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This paper studies the ruin probabilities in a homogeneous continuous compound Poisson risk model which is adapted for the perturbed insurance risk model with standard Brownian motion. In such a model, we construct a martingale in terms of a differentiable exponential function based on the discounted and perturbed surplus process. We obtain the exponential upper bounds for the ruin probabilities using Martingale approach and provide a sharper exponential upper bound for the infinite time ruin probability. Moreover, we derive two asymptotic approximation formulas for the finite time ruin probability when claim size distribution belongs to some heavy-tailed families. Finally, several numerical examples are presented to show the effect of constant force of interest on the ruin probabilities and that our results are excellent and reliable. [ABSTRACT FROM AUTHOR]

The discrete Pareto (DP) distribution studied in this paper is a probability model with a power-law tail, which provides a convenient alternative to the well-known Zipf distribution. While basic characteristics of the DP model are available explicitly, this is not an exponential family and parameter estimation connected with this model is a challenging task. With this in mind we develop a computational approach to this problem, based on the expectation-maximization (EM) algorithm. In the process, we discover an interesting new probability distribution, which is a certain tilted version of the standard gamma model, and we provide a short account of its basic properties. The latter play a crucial role in our EM algorithm. Our computational approach to DP parameter estimation is illustrated by simulations, while a real data example from finance illustrates potential applications of the DP stochastic model. [ABSTRACT FROM AUTHOR]

An α -slash distribution built upon a random variable X is a heavy tailed distribution corresponding to Y = X / U 1 / α , where U is standard uniform random variable, independent of X. We point out and explore a connection between α -slash distributions, which are gaining popularity in statistical practice, and generalized convolutions, which come up in the probability theory as generalizations of the standard concept of the convolution of probability measures and allow for the operation between the measures to be random itself. The stochastic interpretation of Kendall convolution discussed in this work brings this theoretical concept closer to statistical practice, and leads to new results for α -slash distributions connected with extremes. In particular, we show that the maximum of independent random variables with α -slash distributions is also a random variable with an α -slash distribution. Our theoretical results are illustrated by several examples involving standard and novel probability distributions and extremes. [ABSTRACT FROM AUTHOR]

In this follow up article of Arnold and Ghosh, we re-visit and study in details one of the two special cases of multivariate hidden truncation -(a) single variable truncation from above in a trivariate scenario where the component random variables follow an appropriate Pareto (type II) distribution. The case with single variable truncation was not discussed in details earlier in Arnold and Ghosh, in particular the estimation under both the classical and the Bayesian paradigm. It is well known that the Pareto distribution is a simple model for nonnegative data with a power law probability tail. Accumulating experience rapidly pointed out the fact that it is only in the upper tail of the income distributions that Pareto-like behavior can be expected. In practice, it has been observed that in income modeling and in other medical studies, quite often the observed data has been truncated with respect to several other unobserved co-variable(s) attaining certain critical levels. In this paper, we consider both the classical and the Bayesian inference (under the non-informative priors set up) of the resulting hidden truncated model. For illustrative purposes, a real data set on abalone is considered to show the flexibility of a single variable truncated Pareto (type II) distribution in a trivariate setting. Finally, the large sample behavior of the smallest order statistic in the case of a single co-variable truncation from above is provided in the appendix. [ABSTRACT FROM AUTHOR]

The convergence rate in the famous Rényi theorem is studied by means of the Stein method refinement. Namely, it is demonstrated that the new estimate of the convergence rate of the normalized geometric sums to exponential law involving the ideal probability metric of the second order is sharp. Some recent results concerning the convergence rates in Kolmogorov and Kantorovich metrics are extended as well. In contrast to many previous works, there are no assumptions that the summands of geometric sums are positive and have the same distribution. For the first time, an analogue of the Rényi theorem is established for the model of exchangeable random variables. Also within this model, a sharp estimate of convergence rate to a specified mixture of distributions is provided. The convergence rate of the appropriately normalized random sums of random summands to the generalized gamma distribution is estimated. Here, the number of summands follows the generalized negative binomial law. The sharp estimates of the proximity of random sums of random summands distributions to the limit law are established for independent summands and for the model of exchangeable ones. The inverse to the equilibrium transformation of the probability measures is introduced, and in this way a new approximation of the Pareto distributions by exponential laws is proposed. The integral probability metrics and the techniques of integration with respect to sign measures are essentially employed. [ABSTRACT FROM AUTHOR]

Asymptotic Ruin Probabilities in the Cramér-Lundberg model has been widely studied when the claims have light-tailed or heavy-tailed distributions. However, it has not been studied for extreme values over some threshold. In this paper, we investigate the use of the peaks over threshold method, to construct a new estimator of the ruin probability for a risk process with heavy tails claims amounts with infinite variance for the stationary arrival claims in an infinite time. Our approach is based on approximating the sample over some threshold by the generalized Pareto distribution. We prove that the proposed estimator is consistent and asymptotically normal. The performance of our new estimator is illustrated by some results of simulations for some loss models and provides an extensive example application to Danish data on large fire insurance. [ABSTRACT FROM AUTHOR]

The singular probability distributions that exist are insufficient to model naturally occurring events. Including the exponential distribution and the one-parameter Pareto distribution as a model for modelling real data. Therefore, there was a need to generalize and expand the probability distributions to develop the effectiveness of the distributions by adding one or more new parameters. Therefore, in this paper, we will introduce (Topp Leone G-Pareto distribution (TLG-P)) as a new model with two parameters (β,θ) based on the proposed family (Topp Leone G family). Therefore, the sum of the mathematical properties such as the central and eccentric moments and the moment-generating function. We will used maximum likelihood method. Moreover, the practical application of real data based on the proposed method. We can be employed for showing this distribution better compared to other distributions. [ABSTRACT FROM AUTHOR]

The development and application of the Envelope Peak-over-Threshold (EPOT) method is summarized for the roll motion of a ship in irregular waves. The method is based on the conventional Peak-over-Threshold (POT) method, but uses an envelope of the peaks for declustering the data. As suggested by a reduced-order model, a power-law tail should be applicable for the peaks of roll motions, which allows the introduction of physical considerations into the statistical model. A data-driven Generalized Pareto distribution (GPD) is applied to study the behavior of the shape parameter on large-volume samples in order to confirm the theoretical prediction of the tail structure. The physics-informed statistical model approximates data above a suitable threshold with a Pareto distribution, while the threshold is determined with a prediction error criterion. An example is included, where extrapolations with GPD are compared to the physics-informed model. • Application of Peak-over-Threshold method is considered where a time history envelope is used for declustering the data. • The study of behavior of distribution tails of roll motion peaks. • A physics-informed model with power-law, Pareto tail was formulated and applied to amplitudes of roll motions. [ABSTRACT FROM AUTHOR]