Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, 1015 Lausanne, Switzerland
Swiss Finance Institute, Lausanne 1015, Switzerland
Université de Lyon, Université Lyon 1, Institut de Science Financière et d'Assurances,Lyon 69007, France
Department of Mathematics, University of Leuven, Leuven 3001, Belgium
Author to whom correspondence should be addressed.
Received: 25 February 2014; in revised form: 15 June 2014 / Accepted: 15 July 2014 / Published: 31 July 2014
Assume that claims in a portfolio of insurance contracts are described by independent and identically distributed random variables with regularly varying tails and occur according to a near mixed Poisson process. We provide a collection of results pertaining to the joint asymptotic Laplace transforms of the normalised sums of the smallest and largest claims, when the length of the considered time interval tends to infinity. The results crucially depend on the value of the tail index of the claim distribution, as well as on the number of largest claims under consideration.
When dealing with heavy-tailed insurance claims, it is a classical problem to consider and quantify the influence of the largest among the claims on their total sum, see e.g., Ammeter  for an early reference in the actuarial literature. This topic is particularly relevant in non-proportional reinsurance applications when a significant proportion of the sum of claims is consumed by a small number of claims. The influence of the maximum of a sample on the sum has in particular attracted considerable attention over the last fifty years (see Ladoucette and Teugels  for a recent overview of existing literature on the subject). Different modes of convergence of the ratios sum over maximum or maximum over sum have been linked with conditions on additive domain of attractions of a stable law (see e.g., Darling , Bobrov , Chow and Teugels , O’Brien  and Bingham and Teugels ).
It is also of interest to study the joint distribution of normalised smallest and largest claims. We will address this question in this paper under the assumption that the number of claims over time is described by a rather general counting process. These considerations may be helpful for the design of possible reinsurance strategies and risk management in general. In particular, extending the understanding of the influence of the largest claim on the aggregate sum (which is a classical topic in the theory of subexponential distributions) towards the relative influence of several large claims together, can help to assess the potential gain from reinsurance treaties of large claim reinsurance and ECOMOR type (see e.g., Section 1).
In this paper we consider a homogeneous insurance portfolio, where the distribution of the individual claims has a regularly varying tail. The number of claims is generated by a near mixed Poisson process. For this rather general situation we derive a number of limiting results for the joint Laplace transforms of the smallest and largest claims, as the time t tends to infinity. These turn out to be quite explicit and crucially depend on the rule of what is considered to be a large claim as well as on the value of the tail index.
Let be a sequence of independent positive random variables (representing claims) with common distribution function F. For , denote by the corresponding order statistics. We assume that the claim size distribution satisfies the condition
where and ℓ is a slowly varying function at infinity. The tail index is defined as and we will typically express our results in terms of γ. Denote by the tail quantile function of F where . Under (1), , where is again a slowly varying function. For textbook treatments of regularly varying distributions and/or their applications in insurance modelling, see e.g., Bingham et al. , Embrechts et al. , Rolski et al.  and Asmussen and Albrecher .
Denote the number of claims up to time t by with . The probability generating function of is given by
which is defined for . Let
be its derivative of order r with respect to z. In this paper we assume that is a near mixed Poisson (NMP) process, i.e., the claim counting process satisfies the condition
for some random variable Θ, where D denotes convergence in distribution. This condition implies that
Note also that, for and ,
A simple example of an NMP process is the homogeneous Poisson process, which is very popular in claims modelling and plays a crucial role in both actuarial literature and in practice. The class of mixed Poisson processes (for which condition (2) holds not only in the limit, but for any t) has found numerous applications in (re)insurance modelling because of its flexibility, its success in actuarial data fitting and its property of being more dispersed than the Poisson process (see Grandell  for a general account and various characterisations of mixed Poisson processes). The mixing may, e.g., be interpreted as claims coming from a heterogeneity of groups of policyholders or of contract specifications. The more general class of NMP processes is used here mainly because the results hold under this more general setting as well. The NMP distributions (for fixed t) contain the class of infinitely divisible distributions (if the component distribution has finite mean). Moreover any renewal process generated by an interclaim distribution with finite mean ν is an NMP process (note that then by the weak law of large numbers in renewal theory, where Θ is degenerate at the point ).
The aggregate claim up to time t is given by
where it is assumed that is independent of the claims . For and , we define the sum of the smallest and the sum of the s largest claims by
so that . Here Σ refers to small while Λ refers to large.
In this paper we study the limiting behaviour of the triplet with appropriate normalisation coefficients depending on γ, the tail index, and on s, the number of terms in the sum of the largest claims. We will consider three asymptotic cases, as they show a different behaviour: s is fixed, s tends to infinity but slower than the expected number of claims, and s tends to infinity and is asymptotically equal to a proportion of the number of claims.
Example 1.In large claim reinsurance for the s largest claims in a specified time interval , the reinsured amount is given by , so the interpretation of our results to the analysis of such reinsurance treaties is immediate. A variant of large claim reinsurance that also has a flavour of excess-of-loss treaties is the so-called ECOMOR (excedent du cout moyen relatif) treaty with reinsured amount
i.e., the deductible is itself an order statistic (or in other words, the reinsurer pays all exceedances over the s-largest claim). This treaty has some interesting properties, for instance with respect to inflation, see e.g., Ladoucette & Teugels . If the reinsurer accepts to only cover a proportion β () of this amount, the cedent’s claim amount is given by
which is a weighted sum of the quantities , and . The asymptotic results above in terms of the Laplace transform can then be used to approximate the distribution of the cedent’s and reinsurer’s claim amount in such a contract, and correspondingly the design of a suitable contract (including the choice of s) will depend quite substantially on the value of the extreme value index of the underlying claim distribution.
The paper is organised as follows. We first give the joint Laplace transform of the triplet for a fixed t in Section 2. Section 3 deals with asymptotic joint Laplace transforms in the case . We also discuss consequences for moments of ratios of the limiting quantities. The behaviour for depends on whether is finite or not. In the first case, the analysis for applies, whereas in the latter one has to adapt the analysis of Section 3 exploiting the slowly varying function , but we refrain from treating this very special case in detail (see e.g., Albrecher and Teugels  for a similar adaptation in another context). Section 4 and Section 5 treat the case without and with centering, respectively. The proofs of the results in Section 3, Section 4 and Section 5 are given in Section 6. Section 7 concludes.
In this section, we state a versatile formula that will allow us later to derive almost all the desired asymptotic properties of the joint distributions of the triplet . We consider the joint Laplace transform of to study their joint distribution in an easy fashion. For a fixed t, it is denoted by
Then the following representation holds:
Proposition 1.We have
Proof. The proof is standard if we interpret whenever . Indeed, condition on the number of claims at the time epoch t and subdivide the requested expression into three parts:
The conditional expectation in the first term on the right simplifies easily to the form . For the conditional expectations in the second and third term, we condition additionally on the value y of the order statistic ; the order statistics are then distributed independently and identically on the interval yielding the factor . A similar argument works for the s order statistics . Combining the two terms yields
A straightforward calculation finally shows
Consequently, it is possible to easily derive the expectations of products (or ratios) of , , and by differentiating (or integrating) the joint Laplace transform. We only write down their first moment for simplicity.
Corollary 2.We have
Proof. The individual Laplace transforms can be written in the following form:
where . By taking the first derivative, we arrive at the respective expectations. ☐
3. Asymptotics for the Joint Laplace Transforms when
Before giving the asymptotic joint Laplace transform of the sum of the smallest and the sum of the largest claims, we first recall an important result about convergence in distribution of order statistics and derive a characterisation of their asymptotic distribution. All proofs of this section are deferred to Section 6.
It is well-known that there exists a sequence of exponential random variables with unit mean such that
where . Let . It may be shown that converges in distribution to in , where (see Lemma 1 in LePage et al. ). For , the series converges almost surely. Therefore, for a fixed s, we deduce that, as ,
In particular, we derive by the Continuous Mapping Theorem that
Note that the first moment of (but only the first moment) may be easily derived since
where has a Beta distribution. We also recall that F belongs to the (additive) domain of attraction of a stable law with index if and only if
(see e.g., Theorem 1 in Ladoucette and Teugels ).
When is an NMP process, we also have, as ,
(see e.g., Lemma 2.5.6 in Embrechts et al. ). However, note that if the triplet is normalised by instead of in (6), then the asymptotic distribution will differ due to the randomness brought in by the counting process .
The following proposition gives the asymptotic Laplace transform when the triplet is normalised by .
Proposition 3.For a fixed s , as , we havewhere
Ifa.s., this expression simplifies to
We observe that modifies the asymptotic Laplace transform by introducing into the integral (7). However, the moments of do not depend on the law of Θ:
Corollary 4.For , we have
Note that this corollary only provides the moments of . In order to have moment convergence results for the ratios, it is necessary to assume uniform integrability of . It is also possible to use the Laplace transform of the triplet with a fixed t to characterise the moments of the ratios (see Corollary 2), and then to follow the same approach as proposed by Ladoucette  for the ratio of the random sum of squares to the square of the random sum under the condition that and for some .
Remark 1.For , (8) reduces again towhich is (5). Furthermore, for all
Remark 2.is the ratio of the sumover . By taking the derivative of (7), it may be shown that, forand ,
Therefore the mean ofwill only be finite for sufficiently small γ. An alternative interpretation is that for given value of γ, the number s of removed maximal terms in the sum has to be sufficiently large to make the mean of the remaining sum finite. The normalisation of the sum by , on the other hand, ensures the existence of the moments of the ratiofor all values of s and .
Remark 3.It is interesting to compare Formula (8) with the limiting moment of the statistic
For instance, , and the limit of the nth moment can be expressed as an nth-order polynomial in , see Albrecher and Teugels , Ladoucette  and Albrecher et al. . Motivated by this similarity, let us study the link in some more detail. By using once again Lemma 1 in LePage et al. , we deduce that
Recall thatis the weak limit of the ratioand . Using Equation (10) and(which is a straightforward consequence of the fact thathas regularly varying tail with tail index), one then obtains a simple formula for the covariance betweenand :
Determiningby exploiting Equation (8) for , we then arrive at the linear correlation coefficient
Figure 1 depictsas a function of . Note that . The correlation coefficient allows to quantify the negative linear dependence between the two ratios (the dependence becomes weaker when α increases, as the maximum term will then typically be less dominant in the sum).
as a function of α.
as a function of α.
Next, let us consider the case when the number of largest terms also increases as , but slower than the expected number of claims. It is now necessary to change the normalisation coefficients of and .
Proposition 5.Letfor a functionwithand . Thenwhere
Several messages may be derived from (11). First note that the asymptotic distribution of is degenerated for , since as . Second, the asymptotic distribution of the sum of the smallest claims is the distribution of Θ up to a scaling factor, since as .
Finally, for a fixed proportion of maximum terms, it is also necessary to change the normalisation coefficients of and . We have
Proposition 6.Letfor a fixed . Thenwhere
and . Ifa.s.,
As expected, and as . If a.s. and , then has an inverse Gamma distribution with shape parameter equal to .
4. Asymptotics for the Joint Laplace Transforms when
In this section, we assume that and hence the expectation of the claim distribution is finite. We let . The normalisation coefficient of the sum of the smallest claims, , will therefore be as it is the case for for the Law of Large Numbers. In Section 5, we will then consider the sum of the smallest centered claims with another normalisation coefficient.
Again, consider a fixed s first. The normalisation coefficients of and are the same as for the case , but the normalisation coefficient of is now .
Proposition 7.For fixed s , we havewhere
Corollary 8.We have
We first note that
and therefore as for any fixed s . The influence of the largest claims on the sum becomes less and less important as t is large and is asymptotically negligible. This is very different from the case . In Theorem 1 in Downey and Wright , it is moreover shown that, as ,
This result is no more true in our framework when Θ is not degenerate at 1. Assume that . Using (3) and under a uniform integrability condition, one has
Next, we consider the case with varying number of maximum terms. The normalisation coefficients of and now differ.
Proposition 9.Letand , i.e., . Thenwhere
As for the case , as . Moreover the asymptotic distribution of the sum of the largest claim is the distribution of Θ up to a scaling factor since as . Finally note that as as for the case when s was fixed.
Finally we fix p. Only the normalisation coefficient of and its asymptotic distribution differ from the case .
Proposition 10.Letand . Thenwhere
and . Ifa.s.,
We note that the normalisation of is the same as for and that as .
5. Asymptotics for the Joint Laplace Transform for with centered claims when s is fixed
In this section, we consider the sum of the smallest centered claims
instead of the sum of the smallest claims . Like for the Central Limit Theorem, we have to consider two sub-cases: and .
For the sub-case , the normalisation coefficient of is now .
Proposition 11.For fixed sand , we have
If , then
and we see that and are not independent.
Corollary 12.We have
This result is to compare with the one obtained by Bingham and Teugels  for (see also Ladoucette and Teugels ).
For the sub-case , let . The normalisation coefficient of becomes .
Proposition 13.For sfixed and , we havewhere
If and a.s., we note that the maximum, , and the centred sum, , are independent. If and a.s., is independent of .
Proof of Proposition 3. In formula (7), we first use the substitution , i.e., :
Next, the substitution , i.e., leads to
as and also
Note that the integral is well defined since . Moreover and
Proof of Corollary 4. From Proposition 3 we have
This gives indeed, using (3),
which extends (5) to the case of NMP processes. Next, we focus on (8) for general k. We first consider the case . We have
By Proposition 3
By the Faa di Bruno’s formula
with definition (9). This gives
cf. (3), and the result follows.
For the case , we proceed in an analogous way. Equation (12) becomes
Then is replaced by , by , by and, by following the same path as for , we get
Proof of Proposition 5. The proof is similar to the previous one, so we just highlight the differences here: Conditioning on we have
We first replace F by the substitution , i.e., :
The factor involving v converges to
The factor containing w behaves as
and hence for the power
Finally, for the factor containing u, replace F by the substitution , i.e., :
so that, as ,
For the factor with the factorials, we have by Stirling’s formula
Equivalent for the integral in z:
By Laplace’s method, we deduce that
Proof of Proposition 6. Again, we condition on :
For the factor containing w, we have
For the factor involving u one can write
The ratio with factorials behaves, by Stirling’s formula, as
For the integral in y we have the equivalence
By Laplace’s method, we deduce that
Proof of Proposition 7. We first replace F by the substitution , i.e.,
Then we have
Now use analogous arguments as in the proof of Proposition 3 and note that
Proof of Corollary 8. By Proposition 7
By Proposition 7
Proof of Proposition 9. Condition on to see that
Now replace F by the substitution , i.e.,
For the factor with w, one sees that
For the factor with u, replace F by the substitution , i.e., :
The ratio of factorials coincides with (13). Also (14) applies here. Altogether
Proof of Proposition 10. Given
The part involving w coincides with the one in the proof of Proposition 6. For the factor involving u, we have
The rest of the proof is completely analogous to the one for Proposition 6. ☐
Proof of Proposition 11. Use the substitution , i.e., :
We then replace F by the substitution , i.e., :
Now use the same arguments as in the proof of Proposition ☐
Proof of Corollary 12. Note that
Proof of Proposition 13. Use the substitution , i.e., :
Then we have
First note that
and it follows that
which completes the proof. Note that
In this paper we provided a fairly general collection of results on the joint asymptotic Laplace transforms of the normalised sums of smallest and largest among regularly varying claims, when the length of the considered time interval tends to infinity. This extends several classical results in the field. The appropriate scaling of the different quantities is essential. We showed to what extent the type of the near mixed Poisson process counting the number of claim instances influences the limit results, and also identified quantities for which this influence is asymptotically negligible. We further related the dominance of the maximum term in such a random sum to another quantity that exhibits the effect of the tail index on the aggregate claim rather explicitly, namely the ratio of sum of squares of the claims over the sum of the claims squared. The results allow to further quantify the effect of large claims on the total claim amount in an insurance portfolio, and could hence be helpful in the design of appropriate reinsurance programs when facing heavy-tailed claims with regularly varying tail. Particular emphasis is given to the case when the tail index exceeds 1, which corresponds to infinite-mean claims, a situation that is particularly relevant for catastrophe modelling.
H.A. acknowledges support from the Swiss National Science Foundation Project 200021-124635/1.
Three authors contributed to all aspects of this work.
Conflicts of Interest
The authors declare no conflict of interest.
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