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This paper investigates the randomly stopped sums, minima and maxima of heavy- and light-tailed random variables. The conditions on the primary random variables, which are independent but generally not identically distributed, and counting random variable are given in order that the randomly stopped sum, random minimum and maximum is heavy/light tailed. The results generalize some existing ones in the literature. The examples illustrating the results are provided. [ABSTRACT FROM AUTHOR]

In this paper, we find conditions under which distribution functions of randomly stopped minimum, maximum, minimum of sums and maximum of sums belong to the class of generalized subexponential distributions. The results presented in this article complement the closure properties of randomly stopped sums considered in the authors' previous work. In this work, as in the previous one, the primary random variables are supposed to be independent and real-valued, but not necessarily identically distributed. The counting random variable describing the stopping moment of random structures is supposed to be nonnegative, integer-valued and not degenerate at zero. In addition, it is supposed that counting random variable and the sequence of the primary random variables are independent. At the end of the paper, it is demonstrated how randomly stopped structures can be applied to the construction of new generalized subexponential distributions. [ABSTRACT FROM AUTHOR]

In this paper, the asmptotics is considered for the distribution tail of a randomly stopped sum Sν = X1 + ··· + Xν of independent identically distributed consistently varying random variables with zero mean, where ν is a counting random variable independent of {X1,X2, . . .}. The conditions are provided for the relation P(Sν > x) ∼ Eν P(X1 > x) to hold, as x →∞, involving the finiteness of E|X1|. The result improves that of Olvera-Cravioto [14], where the finiteness of a moment E|X1|r for some r > 1 was assumed. [ABSTRACT FROM AUTHOR]

Let { ξ 1 , ξ 2 , ... } be a sequence of independent possibly differently distributed random variables, defined on a probability space (Ω , F , P) with distribution functions { F ξ 1 , F ξ 2 , ... } . Let η be a counting random variable independent of sequence { ξ 1 , ξ 2 , ... } . In this paper, we find conditions under which the distribution function of randomly stopped sum S η = ξ 1 + ξ 2 + ... + ξ η belongs to the class of generalized subexponential distributions. [ABSTRACT FROM AUTHOR]

A new formula for the ultimate ruin probability in the Cramér-Lundberg risk process is provided when the claims are assumed to follow a finite mixture of m Erlang distributions. Using the theory of recurrence sequences, the method proposed here shifts the problem of finding the ruin probability to the study of an associated characteristic polynomial and its roots. The found formula is given by a finite sum of terms, one for each root of the polynomial, and allows for yet another approximation of the ruin probability. No constraints are assumed on the multiplicity of the roots and that is illustrated via a couple of numerical examples. [ABSTRACT FROM AUTHOR]

Suppose that ξ + is the positive part of a random variable defined on the probability space (Ω , F , P) with the distribution function F ξ . When the moment E ξ + p of order p > 0 is finite, then the truncated moment F ¯ ξ , p (x) = min 1 , E ξ p 1 I { ξ > x } , defined for all x ⩾ 0 , is the survival function or, in other words, the distribution tail of the distribution function F ξ , p . In this paper, we examine which regularity properties transfer from the distribution function F ξ to the distribution function F ξ , p and which properties transfer from the function F ξ , p to the function F ξ . The construction of the distribution function F ξ , p describes the truncated moment transformation of the initial distribution function F ξ . Our results show that the subclasses of heavy-tailed distributions, such as regularly varying, dominatedly varying, consistently varying and long-tailed distribution classes, are closed under this truncated moment transformation. We also show that exponential-like-tailed and generalized long-tailed distribution classes, which contain both heavy- and light-tailed distributions, are also closed under the truncated moment transformation. On the other hand, we demonstrate that regularly varying and exponential-like-tailed distribution classes also admit inverse transformation closures, i.e., from the condition that F ξ , p belongs to one of these classes, it follows that F ξ also belongs to the corresponding class. In general, the obtained results complement the known closure properties of distribution regularity classes. [ABSTRACT FROM AUTHOR]

In this paper we consider the product Π n = ∏ k = 1 n ξ k of n independent normally distributed zero mean random variables ξ 1 , ... , ξ n . We derive an asymptotic formula for the survival probability P (Π n > x) , as x → ∞ , with the first remaining term. [ABSTRACT FROM AUTHOR]

Assume that ξ and η are two independent random variables with distribution functions F ξ and F η , respectively. The distribution of a random variable ξ η , denoted by F ξ ⊗ F η , is called the product-convolution of F ξ and F η . It is proved that F ξ ⊗ F η is a generalized subexponential distribution if F ξ belongs to the class of generalized subexponential distributions and η is nonnegative and not degenerated at zero. [ABSTRACT FROM AUTHOR]

In actuarial statistics, distributions with heavy tails are of great interest to actuaries, as they represent a better description of risk exposure through a type of indicator with a certain probability. These risk indicators are used to determine companies' exposure to a particular risk. In this paper, we present a distribution with heavy right tail, studying its properties and the behaviour of the tail. We estimate the parameters using the maximum likelihood method and evaluate the performance of these estimators using Monte Carlo. We analyse one set of simulated data and another set of real data, showing that the distribution studied can be used to model income data. [ABSTRACT FROM AUTHOR]

Probability of ultimate survival is one of the central problems in insurance because it is a management tool that may be used to check on the solvency levels of the insurer. In this article, we numerically compute this probability for an insurer whose portfolio is compounded by investments arising from a risky asset. The uncertainty in the celebrated Cramér-Lundberg model is provided by a standard Brownian motion that is independent of the standard Brownian motion in the model for the risky asset. We apply an order four block-by-block method in conjunction with the Simpson rule to solve the resulting Volterra integral equation of the second kind. The ultimate survival probability is arrived at by taking a linear combination of some two solutions to the Volterra equations. The several numerical examples show that the results are accurate and reliable even when the net profit condition is violated. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the Gerber–Shiu discounted penalty function for a constant interest rate in delayed claim reporting times. Using the Poisson claim arrival scenario, we derive the differential equation of the Laplace transform of the generalized Gerber–Shiu function and show that the differential equation can be transformed to a Volterra equation of the second kind with a degenerated kernel. In the case of an exponential claim distribution, a closed-expression for the Gerber–Shiu function is obtained via sequence expansion. This result allows us to calculate the absolute (relative) ruin probability. Additionally, we discuss a method of solving the Volterra equation numerically and provide an illustration of the ruin's probability to support the finding. [ABSTRACT FROM AUTHOR]

Catastrophe risks lead to severe problems of insurance and reinsurance industry. In order to reduce the underwriting risk, the insurer would seek protection by transferring part of its risk exposure to the reinsurer. A framework for valuing multirisk catastrophe reinsurance under stochastic interest rates driven by the CIR model shall be discussed. To evaluate the distribution and the dependence of catastrophe variables, the Peaks over Threshold model and Copula function are used to measure them, respectively. Furthermore, the parameters of the valuing model are estimated and calibrated by using the Global Flood Date provided by Dartmouth College from 2000 to 2016. Finally, the value of catastrophe reinsurance is derived and a sensitivity analysis of how stochastic interest rates and catastrophe dependence affect the values is performed via Monte Carlo simulations. The results obtained show that the catastrophe reinsurance value is the inverse relation between initial value of interest rate and average interest rate in the long run. Additionally, a high level of dependence between catastrophe variables increases the catastrophe reinsurance value. The findings of this paper may be interesting to (re)insurance companies and other financial institutions that want to transfer catastrophic risks. [ABSTRACT FROM AUTHOR]

Consider a general bivariate Lévy-driven risk model. The surplus process Y, starting with Y0 = x > 0, evolves according to dYt = Yt-dRt - dPt for t > 0, where P and R are two independent Lévy processes respectively representing a loss process in a world without economic factors and a process describing the return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x → ∞, which confirms Paulsen's conjecture. [ABSTRACT FROM AUTHOR]

Consider a discrete-time insurance risk model. Within period i, the net insurance loss is denoted by a real-valued random variable Xi. The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factor Yi from time i to time i -- 1. Assume that (Xi, Yi), i ∈ ℕ, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functions F and G. When F is subexponential and G fulfills some constraints in order for the product convolution of F and G to be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in which F belongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent. [ABSTRACT FROM AUTHOR]

In this paper we consider the probabilities of finite- and infinite-time absolute ruins in the renewal risk model with constant premium rate and constant force of interest. In the particular case of the compound Poisson model, explicit asymptotic expressions for the finite- and infinite-time absolute ruin probabilities are given. For the general renewal risk model, we present an asymptotic expression for the infinite-time absolute ruin probability. Conditional distributions of Poisson processes and probabilistic techniques regarding randomly weighted sums are employed in the course of this study. [ABSTRACT FROM AUTHOR]

The generalized Pareto distribution (GPD) is widely used to model extreme values, for example, exceedences over thresholds, in modeling floods. Existing methods for estimating parameters have theoretical or computational defects. An efficient new estimator is proposed, which is computationally easy, free from the problems observed in traditional approaches, and performs well compared with existing estimators. A numerical example involving heights of waves is used to illustrate the various methods and tests of fit are performed to compare them. [ABSTRACT FROM AUTHOR]

New systems have emerged that deliver hundreds of megabits per second of video to the home. This, coupled with deeper penetration of in-home video networking, will create demand for digital video recorder (DVR) performance to scale up dramatically as well. This paper outlines a simple analytic framework that can be used to estimate the performance of any DVR system, using either hard-disk drives or solid-state disk drives, in terms of both megabits per second and number of video streams. The framework also highlights the extent to which the maximum performance is constrained by the disk, by the host hardware, or by the host software. [ABSTRACT FROM AUTHOR]

In this paper, a modified exponentiated family of distributions is introduced. The new model was built from a continuous parent cumulative distribution function and depends on a shape parameter. Its most relevant characteristics have been obtained: the probability density function, quantile function, moments, stochastic ordering, Poisson mixture with our proposal as the mixing distribution, order statistics, tail behavior and estimates of parameters. We highlight the particular model based on the classical exponential distribution, which is an alternative to the exponentiated exponential, gamma and Weibull. A simulation study and a real application are presented. It is shown that the proposed family of distributions is of interest to applied areas, such as economics, reliability and finances. [ABSTRACT FROM AUTHOR]

In this paper we consider a compound Poisson risk model where the insurer earns credit interest at a constant rate if the surplus is positive and pays out debit interest at another constant rate if the surplus is negative. Absolute ruin occurs at the moment when the surplus first drops below a critical value (a negative constant). We study the asymptotic properties of the absolute ruin probability of this model. First we investigate the asymptotic behavior of the absolute ruin probability when the claim size distribution is light tailed. Then we study the case where the common distribution of claim sizes are heavy tailed. [ABSTRACT FROM AUTHOR]

The asymptotic behaviour of the tail expectation E (S n ξ) α { S n ξ > x } is investigated, where exponent α is a nonnegative real number and S n ξ = ξ 1 + ... + ξ n is a sum of dominatedly varying and not necessarily identically distributed random summands, following a specific dependence structure. It turns out that the tail expectation of such a sum can be asymptotically bounded from above and below by the sums of expectations E ξ i α { ξ i > x } with correcting constants. The obtained results are extended to the case of randomly weighted sums, where collections of random weights and primary random variables are independent. For illustration of the results obtained, some particular examples are given, where dependence between random variables is modelled in copulas framework. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of the ruin probability in the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. We obtain an exponential bound for the ruin probability and investigate conditions, under which it holds for a number of distributions of the premium and claim sizes. Next, we use the exponential bound to construct non-exponential bounds for the ruin probability. We show that the non-exponential bounds turn out to be tighter than the exponential one in some cases. Moreover, we derive explicit formulas for the ruin probability when the premium and claim sizes have either the hyperexponential or the Erlang distributions and apply them to investigate how tight the bounds are. To illustrate and analyze the results obtained, we give numerical examples. [ABSTRACT FROM AUTHOR]

Singular spectrum analysis (SSA) is a non-parametric method that breaks down a time series into a set of components that can be interpreted and grouped as trend, periodicity, and noise, emphasizing the separability of the underlying components and separate periodicities that occur at different time scales. The original time series can be recovered by summing all components. However, only the components associated to the signal should be considered for the reconstruction of the noise-free time series and to conduct forecasts. When the time series data has the presence of outliers, SSA and other classic parametric and non-parametric methods might result in misleading conclusions and robust methodologies should be used. In this paper we consider the use of two robust SSA algorithms for model fit and one for model forecasting. The classic SSA model, the robust SSA alternatives, and the autoregressive integrated moving average (ARIMA) model are compared in terms of computational time and accuracy for model fit and model forecast, using a simulation example and time series data from the quotas and returns of six mutual investment funds. When outliers are present in the data, the simulation study shows that the robust SSA algorithms outperform the classical ARIMA and SSA models. [ABSTRACT FROM AUTHOR]

We consider a dependent compound renewal risk model, where the interarrival times of accidents and the claim numbers follow a dependence structure characterized by a conditional tail probability and the claim sizes have a pairwise negatively quadrant dependence structure or a related dependence structure with the upper tail asymptotical dependence structure. When the distributions of the claim sizes belong to the dominated variation distribution class, we give the asymptotic lower and upper bounds for the precise large deviations of the aggregate claims. [ABSTRACT FROM AUTHOR]