On the Appropriate Choice of the Risk Model

Recently the classical actuarial risk model for the evaluation of the ruin probability has been generalized to include dependency relations between the claim occurrences times and the claim amounts. For instance it has been incorporated into the stochastic model that the distributions of the times between consecutive claim occurrences times may depend on the last previous claim amount [1] or that claim sizes depend on the past of the point process of instants when claims are presented [2,3]. Results on ruin probabilities and related quantities have been published in several papers under such assumptions. This evolution implies that in a concrete application we have to choose between different versions of the actuarial risk model. This choice should be performed in a reasonable and objective way taking into account our empirical knowledge of the risk process. This leads us naturally to develop statistical tests able to distinguish certain classes of marked point processes. In order to assure the optimal use of the data it seems indicated to look for locally most powerful tests [4] and a clever choice of the period of observation of the risk process greatly facilitates the task and leads to simple procedures which are easy to implement.


Introduction
Recently the classical risk models for the evaluation of ruin probabilities have been generalized to include dependencies between claim occurrence times and claim sizes.Several authors have developed results on ruin probabilities and related relevant quantities under these less restrictive conditions.In view of these extensions the need arises to choose in concrete applications in a reasonable and objective way between the classical models and their generalizations.Since the stochastic part of the risk process can be described by a marked point process the choice among different ruin models amounts to perform a test discriminating different classes of marked point processes based on the observation of the investigated risk process in a given time interval.
To illustrate these ideas we present in the sequel statistical tests opposing the classical compound Poisson ruin model and one of four generalizations of it introduced in the scientific literature.The following notation is used in this article: We designate random variables by capital letters and their realized values by the corresponding lower case letters.Let T (i) be the occurrence time of the ith claim after the instant t = 0 and B(i) the associated claim size.We denote by U (i) := T (i) − T (i − 1)[i = 1, 2, . ..] (with the convention P (T (0) = 0) = 1) the ith gap between consecutive point events.Φ stands for the distribution function of the standard normal distribution, Exp(λ) for the exponential distribution with parameter λ[λ > 0] and ∼ means "distributed according to".R + is the set of positive real numbers, N the set of natural numbers and the function 1 A is given by 1 A (x) = 1 for x ∈ A and 1 A (x) = 0 for x ∈ A.

Should the risk model of Albrecher and Boxma be used?
In [1, Model 1 and 3] Albrecher and Boxma propose a ruin model with the property that if the claim B(i) is larger than the random threshold R(i) then the time until the next claim is independent exponentially distributed with parameter λ 1 , otherwise it is independent exponentially distributed with parameter λ 2 .The quantities R(i) are assumed to be independent identically distributed with distribution function F R and the claim sizes B(i) are supposed to be independent, identically distributed with density function f B , both distributions concentrated on R + and independent.We choose as observation period of the investigated risk process the interval (0, t n 0 ] from t = 0 to the occurrence time of the n 0 th claim [n 0 ∈ N ].Our observation of the associated marked point process is then specified by the data (t 1 , b 1 ; . . .; t n 0 , b n 0 ).We seek a locally most powerful test [4] of H 0 : g = 0 versus H 1 : g > 0, choosing λ 1 = λ + g λ 2 = λ [λ, g > 0] and assuming that the initial condition U (1) ∼ Exp(λ) is satisfied.The likelihood function is given by and for its derivative near the null hypothesis Thus we find as score statistic [4] Under H 0 the additive terms of T λ,f B ,F R ;n 0 (0) are independent, identically distributed.According to the standard central limit theorem we have The critical region of the test of size α and for large n 0 is formed by the large values of the standardized score statistic and specified by the condition For the special case of a deterministic threshold R * (i.e.
, where F B denotes the distribution function of B.

Should the risk model of Boudreault et alia be used?
In [ where f 1 and f 2 are known density functions on R + .We check whether the assumptions of this stochastic model are realistic for the observed data in the period (0, t n 0 ] by means of a locally most powerful test of the hypotheses H 0 : g = 0 and H 1 : g > 0. The likelihood function reads Derivation of the log likelihood function with respect to g at g = 0 yields the score statistic ].This statistic has under H 0 mean 0 and variance V ar[T λ,f 1 ,f 2 ;n 0 (0) : If this quantity is finite and different from 0, then according to the standard central limit theorem for large n 0 and H 0 is rejected for large values of this standardized test statistic.

Should the alternative risk model of Albrecher and Boxma be used?
In [1,Model 2] Albrecher and Boxma propose a risk model where the time until the next claim is exponentially distributed with either parameter λ or λ + g [g > 0].If a claim B(i) is smaller or equal than the random threshold R(i) the λ-value of the distribution of the next gap changes from one of these λ-values to the other, if this condition is not satisfied the λ-value stays unaltered.The quantities R(i) are assumed to be independent, identically distributed with distribution F R and the claim sizes are supposed to be independent identically distributed with density function f B , both distributions concentrated on R + and independent.Let t νs be the occurrence time of the sth "small claim" after the instant t = 0, where a "small claim" satisfies by definition the condition {b k ≤ r k }, while we have {b k > r k } for k ∈ N − {ν 1 , ν 2 , . ..}.We note that the increments of the claim aggregated process in the periods (0, t ν 2 ], (t ν 2 , t ν 4 ], . . .are independent identically distributed and form thus a renewal process.For a prefixed m ∈ N the choice of (0, t ν 2m ] as observation period is adopted, since it leads to a considerable simplification of the procedure.We put λ(1) = λ(3) = . . .= λ and λ(2) = λ(4) = . . .= λ + g for the successively utilised λ-values and check by means of the locally most powerful test of the hypotheses H 0 : g = 0 and H 1 : g > 0 whether the assumptions underlying model 2 are fulfilled for the observed risk process.The likelihood function is given by ), and the log likelihood function takes the form ln(L(λ, F R , f B , g : Therefore the test statistic is given by Since under H 0 the (B(i), R(i))'s are independent of the U (i+1)'s and E[U (i+ 1) : H 0 ] = 1/λ, we find E[T λ,F R ,f B ;ν 2m (0) : g = 0] = 0.For P (B ≤ R) =: p and given (ν 1 , . . ., ν 2m ) the probability of the realization of an observation with these ν-values is If we subdivise the typical observation in m cycles stretching from t ν 2(l−1) to t ν 2l for l = 1, . . ., m and putting t ν 0 = 0 the successive increments of these cycles form an embedded renewal process and the probability of the lth cycle is and thus the contribution of the cycle corresponding to the time interval (t ν 2(l−1) , t ν 2l ] towards the variance of The contribution of all the cycles starting at t ν 2(l−1) and ending at t ν 2l towards this variance is therefore The overall contribution of a cycle to this variance becomes

Should the risk model of Asmussen and Biard be used?
In this model [3] the point process of claim amount arrival instants is a homogeneous Poisson process with global rate λ.A threshold d is introduced for claim interarrival times, such that if the two immediately preceding consecutive both newly observed interarrival times are larger than d the claim size distribution has density f 2 in all other cases the density function is f 1 .At t = 0 or just after the appearence of a f 2 -distributed claim a cycle of an embedded renewal process starts and lasts until the end of a sequence of two consecutive new gaps larger than d.Let (t c 1 , t c 2 , . ..) be the ordered sequence of instances of appearence of a f 2 -distributed claim.Assume that our observation of the aggregated risk process consists of all t i and the associated claim sizes b i until the rth cycle of the embedded renewal process ends, i.e. covering the period (0, t cr ].The likelihood function is given by L

Conclusion
As we have seen the recent research achievements in assessing the actuarial risk lead in a natural way to new applications of the methodology of optimal statistical tests for marked point processes.