Actuarial comparisons for aggregate claims with randomly right-truncated claims

In this note, we consider an extension of the largest claims reinsurance treaty (LCR) with random upper thresholds for the claim sizes, that we call retention levels. The Laplace transform order for insurer’s aggregate claims is obtained assuming dependence among the random retention levels. Different results about the influence of dependence on the insurer total claim amount are also given including the connections with LCR and the case of combination with quota-share. Algebraic bounds for the insurer aggregate claims are obtained when a common fixed threshold is considered.     

Distribution-free upper bounds on the premiums of the LCR and ECOMOR treaties

The problem of calculating a premium for the largest claims and ECOMOR reinsurance covers is investigated and simple, distribution-free upper premium-bounds are given. The results can be regarded as counterparts to Bowers' bound (see Bowers (1969)) for the premium of a stop-loss treaty.     

<Emphasis Type="Italic">l</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis>,<Emphasis Type="Italic">s</Emphasis></Superscript>-Singular values and spectral radius of partially symmetric rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties of l ^k,s -singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest l ^k,s -singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest l ^k,s -singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.     

Tail fitting for truncated and non-truncated Pareto-type distributions

In extreme value analysis, natural upper bounds can appear that truncate the probability tail. At other instances ultimately at the largest data, deviations from a Pareto tail behaviour become apparent. This matter is especially important when extrapolation outside the sample is required. Given that in practice one does not always know whether the distribution is truncated or not, we consider estimators for extreme quantiles both under truncated and non-truncated Pareto-type distributions. We make use of the estimator of the tail index for the truncated Pareto distribution first proposed in Aban et al. (J. Amer. Statist. Assoc. 101(473), 270–277, 2006). We also propose a truncated Pareto QQ-plot and a formal test for truncation in order to help deciding between a truncated and a non-truncated case. In this way we enlarge the possibilities of extreme value modelling using Pareto tails, offering an alternative scenario by adding a truncation point T that is large with respect to the available data. In the mathematical modelling we hence let T→∞ at different speeds compared to the limiting fraction (k/n→0) of data used in the extreme value estimation. This work is motivated using practical examples from different fields, simulation results, and some asymptotic results.     

Lattice-ordered fields

Mainly archimedean lattice-ordered fields (l-fields) are investigated in this paper. An archimedean l-field has a largest subfield (its o-subfield) which can be totally ordered in such a way that the l-field is a partially ordered vector space over this subfield. For archimedean l-fields which are algebraic over their o-subfields the following questions are investigated: What is the structure of the additive l-group of an l-field? Can the lattice order of an l-field be extended to a total order? Are the intermediate fields of an l-field and its o-subfield also l-fields with the induced partial order?     

ECOMOR and LCR reinsurance with gamma-like claims

Assuming that the claim sizes of an insurance company have a common distribution with gamma-like tail, we study the asymptotic tail behaviour of the reinsured amounts under the ECOMOR and LCR reinsurance treaties, respectively. Our novel results include a precise asymptotic expansion for the tail probability of the reinsured amounts under the ECOMOR treaty and tight asymptotic bounds for the LCR case. As a by-product we derive a precise asymptotic expansion for the tail of the product of independent regularly varying random variables.     

The Poisson random effect model for experience ratemaking: Limitations and alternative solutions

Poisson random effect models with a shared random effect have been widely used in actuarial science for analyzing the number of claims. In particular, the random effect is a key factor in a posteriori risk classification. However, the necessity of the random effect may not be properly assessed due to the dual role of the random effect; it affects both the marginal distribution of the number of claims and the dependence among the numbers of claims obtained from an individual over time. We first show that the score test for the nullity of the variance of the shared random effect can falsely indicate significant dependence among the numbers of claims even though they are independent. To mitigate this problem, we propose to separate the dual role of the random effect by introducing additional random effects to capture the overdispersion part, which are called saturated random effects. In order to circumvent heavy computational issues by the saturated random effects, we choose a gamma distribution for the saturated random effects because it gives the closed form of marginal distribution. In fact, this choice leads to the negative binomial random effect model that has been widely used for the analysis of frequency data. We show that safer conclusions about the a posteriori risk classification can be made based on the negative binomial mixed model under various situations. We also derive the score test as a sufficient condition for the existence of the a posteriori risk classification based on the proposed model.     

How retention levels influence the variability of the total risk under reinsurance

Recently, Escudero and Ortega (Insur. Math. Econ. 43:255–262, 2008) have considered an extension of the largest claims reinsurance with arbitrary random retention levels. They have analyzed the effect of some dependencies on the Laplace transform of the retained total claim amount. In this note, we study how dependencies influence the variability of the retained and the reinsured total claim amount, under excess-loss and stop-loss reinsurance policies, with stochastic retention levels. Stochastic directional convexity properties, variability orderings, and bounds for the retained and the reinsured total risk are given. Some examples on the calculation of bounds for stop-loss premiums (i.e., the expected value of the reinsured total risk under this treaty) and for net premiums for the cedent company under excess-loss, and complementary results on convex comparisons of discounted values of benefits for the insurer from a portfolio with risks having random policy limits (deductibles) are derived.      

The expected discounted penalty function under a risk model with stochastic income

Quantities of interest in ruin theory are investigated under the general framework of the expected discounted penalty function, assuming a risk model where both premiums and claims follow compound Poisson processes. Both a defective renewal equation and an integral equation satisfied by the expected discounted penalty function are established. Some implications that these equations have on particular quantities such as the discounted deficit and the probability of ultimate ruin are illustrated. Finally, the case when premiums have Erlang(n,β) distribution and the distribution of the claims is arbitrary is investigated in more depth. Throughout the paper specific examples where claims and premiums have particular distributions are provided.     

Random walks with non-convolution equivalent increments and their applications

This paper mainly presents some global and local asymptotic estimates for the tail probabilities of the supremum and overshoot of a random walk in “the intermediate case”, where the related distributions of the increments of the random walk may not belong to the convolution equivalent distribution class. Some of the obtained results can include the classical results. For this, the paper first introduces some new distribution classes using the γ-transform of distributions, and investigates their properties and relations with some other existing distribution classes. Based on the above results, some equivalent conditions for the global and local asymptotics of the γ-transform of the distribution of the supremum of the above random walk are given. Applying these results to risk theory and infinitely divisible laws, the paper obtains some asymptotic estimates for the ruin probability and the local ruin probability of the renewal risk model with non-convolution equivalent claims, and the global and local asymptotics of an infinitely divisible law with a non-convolution equivalent Lévy measure.     

On increasing subsequences of minimal Erdös-Szekeres permutations

Let π be a minimal Erdös-Szekeres permutation of 1, 2, ..., n ², and let l _n,k be the length of the longest increasing subsequence in the segment (π(1), ..., π(k)). Under uniform measure we establish an exponentially decaying bound of the upper tail probability for l _n,k , and as a consequence we obtain a complete convergence, which is an improvement of Romik’s recent result. We also give a precise lower exponential tail for l _n,k .     

On RSA moduli with almost half of the bits prescribed

We show that using character sum estimates due to H. Iwaniec leads to an improvement of recent results about the distribution and finding RSA moduli M=pl, where p and l are primes, with prescribed bit patterns. We are now able to specify about n bits instead of about n/2 bits as in the previous work. We also show that the same result of H. Iwaniec can be used to obtain an unconditional version of a combinatorial result of W. de Launey and D. Gordon that was originally derived under the Extended Riemann Hypothesis.     

Special-valued l-groups and Abelian covers

This paper presents a new and independent proof of the theorem (proven first by Kopytov and Gurchenkov [7] and again by Reilly [10]) that covers of the Abelian l-variety are either representable or are Scrimger covers. The proof in this paper is based upon the l-Cauchy constructions of Ball [1]; once these are applied to the problem, the proof becomes elementary.     

Optimal ordering policies with convertible lead times

A firm receives orders that will be required at an uncertain time given by an Erlang distribution, and over time observes the associated independent exponential events. The firm, in turn, places orders at a linear cost from a supplier with fixed lead time l and has the option of converting (expediting) each order, at a cost, over a certain time interval after the order is originally placed. A converted order arrives l_e < l units of time after it is converted. We show that a threshold policy is optimal. Under such a policy the firm places an order after a certain number of exponential events have been observed. An order is converted the first time, if any, when the residual lead time exceeds a time threshold related to the number of exponential events realized since the order was placed.     

关于溢额损失再保险的条件分布的递推方程

在索赔数目服从Poisson分布、二项分布或负二项分布,以及索赔额分布的密度函数连续且有界的条件下,研究了溢额损失再保险条款的总体损失分布的条件递推方程．在再保险人或分出人的索赔数目给定的条件下,得到了再保险人以及分出人的总赔付额分布的递推方程．     

Covering the alternating groups by products of cycle classes

Given integers k,l?2, where either l is odd or k is even, we denote by n=n(k,l) the largest integer such that each element of An is a product of k cycles of length l. For an odd l, k is the diameter of the undirected Cayley graph Cay(An,Cl), where Cl is the set of all l-cycles in An. We prove that if k?2 and l?9 is odd and divisible by 3, then . This extends earlier results by Bertram [E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory 12 (1972) 368-380] and Bertram and Herzog [E. Bertram, M. Herzog, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (2001) 87-99].     

The Maximum Queue Length for Heavy-Tailed Service Times in the M/G/1 FB Queue

This paper treats the maximum queue length M, in terms of the number of customers present, in a busy cycle in the M/G/1 queue. The distribution of M depends both on the service time distribution and on the service discipline. Assume that the service times have a logconvex density and the discipline is Foreground Background (FB). The FB service discipline gives service to the customer(s) that have received the least amount of service so far. It is shown that under these assumptions the tail of M is bounded by an exponential tail. This bound is used to calculate the time to overflow of a buffer, both in stable and unstable queues.     

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

The ratio of the largest eigenvalue divided by the trace of a p×p random Wishart matrix with n degrees of freedom and an identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p,n→∞ with p/n→c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p,n.     

Connector Families of Graphs

For every pair of fixed natural numbers k > l we consider families of subgraphs of the complete graph K _n such that each graph in the family has at least k connected components while the union of any two has at most l. We show that the cardinality of such a family is at most exponential in n and determine the exact exponential growth of the largest such families for every value of k and l = 1.