Class Of Subexponential Distributions Research Articles

An Asymptotic Result on Catastrophe Insurance Losses

Consider an insurer who both sells catastrophe insurance policies and makes risky investments. Suppose that insurance claims arrive according to a Poisson process and the price of the investment portfolio evolves according to a general stochastic process independent of the insurance claims. In the focus of catastrophe risk management are catastrophe insurance losses. For the case of heavy-tailed claims, we derive a simple asymptotic formula for the tail probability of the present value of future claims. The transparent expression of our formula explicitly reflects the different roles of the various underlying risks in driving catastrophic losses. Our work is distinguished from most other works in this strand of research in that we carry out the asymptotic study over the whole class of subexponential distributions. Thus, our work allows both very heavy-tailed distributions such as Pareto-type distributions and moderately heavy-tailed distributions such as Lognormal and Weibull distributions.

Uniform Asymptotic Probability for Multi Renewal Risk Model with Strong Subexponential Tailed Claims

In this paper, we study the uniform asymptotic behavior for the ruin probability in a continuous time renewal counting process. For the proposed model, we assume that the financial claims for each extreme event are compensated by a finite number of independent insurance companies. Moreover, it is assumed that the claims of each insurance company is a sequence of random variables the tail distribution of which belongs to the class of subexponential distributions with finite mean. More specifically the objective of this work is the study of the uniform asymptotic behavior of ruin probability within the class of strongly subexponential distributions, a subclass of subexponential distributions, which provides a convenient framework for investigating heavy tailed distributions. The results are established for two types of asymptotic relations, namely for the common ruin probability of insurance companies and also for the ruin probability of at least one insurance company.

On a new generalization of Pareto distribution and its applications

Pareto distribution is one of the well known distributions used to fit heavy-tailed data. Various generalizations of this distribution have been reported in the literature by several authors. An obvious reason for generalizing a standard distribution is that the generalized form provides greater flexibility in modeling real data. In this article, we introduce a new four parameter distribution called New Generalized Pareto distribution, which is a generalization of the classical Pareto distribution. The structural properties of the new distribution are discussed. It is shown that the distribution is heavy-tailed and belongs to the class of subexponential distributions. Certain characterizations of the new distribution are obtained. We also derive the distribution of order statistics. The method of maximum likelihood is used for estimating the model parameters. We provide simulation results to assess the performance of the maximum likelihood estimates of the model parameters. Autoregressive time series model with the new generalized Pareto distribution as marginal is developed. An application of a real data set shows the performance of the new distribution over other generalizations of Pareto distribution.

A new class of large claim size distributions: Definition, properties, and ruin theory

We investigate a new natural class $\mathcal{J}$ of probability distributions modeling large claim sizes, motivated by the `principle of one big jump'. Though significantly more general than the (sub-)class of subexponential distributions $\mathcal{S}$, many important and desirable structural properties can still be derived. We establish relations to many other important large claim distribution classes (such as $\mathcal{D}$, $\mathcal{S}$, $\mathcal{L}$, $\mathcal {K}$, $\mathcal{OS}$ and $\mathcal{OL}$), discuss the stability of $\mathcal{J}$ under tail-equivalence, convolution, convolution roots, random sums and mixture, and then apply these results to derive a partial analogue of the famous Pakes-Veraverbeke-Embrechts theorem from ruin theory for $\mathcal{J}$. Finally, we discuss the (weak) tail-equivalence of infinitely-divisible distributions in $\mathcal{J}$ with their L\'{e}vy measure.

Asymptotic optimal investment under interest rate for a class of subexponential distributions

We consider a classical risk model with the possibility of investment and positive interest rate for the riskless bond. The stock price movement is modelled as a geometric Brownian motion, the claim sizes are assumed to have a distribution belonging to a certain subclass of subexponential distributions. In this setting, we study the asymptotic behaviour of the optimal investment strategy under the ruin probability as a risk measure. This problem has been already considered before, but no results were obtained, for instance, for Weibull and Benktander-type-II distributions with certain parameters. We introduce a method which closes this gap.

The Joint Density of the Surplus Before and After Ruin in the Sparre Andersen Model

Gerber and Shiu (1997) have studied the joint density of the time of ruin, the surplus immediately before ruin, and the deficit at ruin in the classical model of collective risk theory. More recently, their results have been generalised for risk models where the interarrival density for claims is nonexponential, but belongs to the Erlang family. Here we obtain generalisations of the Gerber-Shiu (1997) results that are valid in a general Sparre Andersen model, i.e. for any interclaim density. In particular, we obtain a generalisation of the key formula in that paper. Our results are made more concrete for the case where the distribution between claim arrivals is phase-type or the integrated tail distribution associated with the claim size distribution belongs to the class of subexponential distributions. Furthermore, we obtain conditions for finiteness of the joint moments of the surplus before ruin and the deficit at ruin in the Sparre Andersen model.

The Joint Density of the Surplus Before and After Ruin in the Sparre Andersen Model

Gerber and Shiu (1997) have studied the joint density of the time of ruin, the surplus immediately before ruin, and the deficit at ruin in the classical model of collective risk theory. More recently, their results have been generalised for risk models where the interarrival density for claims is nonexponential, but belongs to the Erlang family. Here we obtain generalisations of the Gerber-Shiu (1997) results that are valid in a general Sparre Andersen model, i.e. for any interclaim density. In particular, we obtain a generalisation of the key formula in that paper. Our results are made more concrete for the case where the distribution between claim arrivals is phase-type or the integrated tail distribution associated with the claim size distribution belongs to the class of subexponential distributions. Furthermore, we obtain conditions for finiteness of the joint moments of the surplus before ruin and the deficit at ruin in the Sparre Andersen model.

Preventive maintenance for inspected systems with additive subexponential shock magnitudes

AbstractWe examine the long‐run average availability and cost rate of a maintained system which deteriorates according to a random‐shock process. Shocks arrive according to a Poisson process. The system fails whenever the cumulative damage exceeds a given threshold. The system's failures are not self‐announcing, hence, failures must be detected via inspections. The system is inspected at periodic or exponentially distributed intervals. Systems are replaced by preventive maintenance or after failure (corrective maintenance), whichever occurs first. When the distribution function of the shock magnitudes belongs to the class of subexponential distributions, we obtain simple approximations for the availability and the cost rate. Copyright © 2007 John Wiley & Sons, Ltd.

On the Gerber-Shiu discounted penalty function for subexponential claims

We obtain the asymptotics of the Gerber-Shiu discounted penalty function in the classical Lundberg model. We cosider claims from a class of subexponential distributions and find the asymptotics as the initial surplus x tends to infinity. The main term of the discounted penalty function ?(x, d) has different expressions in the cases where the interest rate d > 0 and where d = 0.

Minimal ruin probabilities and investment under interest force for a class of subexponential distributions

We consider the infinite-time ruin probability of an insurance company investing in the stock market. This is done under the following assumptions: For the claim surplus process we use the classical Cramér-Lundberg model, where the claims have a distribution function, belonging to certain subclasses of the class of subexponential distributions. The stock price movement is modeled by geometric Brownian motion, and we allow positive interest rates for the riskless bond. In this setting we analyze the Hamilton-Jacobi-Bellman equation for the minimal ruin probability. We give asymptotic expressions for the minimal ruin probability, as well as for the optimal investment strategy. It turns out that the asymptotic order of the minimal ruin probability is different from the previous considered case of zero interest on the bond.

Estimates for the Distributions of the Sums of Subexponential Random Variables

Let \(\left\{ {S_n } \right\}_{n \geqslant 1} \) be a random walk with independent identically distributed increments \(\left\{ {\xi _i } \right\}_{i \geqslant 1} \). We study the ratios of the probabilities P(Sn>x) / P(ξ1 > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(Sτ > x) ∼ E τ P(ξ1 > x) as x → ∞. Here τ is a positive integer-valued random variable independent of \(\left\{ {\xi _i } \right\}_{i \geqslant 1} \). The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.

A Karamata-type theorem and ruin probabilities for an insurer investing proportionally in the stock market

We study the infinite time ruin probability in the classical Cramér–Lundberg model, where the company invests a constant fraction of its money in a stock, which is described by geometric Brownian motion. We prove that a certain integro-differential equation describes the ruin probability under weak regularity and integrability assumptions on the claim size distribution. Furthermore we show that, within the class of subexponential distributions, the claim size distributions with regularly varying tail distribution are the only ones, for which the ruin probability is proportional to the tail distribution.

Large Deviation Analysis of Subexponential Waiting Times in a Processor-Sharing Queue

We investigate the distribution of the waiting time V in a stable M/G/1 processor-sharing queue with traffic intensity ρ < 1. When the distribution of a customer service request B belongs to a large class of subexponential distributions with tails heavier than e−√x, it is shown that [Formula: see text] Furthermore, we demonstrate that the preceding relationship does not hold if the service distribution has a lighter tail than e−√x.

On a Property of Subexponential Distributions

Let F be a distribution function (d.f.) on [0,∞ ) with finite first moment m >0. We define the integrated tail distribution function F1 of F by F1(t)=m-1∩0t(1- F(u))du, t≥0. In this paper, we obtain sufficient conditions under which implications F∈S⇒F1∈S and F1∈S ⇒ F∈S hold, where S is the class of subexponential distributions.

On the Constant in the Definition of Subexponential Distributions

Previous article Next article On the Constant in the Definition of Subexponential DistributionsB. A. RogozinB. A. Rogozinhttps://doi.org/10.1137/S0040585X97977665PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe prove that the constant in the definition of subexponential distributions is equal to 2.[1] V. P. Chistyakov, A theorem on sums of independent positive random variables and its applications to branching random processes, Theory Probab. Appl., 9 (1964), pp. 640–648. tba TPRBAU 0040-585X Theor. Probab. Appl. LinkGoogle Scholar[2] J. Chover, , P. Ney and , and S. Wainger, Degeneracy properties of subcritical branching processes, Ann. Probab., 1 (1973), pp. 663–673. anb APBYAE 0091-1798 Ann. Probab. CrossrefGoogle Scholar[3] J. Chover, , P. Ney and , and S. Wainger, Functions of probability measures, J. Anal. Math., 26 (1973), pp. 255–302. 278 JOAMAV 0099-698X J. Anal. Math. 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CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Subexponential Densities of Infinitely Divisible Distributions on the Half-Line17 September 2020 | Lithuanian Mathematical Journal, Vol. 60, No. 4 Cross Ref Introduction24 April 2013 Cross Ref Heavy-Tailed and Long-Tailed Distributions24 April 2013 Cross Ref Subexponential Distributions24 April 2013 Cross Ref Densities and Local Probabilities24 April 2013 Cross Ref Maximum of Random Walk24 April 2013 Cross Ref Asymptotic estimates of Gerber-Shiu functions in the renewal risk model with exponential claims13 December 2011 | Acta Mathematicae Applicatae Sinica, English Series, Vol. 28, No. 1 Cross Ref Asymptotic aspects of the Gerber–Shiu function in the renewal risk model using Wiener–Hopf factorization and convolution equivalenceInsurance: Mathematics and Economics, Vol. 46, No. 1 Cross Ref Asymptotic Ruin Probabilities of the Lévy Insurance Model under Periodic Taxation9 August 2013 | ASTIN Bulletin, Vol. 39, No. 2 Cross Ref Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claimsInsurance: Mathematics and Economics, Vol. 43, No. 3 Cross Ref Convolution equivalence and distributions of random sums30 November 2007 | Probability Theory and Related Fields, Vol. 142, No. 3-4 Cross Ref On lower limits and equivalences for distribution tails of randomly stopped sumsBernoulli, Vol. 14, No. 2 Cross Ref Lower Limits for Distribution Tails of Randomly Stopped SumsD. E. Denisov, D. A. Korshunov, and S. G. Foss19 November 2008 | Theory of Probability & Its Applications, Vol. 52, No. 4AbstractPDF (157 KB)Convolution Equivalence and Infinite Divisibility: Corrections and Corollaries14 July 2016 | Journal of Applied Probability, Vol. 44, No. 2 Cross Ref The overshoot of a random walk with negative driftStatistics & Probability Letters, Vol. 77, No. 2 Cross Ref Lower limits and equivalences for convolution tailsThe Annals of Probability, Vol. 35, No. 1 Cross Ref On convolution equivalence with applicationsBernoulli, Vol. 12, No. 3 Cross Ref Overshoots and undershoots of Lévy processesThe Annals of Applied Probability, Vol. 16, No. 1 Cross Ref Maxima of Independent Sums in the Presence of Heavy TailsA. V. Lebedev25 July 2006 | Theory of Probability & Its Applications, Vol. 49, No. 4AbstractPDF (100 KB)Ruin probabilities and overshoots for general Lévy insurance risk processesThe Annals of Applied Probability, Vol. 14, No. 4 Cross Ref Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risksStochastic Processes and their Applications, Vol. 108, No. 2 Cross Ref Volume 44, Issue 2| 2000Theory of Probability & Its Applications History Published online:25 July 2006 InformationCopyright © 2000 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/S0040585X97977665Article page range:pp. 409-412ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Stationary M/G/1 excursions in the presence of heavy tails

It is shown that the stationary excursions above level x for the stationary M/G/1 queue with the service time distribution belonging to a certain class of subexponential distributions are asymptotically of two types as x →∞: either the excursion starts with a jump from a level which is O(1) and the initial excess over x converges to ∞, or it starts from a level of the form x – O(1) and the excess has a proper limit distribution. The two types occur with probabilities ρ, resp. 1 – ρ.

Stationary M/G/1 excursions in the presence of heavy tails

It is shown that the stationary excursions above level x for the stationary M/G/1 queue with the service time distribution belonging to a certain class of subexponential distributions are asymptotically of two types as x →∞: either the excursion starts with a jump from a level which is O(1) and the initial excess over x converges to ∞, or it starts from a level of the form x – O(1) and the excess has a proper limit distribution. The two types occur with probabilities ρ, resp. 1 – ρ.

Second order subexponential distributions

AbstractThe class of subexponential distributionsSis characterized byF(0) = 0, 1 −F(2)(x) ~ 2(1 −F(x)) asx→ ∞. In this paper we consider a subclass ofSfor which the relation 1 −F(2)(x) − 2(1 −F(x)) + (1 −F(x))2=o(a(x)) asx→ ∞ holds, where α is a positive function satisfying α(X) = 0(1 −F(x)) (x→ ∞).

The structure of the class of subexponential distributions

LetX 1,X 2, ...,X n be a sequence of positive, independent, identically distributed random variables with the same distribution function (d.f.)F and denote byX 1:n ≦X 2:n ≦...≦X n:n the order statistics of the sample. We characterize the class of d.f.F for which $$P(X_{1:n} + X_{2:n} + \ldots + X_{n - i:n} > x) \sim P(X_{n - i:n} > x) as x \to \infty $$ for fixedn andi (i≦n-1), and we show that it is independent ofn. This leads to the genesis of a new class of d.f.L i ; we show that the sequence (L i ) ∞ =0 is strictly decreasing and we illustrate how the classesL i determine the probabilistic structure of the classL of subexponential distributions.

The Class of Subexponential Distributions

The class $\mathscr{J}$ of subexponential distributions is characterized by $F(0) = 0, 1 - F^{(2)} (x) \sim 2\{1 - F(x)\}$ as $x \rightarrow \infty$. New properties of the class $\mathscr{J}$ are derived as well as for the more general case where $1 - F^{(2)} (x) \sim \beta\{1 - F(x)\}$. An application to transient renewal theory illustrates these results as does an adaptation of a result of Greenwood on randomly stopped sums of subexponentially distributed random variables.