Kernel-type estimator of the reinsurance premium for heavy-tailed loss distributions

In this paper, we generalize the classical estimator of the reinsurance premium for heavy-tailed loss distributions with a kernel-type estimator. Since this estimator exhibits a bias, we propose its bias-reduced version by using a least-squares method. The asymptotic normality of the proposed estimators is established under suitable assumptions. A small simulation study is carried out to prove the performance of our approach.     

Bounds and inequalities for single server loss systems

We consider a single server loss system in which arrivals occur according to a doubly stochastic Poisson process with a stationary ergodic intensity functionλ _t . The service times are independent, exponentially distributed r.v.'s with meanμ ⁻¹, and are independent of arrivals. We obtain monotonicity results for loss probabilities under time scaling as well as under amplitude scaling ofλ _t . Moreover, using these results we obtain both lower and upper bounds for the loss probability.     

Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Numerical evaluation of performance measures in heavy-tailed risk models is an important and challenging problem. In this paper, we construct very accurate approximations of such performance measures that provide small absolute and relative errors. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution and with the aid of perturbation analysis we derive a series expansion for the performance measure under consideration. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a phase-type approximation of our measure. We refer to our approximations collectively as corrected phase-type approximations. We show that the corrected phase-type approximations exhibit a nice behavior both in finite and infinite time horizon, and we check their accuracy through numerical experiments.     

Asymptotic analysis of risk quantities conditional on ruin for multidimensional heavy-tailed random walks

In this paper we consider a multidimensional renewal risk model with regularly varying claims. This model may be used to describe the surplus of an insurance company possessing several lines of business where a large claim possibly puts multiple lines in a risky condition. Conditional on the occurrence of ruin, we develop asymptotic approximations for the average accumulated number of claims leading the process to a rare set, and the expected total amount of shortfalls to this set in finite and infinite horizons. Furthermore, for the continuous time case, asymptotic results regarding the total occupation time of the process in a rare set and time-integrated amount of shortfalls to a rare set are obtained.     

Bounds for the regret loss in dynamic programming under adaptive control

We consider a Markovian dynamic programming model in which the transition probabilities depend on an unknown parameterθ. We estimate the unknownθ and adapt the control action to the estimated value. Bounds are given for the expected regret loss under this adaptive procedure, i.e. for the loss caused by using the adaptive procedure instead of an (unknown) optimal one. We assume that the dependence of the model onθ is Lipschitz continuous. The bounds depend on the expected estimation error. When confidence intervals forθ with fixed width are available, we obtain bounds for the expected regret loss that hold uniformly inθ.     

Bayesian total loss estimation using shared random effects

The pricing of insurance policies requires estimates of the total loss. The traditional compound model imposes an independence assumption on the number of claims and their individual sizes. Bivariate models, which model both variables jointly, eliminate this assumption. A regression approach allows policy holder characteristics and product features to be included in the model. This article presents a bivariate model that uses joint random effects across both response variables to induce dependence effects. Bayesian posterior estimation is done using Markov Chain Monte Carlo (MCMC) methods. A real data example demonstrates that our proposed model exhibits better fitting and forecasting capabilities than existing models.     

A multicast problem with shared risk cost

In this paper, we study a minimum cost multicast problem on a network with shared risk link groups (SRLGs). Each SRLG contains a set of arcs with a common risk, and there is a cost associated with it. The objective of the problem is to find a multicast tree from the source to a set of destinations with minimum transmission cost and risk cost. We present a basic model for the multicast problem with shared risk cost (MCSR) based on the well-known multicommodity flow formulation for the Steiner tree problem (Goemans and Myung in Networks 1:19–28, 1993; Polzin and Daneshmand in Discrete Applied Mathematics 112(1–3): 241–261, 2001). We propose a set of strong valid inequalities to tighten the linear relaxation of the basic model. We also present a mathematical model for undirected MCSR. The computational results of real life test instances demonstrate that the new valid inequalities significantly improve the linear relaxation bounds of the basic model, and reduce the total computation time by half in average.     

A new extreme quantile estimator for heavy-tailed distributions

The classical estimation method for extreme quantiles of heavy-tailed distributions was presented by Weissman (J. Amer. Statist. Assoc. 73 (1978) 812–815) and makes use of the Hill estimator (Ann. Statist. 3 (1975) 1163–1174) for the positive extreme value index. This index estimator can be interpreted as an estimator of the slope in the Pareto quantile plot in case one considers regression lines passing through a fixed anchor point. In this Note we propose a new extreme quantile estimator based on an unconstrained least squares estimator of the index, introduced by Kratz and Resnick (Comm. Statist. Stochastic Models 12 (1996) 699–724) and Schultze and Steinebach (Statist. Decisions 14 (1996) 353–372) and we study its asymptotic behavior. To cite this article: A. Fils, A. Guillou, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory

This paper obtains the uniform estimate for maximum of sums of independent and heavy-tailed random variables with nonnegative random weights,which can be arbi- trarily dependent of each other.Then the applications to ruin probabilities in a discrete time risk model with dependent stochastic returns are considered.     

On the random max-closure for heavy-tailed random variables

In this paper, we study the random max-closure property for not necessarily identically distributed real-valued random variables X ₁ ,X ₂ , . . . , which states that, given distributions \( {F}_{X_1} \) , \( {F}_{X_2} \) , . . . from some class of heavy-tailed distributions, the distribution of the random maximum X( _η) := max{0,X ₁ , . . . , X _η } or random maximum S _(η) := max{0, S ₁ , . . . , S _η } belongs to the same class of heavy-tailed distributions. Here, S _n = X ₁ + · · · + X _n , n ≥ 1, and η is a counting random variable, independent of {X ₁ ,X ₂ , . . . }. We provide the conditions for the random max-closure property in the case of classes Open image in new window and Open image in new window .     

Bounds for multiplicities

Let and be a homogeneous -algebra. We establish bounds for the multiplicity of certain homogeneous -algebras in terms of the shifts in a free resolution of over . Huneke and we conjectured these bounds as they generalize the formula of Huneke and Miller for the algebras with pure resolution, the simplest case. We prove these conjectured bounds for various algebras including algebras with quasi-pure resolutions. Our proof for this case gives a new and simple proof of the Huneke-Miller formula. We also settle these conjectures for stable and square free strongly stable monomial ideals . As a consequence, we get a bound for the regularity of . Further, when is not Cohen-Macaulay, we show that the conjectured lower bound fails and prove the upper bound for almost Cohen-Macaulay algebras as well as algebras with a -linear resolution.

     

Large deviations for heavy-tailed random elements in convex cones

We prove large deviation results for sums of heavy-tailed random elements in rather general convex cones being semigroups equipped with a rescaling operation by positive real numbers. In difference to previous results for the cone of convex sets, our technique does not use the embedding of cones in linear spaces. Examples include the cone of convex sets with the Minkowski addition, positive half-line with maximum operation and the family of square integrable functions with arithmetic addition and argument rescaling.     

伪Smarandache函数的上下界

对于正整数n,设Z(n)=min{m|m∈N,1/2m(m+1)≡0(modn)},称为n的伪Smarandache函数.设r是正整数.根据广义Ramanujan-Nagell方程的结果,运用初等数论方法证明了下列结果:i)1/2(-1+(8n+1)≤Z(n)≤2n-1.ii)当r≠1,2,3或5时,Z(2~r+1)≥1/2(-1+(2~(r+3)·5+41)).iii)当r≠1,2,3,4或12时,Z(2~r-1)≥1/2(-1+(2~(r+3)·3-23).     

Localization and delocalization of eigenvectors for heavy-tailed random matrices

Consider an $n \times n$ Hermitian random matrix with, above the diagonal, independent entries with $\alpha $ -stable symmetric distribution and $0 < \alpha < 2$ . We establish new bounds on the rate of convergence of the empirical spectral distribution of this random matrix as $n$ goes to infinity. When $1 < \alpha < 2$ and $ p > 2$ , we give vanishing bounds on the $L^p$ -norm of the eigenvectors normalized to have unit $L^2$ -norm. On the contrary, when $0 < \alpha < 2/3$ , we prove that these eigenvectors are localized.