Exactly Computing the Tail of the Poisson-Binomial Distribution

We present ShiftConvolvePoibin, a fast exact method to compute the tail of a Poisson-binomial distribution (PBD). Our method employs an exponential shift to retain its accuracy when computing a tail probability, and in practice we find that it is immune to the significant relative errors that other methods, exact or approximate, can suffer from when computing very small tail probabilities of the PBD. The accompanying R package is also competitive with the fastest implementations for computing the entire PBD.

Computing minimal signature of coherent systems through matrix-geometric distributions

Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.

NEW EXPONENTIAL PROBABILITY INEQUALITY AND COMPLETE CONVERGENCE FOR F-LNQD RANDOM VARIABLES SEQUENCE WITH APPLICATION TO AR(1)MODEL GENERATED BY F-LNQD ERRORS

The exponential probability inequalities have been important tools in probability and statistics. In this paper, we prove a new tail probability inequality for the distri-butions of sums of conditionally linearly negative quadrant dependent (F-LNQD , in short) random variables, and obtain a result dealing with conditionally complete con-vergence of first-order autoregressive processes with identically distributed (F-LNQD) innovations.

Asymptotic Tail Probability of the Discounted Aggregate Claims under Homogeneous, Non-Homogeneous and Mixed Poisson Risk Model

In this paper, we derive a closed-form expression of the tail probability of the aggregate discounted claims under homogeneous, non-homogeneous and mixed Poisson risk models with constant force of interest by using a general dependence structure between the inter-occurrence time and the claim sizes. This dependence structure is relevant since it is well known that under catastrophic or extreme events the inter-occurrence time and the claim severities are dependent.

Exact Computation of Maximum Rank Correlation Estimator

In this paper we provide a computation algorithm to get a global solution for the maximum rank correlation estimator using the mixed integer programming (MIP) approach. We construct a new constrained optimization problem by transforming all indicator functions into binary parameters to be estimated and show that it is equivalent to the original problem. We also consider an application of the best subset rank prediction and show that the original optimization problem can be reformulated as MIP. We derive the non-asymptotic bound for the tail probability of the predictive performance measure. We investigate the performance of the MIP algorithm by an empirical example and Monte Carlo simulations.

Computing Sensitivities for Distortion Risk Measures

Distortion risk measure, defined by an integral of a distorted tail probability, has been widely used in behavioral economics and risk management as an alternative to expected utility. The sensitivity of the distortion risk measure is a functional of certain distribution sensitivities. We propose a new sensitivity estimator for the distortion risk measure that uses generalized likelihood ratio estimators for distribution sensitivities as input and establish a central limit theorem for the new estimator. The proposed estimator can handle discontinuous sample paths and distortion functions.

Sharp Error Term in Local Limit Theorems and Mixing for Lorentz Gases with Infinite Horizon

We obtain sharp error rates in the local limit theorem for the Sinai billiard map (one and two dimensional) with infinite horizon. This result allows us to further obtain higher order terms and thus, sharp mixing rates in the speed of mixing of dynamically Hölder observables for the planar and tubular infinite horizon Lorentz gases in the map (discrete time) case. We also obtain an asymptotic estimate for the tail probability of the first return time to the initial cell. In the process, we study families of transfer operators for infinite horizon Sinai billiards perturbed with the free flight function and obtain higher order expansions for the associated families of eigenvalues and eigenprojectors.

Evolving extreme events caused by climate change: A tail based Bayesian approach for extreme event risk analysis

Natural hazards are of significant concern for engineering development in the offshore environment. Climate change phenomena are making these concerns even greater. The frequency and extent of natural hazards are undesirably evolving over time; so risk estimation for such events require special consideration. In most cases the existing extreme models (based on the extreme value theory) are unable to capture the changing frequency and extremeness of natural hazards. To capture the evolving frequency and extremeness of natural hazards and their effects on offshore process operations, an advanced probabilistic approach is proposed in this paper. The approach considers a heavy right tail probability model. The model parameter is estimated through the Bayesian inference. Hill and the SmooHill estimators are used to evaluate the lowest and highest exponent of the probability model. The application of the approach is demonstrated through extreme iceberg risk analysis for the Jeanne d’Arc basin. This study shows climate change or global warming is causing to appear a significant number of icebergs every year in the study area. Offshore structures are often designed to withstand the impact of 1 MT icebergs weight; however, the study observes large icebergs (10 MT weight) are sighted in recent years (14% of the total number of cited icebergs for the period of 2002–2017). As a result, the design philosophy needs to be revised. The proposed risk-based approach provides a robust design criterion for offshore structures.