arXiv subjects

Enkelejd Hashorva

Publications and source records attributed to Enkelejd Hashorva.

At least 55 records · Page 3Linked to original sources

arXiv2016

Insurance Applications of Some New Dependence Models derived from Multivariate Collective Models

Consider two different portfolios which have claims triggered by the same events. Their corresponding collective model over a fixed time period is given in terms of individual claim sizes $(X_i,Y_i), i\ge 1$ and a claim counting random variable $N$. In this paper we are concerned with the joint distribution function $F$ of the \ece{largest claim sizes} $(X_{N:N}, Y_{N:N})$. By allowing $N$ to depend on some parameter, say $θ$, then $F=F(θ)$ is for various choices of $N$ a tractable parametric family of bivariate distribution functions. We present three applications of the implied parametric models to some data from the literature and a new data set from a Swiss insurance company. Furthermore, we investigate both distributional and asymptotic properties of $(X_{N:N}, Y_{N:N})$.

stat.AP↗

arXiv2016

Extremes of Gaussian Random Fields with regularly varying dependence structure

Let $X(t), t\in \mathcal{T}$ be a centered Gaussian random field with variance function $σ^2(\cdot)$ that attains its maximum at the unique point $t_0\in \mathcal{T}$, and let $M(\mathcal{T}):=\sup_{t\in \mathcal{T}} X(t)$. For $\mathcal{T}$ a compact subset of $\R$, the current literature explains the asymptotic tail behaviour of $M(\mathcal{T})$ under some regularity conditions including that $1- σ(t)$ has a polynomial decrease to 0 as $t \to t_0$. In this contribution we consider more general case that $1- σ(t)$ is regularly varying at $t_0$. We extend our analysis to random fields defined on some compact $\mathcal{T}\subset \R^2$, deriving the exact tail asymptotics of $M(\mathcal{T})$ for the class of Gaussian random fields with variance and correlation functions being regularly varying at $t_0$. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.

math.PR↗

arXiv2016

Extremes of a class of nonhomogeneous Gaussian random fields

This contribution establishes exact tail asymptotics of $\sup_{(s,t)\in\mathbf{E}}$ $X(s,t)$ for a large class of nonhomogeneous Gaussian random fields $X$ on a bounded convex set $\mathbf{E}\subset\mathbb{R}^2$, with variance function that attains its maximum on a segment on $\mathbf{E}$. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes.

math.PR↗

arXiv2016

Generalized Pickands constants and stationary max-stable processes

Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker-Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Lévy processes. We furthermore provide a link to spatial extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.

math.PR↗

arXiv2016

On bivariate lifetime modelling in life insurance applications

Insurance and annuity products covering several lives require the modelling of the joint distribution of future lifetimes. In the interest of simplifying calculations, it is common in practice to assume that the future lifetimes among a group of people are independent. However, extensive research over the past decades suggests otherwise. In this paper, a copula approach is used to model the dependence between lifetimes within a married couple \eH{using data from a large Canadian insurance company}. As a novelty, the age difference and the \eH{gender} of the elder partner are introduced as an argument of the dependence parameter. \green{Maximum likelihood techniques are} thus implemented for the parameter estimation. Not only do the results make clear that the correlation decreases with age difference, but also the dependence between the lifetimes is higher when husband is older than wife. A goodness-of-fit procedure is applied in order to assess the validity of the model. Finally, considering several products available on the life insurance market, the paper concludes with practical illustrations.

q-fin.RM↗

arXiv2015

Extremes of vector-valued Gaussian processes: exact asymptotics

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of $$ P\left(\exists_{t \in [0,T]} \forall_{i=1 ... n} X_i(t)> u \right) $$ as $u\to\infty$, for both locally stationary $X_i$'s and $X_i$'s with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants, that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell-TIS inequality, the Slepian lemma and the Pickands-Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.

math.PR↗

arXiv2015

On Parisian ruin over a finite-time horizon

For a risk process $R_u(t)=u+ct-X(t), t\ge 0$, where $u\ge 0$ is the initial capital, $c>0$ is the premium rate and $X(t),t\ge 0$ is an aggregate claim process, we investigate the probability of the Parisian ruin \[ \mathcal{P}_S(u,T_u)=\mathbb{P}\{\inf_{t\in[0,S]} \sup_{s\in[t,t+T_u]} R_u(s)<0\}, \] with a given positive constant $S$ and a positive measurable function $T_u$. We derive asymptotic expansion of $\mathcal{P}_S(u,T_u)$, as $u\to\infty$, for the aggregate claim process $X$ modeled by Gaussian processes. As a by-product, we derive the exact tail asymptotics of the infimum of a standard Brownian motion with drift over a finite-time interval.

math.PR↗

arXiv2015

Asymptotic Expansion of Gaussian Chaos via Probabilistic Approach

For a centered $d$-dimensional Gaussian random vector $ξ=(ξ_1,\ldots,ξ_d)$ and a homogeneous function $h:R^d\to R$ we derive asymptotic expansions for the tail of the Gaussian chaos $h(ξ)$ given the function $h$ is sufficiently smooth. Three challenging instances of the Gaussian chaos are the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble and the diameter of random Gaussian clouds. Using a direct probabilistic asymptotic method, we investigate both the asymptotic behaviour of the tail distribution of $h(ξ)$ and its density at infinity and then discuss possible extensions for some general $ξ$ with polar representation.

math.PR↗

arXiv2014

Asymptotics of the convex hull of spherical samples

In this paper we consider the convex hull of a spherically symmetric sample in $R^d$. Our main contributions are some new asymptotic results for the expectation of the number of vertices, number of facets, area and the volume of the convex hull assuming that the marginal distributions are in the Gumbel max-domain of attraction. Further, we briefly discuss two other models assuming that the marginal distributions are regularly varying or $O$-regularly varying.

math.PR↗

arXiv2014

Extremes of Aggregated Dirichlet Risks

The class of Dirichlet random vectors is central in numerous probabilistic and statistical applications. The main result of this paper derives the exact tail asymptotics of the aggregated risk of powers of Dirichlet random vectors when the radial component has df in the Gumbel or the Weibull max-domain of attraction. We present further results for the joint asymptotic independence and the max-sum equivalence.

math.PR↗

arXiv2014

Extremes of Order Statistics of Stationary Processes

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a stationary process $\{X(t), t\ge0\}$. For given positive constants $u,T$, define the set of $r$th conjunctions $ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$ with $X_{r:n}(t)$ the $r$th largest order statistics of $X_1(t), \ldots , X_n(t), t\ge 0$. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions $C_{r,T,u}$ is not empty. Imposing the Albin's conditions on $X$, in this paper we obtain an exact asymptotic expansion of this probability as $u$ tends to infinity. Further, we establish the tail asymptotics of the supremum of a generalized skew-Gaussian process and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes. As a by-product we derive a version of Li and Shao's normal comparison lemma for the minimum and the maximum of Gaussian random vectors.

math.PR↗

arXiv2014

Tail Behaviour of Weighted Sums of Order Statistics of Dependent Risks

Let $X_{1},\ldots ,X_{n}$ be $n$ real-valued dependent random variables. With motivation from Mitra and Resnick (2009), we derive the tail asymptotic expansion for the weighted sum of order statistics $X_{1:n}\leq \cdots \leq X_{n:n}$ of $X_{1},\ldots ,X_{n}$ under the general case in which the distribution function of $X_{n:n}$ is long-tailed or rapidly varying and $% X_{1},\ldots ,X_{n}$ may not be comparable in terms of their tail probability. We also present two examples and an application of our results in risk theory.

math.PR↗

arXiv2014

Boundary Non-Crossings of Additive Wiener Fields

Let $W_i=\{W_i(t), t\in \mathbb{R}_+\}, i=1,2$ be two Wiener processes and $W_3=\{W_3(\mathbf{t}), \mathbf{t}\in \mathbb{R}_+^2\}$ be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary non-crossing probability $$P_f=P\{W_1(t_1)+W_2(t_2)+W_3(\mathbf{t})+h(\mathbf{t})\leq u(\mathbf{t}), \mathbf{t}\in\mathbb{R}_+^2\},$$ where $h, u: \mathbb{R}_+^2\rightarrow \mathbb{R}_+$ are two measurable functions. We show further that for large trend functions $γf>0$ asymptotically when $γ\to \infty$ we have that $\ln P_{γf}$ is the same as $\ln P_{γ\underline{f}}$ where $\underline{f}$ is the projection of $f$ on some closed convex set of the reproducing kernel Hilbert Space of $W$. It turns out that our approach is applicable also for the additive Brownian pillow.

math.PR↗

arXiv2014

Random Scaling of Gumbel Risks

In this paper we consider the product of two positive independent risks $Y_1$ and $Y_2$. If $Y_1$ is bounded and $Y_2$ has distribution in the Gumbel max-domain of attraction with some auxiliary function which is regularly varying at infinity, then we show that $Y_1Y_2$ has also distribution in the Gumbel max-domain of attraction. Additionally, if both $Y_1,Y_2$ have log-Weibullian or Weibullian tail behavior, we show that $Y_1Y_2$ has log-Weibullian or Weibullian asymptotic tail behavior, respectively. We present two applications of our results.

math.PR↗

arXiv2014

Tail asymptotics of randomly weighted large risks

In this paper we are concerned with a sample of asymptotically independent risks. Tail asymptotic probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions, where the individual tail behaviour has a crucial role. An application is provided for Log-Normal risks.

math.PR↗

arXiv2014

Parisian Ruin of Self-similar Gaussian Risk Processes

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.

math.PR↗

arXiv2014

Tail approximation for reinsurance portfolios of Gaussian-like risks

We consider two different portfolios of proportional reinsurance of the same pool of risks. This contribution is concerned with Gaussian-like risks, which means that for large values the survival function of such risks is, up to a multiplier, the same as that of a standard Gaussian risk. We establish the tail asymptotic behavior of the total loss of each of the reinsurance portfolios and determine also the relation between randomly scaled Gaussian-like portfolios and unscaled ones. Further we show that jointly two portfolios of Gaussian-like risks exhibit asymptotic independence and their weak tail dependence coefficient is non-negative.

math.PR↗

arXiv2014

Berman's inequality under random scaling

Berman's inequality is the key for establishing asymptotic properties of maxima of Gaussian random sequences and supremum of Gaussian random fields. This contribution shows that, asymptotically an extended version of Berman's inequality can be established for randomly scaled Gaussian random vectors. Two applications presented in this paper demonstrate the use of Berman's inequality under random scaling.

math.PR↗