An Improved Hazard Rate Twisting Approach for the Statistic of the Sum of Subexponential Variates

In this letter, we present an improved hazard rate twisting technique for the estimation of the probability that a sum of independent but not necessarily identically distributed subexponential Random Variables (RVs) exceeds a given threshold. Instead of twisting all the components in the summation, we propose to twist only the RVs which have the biggest impact on the right-tail of the sum distribution and keep the other RVs unchanged. A minmax approach is performed to determine the optimal twisting parameter which leads to an asymptotic optimality criterion. Moreover, we show through some selected simulation results that our proposed approach results in a variance reduction compared to the technique where all the components are twisted

On the generalization of the hazard rate twisting-based simulation approach

Estimating the probability that a sum of random variables (RVs) exceeds a given threshold is a well-known challenging problem. A naive Monte Carlo simulation is the standard technique for the estimation of this type of probability. However, this approach is computationally expensive, especially when dealing with rare events. An alternative approach is represented by the use of variance reduction techniques, known for their efficiency in requiring less computations for achieving the same accuracy requirement. Most of these methods have thus far been proposed to deal with specific settings under which the RVs belong to particular classes of distributions. In this paper, we propose a generalization of the well-known hazard rate twisting Importance Sampling-based approach that presents the advantage of being logarithmic efficient for arbitrary sums of RVs. The wide scope of applicability of the proposed method is mainly due to our particular way of selecting the twisting parameter. It is worth observing that this interesting feature is rarely satisfied by variance reduction algorithms whose performances were only proven under some restrictive assumptions. It comes along with a good efficiency, illustrated by some selected simulation results comparing the performance of the proposed method with some existing techniques

A review of conditional rare event simulation for tail probabilities of heavy tailed random variables

Approximating the tail probability of a sum of heavy-tailed random variables is a difficult problem. In this review we exhibit the challenges of approximating such probabilities and concentrate on a rare event simulation methodology capable of delivering the most reliable results: Conditional Monte Carlo. To provide a better flavor of this topic we further specialize on two algorithms which were specifically designed for tackling this problem: the Asmussen-Binswanger estimator and the Asmussen-Kroese estimator. We extend the applicability of these estimators to the non-independent case and prove their efficiency

Unified Importance Sampling Schemes for Efficient Simulation of Outage Capacity Over Generalized Fading Channels

The outage capacity (OC) is among the most important performance metrics of communication systems operating over fading channels. Of interest in the present paper is the evaluation of the OC at the output of the Equal Gain Combining (EGC) and the Maximum Ratio Combining (MRC) receivers. In this case, it can be seen that this problem turns out to be that of computing the Cumulative Distribution Function (CDF) for the sum of independent random variables. Since finding a closed-form expression for the CDF of the sum distribution is out of reach for a wide class of commonly used distributions, methods based on Monte Carlo (MC) simulations take pride of price. In order to allow for the estimation of the operating range of small outage probabilities, it is of paramount importance to develop fast and efficient estimation methods as naive MC simulations would require high computational complexity. In this line, we propose in this work two unified, yet efficient, hazard rate twisting Importance Sampling (IS) based approaches that efficiently estimate the OC of MRC or EGC diversity techniques over generalized independent fading channels. The first estimator is shown to possess the asymptotic optimality criterion and applies for arbitrary fading models, whereas the second one achieves the well-desired bounded relative error property for the majority of the well-known fading variates. Moreover, the second estimator is shown to achieve the asymptotic optimality property under the particular Log-normal environment. Some selected simulation results are finally provided in order to illustrate the substantial computational gain achieved by the proposed IS schemes over naive MC simulations

Achieving Efficiency in Black Box Simulation of Distribution Tails with Self-structuring Importance Samplers

Motivated by the increasing adoption of models which facilitate greater automation in risk management and decision-making, this paper presents a novel Importance Sampling (IS) scheme for measuring distribution tails of objectives modelled with enabling tools such as feature-based decision rules, mixed integer linear programs, deep neural networks, etc. Conventional efficient IS approaches suffer from feasibility and scalability concerns due to the need to intricately tailor the sampler to the underlying probability distribution and the objective. This challenge is overcome in the proposed black-box scheme by automating the selection of an effective IS distribution with a transformation that implicitly learns and replicates the concentration properties observed in less rare samples. This novel approach is guided by a large deviations principle that brings out the phenomenon of self-similarity of optimal IS distributions. The proposed sampler is the first to attain asymptotically optimal variance reduction across a spectrum of multivariate distributions despite being oblivious to the underlying structure. The large deviations principle additionally results in new distribution tail asymptotics capable of yielding operational insights. The applicability is illustrated by considering product distribution networks and portfolio credit risk models informed by neural networks as examples.Comment: 51 page

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal