rare events).

Let us illustrate the above procedure for a number of examples that have been introduced in the previous part.

1. Our first example refers to the mixed Poisson process. If we look at the reinsured part then clearly,
   
   This simple relation shows that we only need to replace the time‐variable t by s_ut (i.e., a “slowing down” of the original claim number process). All other ingredients of the original mixed Poisson process remain the same. For example, generally
   
   Let us illustrate the above with two special cases.
   • For the Pólya process we have
     
   • For the Sichel process we obtain the generating function
     
2. For an infinitely divisible process we again look at what happens with the thinned process. From (5.6.38) we get
   
   which can be rewritten in the form
   
   where
   
   and
   
   It takes a little extra calculation to see that where
   
   The latter formula can in itself be interpreted as a thinning of the original variable G. We therefore see that the thinned process remains infinitely divisible.
   • For the Neyman‐type A‐process, the thinned process is of the same type but needs as a new parameter.
   • Similarly for the Pólya–Aeppli process, the thinned process is of the same type with .
3. Next we deal with the Sparre Andersen model. In renewal theory thinning occurs by deleting each epoch of the renewal process with a fixed probability s_u, independently of the renewal process. The thinned process is a counting process that jumps at the time points and for n ≥ 1, . This thinned process is again a renewal process, now generated by the mixture
   
4. Finally we return to the (a,b,1) class of Section 5.4. From the equations (5.6.38) and (5.4.24), some elementary algebra leads to
   (5.6.39)
   where remains the same as before and where
   
   
   and
   
   Hence the three subcases in Section 5.4 remain invariant under the given type of thinning, one just needs to adapt the parameters. Also, the binomial and the Engen cases remain invariant. Correspondingly, the distributions of claim numbers for the insurer and for the reinsurer then only differ by the applied parameters (and for this parameter change the quantity s_u plays a crucuial role). This is a further reason for the popularity of these claim count models in insurance and reinsurance modelling. A convenient general class of probability generating functions that remain invariant under thinning (including many zero inflated models) is discussed in Klugman et al. [492, Sec. 8.1], where also strong mixed Poisson characterizations of the class are given.