Modelling losses and locating the tail with the Pareto Positive Stable distribution

This paper focuses on modelling the severity distribution. We directly model the small, moderate and large losses with the Pareto Positive Stable (PPS) distribution and thus it is not necessary to fix a threshold for the tail behaviour. Estimation with the method of moments is straightforward. Properties, graphical tests and expressions for value-at risk and tail value-at-risk are presented. Furthermore, we show that the PPS distribution can be used to construct a statistical test for the Pareto distribution and to determine the threshold for the Pareto shape if required. An application to loss data is presented. We conclude that the PPS distribution can perform better than commonly used distributions when modelling a single loss distribution for moderate and large losses. This approach avoids the pitfalls of cut-off selection and it is very simple to implement for quantitative risk analysis.     

Bayesian estimation of finite time ruin probabilities

In this paper, we consider Bayesian inference and estimation of finite time ruin probabilities for the Sparre Andersen risk model. The dense family of Coxian distributions is considered for the approximation of both the inter‐claim time and claim size distributions. We illustrate that the Coxian model can be well fitted to real, long‐tailed claims data and that this compares well with the generalized Pareto model. The main advantage of using the Coxian model for inter‐claim times and claim sizes is that it is possible to compute finite time ruin probabilities making use of recent results from queueing theory. In practice, finite time ruin probabilities are much more useful than infinite time ruin probabilities as insurance companies are usually interested in predictions for short periods of future time and not just in the limit. We show how to obtain predictive distributions of these finite time ruin probabilities, which are more informative than simple point estimations and take account of model and parameter uncertainty. We illustrate the procedure with simulated data and the well‐known Danish fire loss data set. Copyright © 2009 John Wiley & Sons, Ltd.     

Exact and approximate numerical solutions to steady-state single-server queues:M/G/1 — a unified approach

This paper presents a unified approach for the numerical solutions of anM/G/1 queue. On the assumption that the service-time distribution has a rational Laplace-Stieltjes transform (LST), explicit closed-form expressions have been obtained for moments, distributions of system length and waiting time (in queue) in terms of the roots of associated characteristic equations (c.e.'s). Approximate analyses for the tails of the distributions based on one or more roots are also discussed. Numerical aspects have been tested for a variety of complex service-time distributions including but not restricted to only mixed generalized Erlang and generalized hyperexponential. A sample of numerical computations is also included. It is hoped that the results obtained would prove to be beneficial to both practitioners and theorists dealing with bounds, inequalities, approximations, and other aspects.     

Computing steady-state queueing-time distributions of single-server queues:GI X /M/1

This paper presents a computationally efficient method to find the steady-state distributions of actual queueing times of the first customer, as well as of a randomly selected customer, of an arrival group for the queueing systemGI ^X /M/1, and hence the queueing-time distribution of a customer for the systemGI/E _X /1. The distribution of virtual queueing time is also obtained. Approximate analysis based on one or more roots is also discussed. Though the exact detailed as well as approximate computations for a variety of interarrival-time distributions such as generalized Erlang, mixed generalized Erlang, hyperexponential, generalized hyperexponential, and deterministic have been carried out, only representative results in the form of tables have been appended. The results obtained should prove useful to queueing theorists, practitioners, and others.     

A recursive method for the N policy G/M/1 queueing system with finite capacity

This paper studies a single removable server in a G/M/1 queueing system with finite capacity operating under the N policy. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distributions of the number of customers in the system. The method is illustrated analytically for exponential interarrival time distribution. Numerical results for various system performance measures are presented for four different interarrival time distributions such as exponential, 2-stage hyperexponential, 4-stage Erlang, and deterministic.     

A hybrid Pareto model for asymmetric fat-tailed data: the univariate case

Density estimators that can adapt to asymmetric heavy tails are required in many applications such as finance and insurance. Extreme value theory (EVT) has developed principled methods based on asymptotic results to estimate the tails of most distributions. However, the finite sample approximation might introduce a severe bias in many cases. Moreover, the full range of the distribution is often needed, not only the tail area. On the other hand, non-parametric methods, while being powerful where data are abundant, fail to extrapolate properly in the tail area. We put forward a non-parametric density estimator that brings together the strengths of non-parametric density estimation and of EVT. A hybrid Pareto distribution that can be used in a mixture model is proposed to extend the generalized Pareto (GP) to the whole real axis. Experiments on simulated data show the following. On one hand, the mixture of hybrid Paretos converges faster in terms of log-likelihood and provides good estimates of the tail of the distributions when compared with other density estimators including the GP distribution. On the other hand, the mixture of hybrid Paretos offers an alternate way to estimate the tail index which is comparable to the one estimated with the standard GP methodology. The mixture of hybrids is also evaluated on the Danish fire insurance data set.      

Bayesian analysis of fixed-interval preventive-maintenance models

This article develops methods for making accurate decisionswhen scheduling preventive maintenance in systems where inter-eventtimes can be modelled by a delayed renewal process or delayedalternating renewal process. A practical application, relatingto the reliability and maintenance of a relatively low-levelcomponent (valve) in a continuous-process industry over a periodof six years is presented to demonstrate and compare the differentapproaches. Our analyses indicate a cost-effective recommendationfor maintenance practice in this context. Our main thrust relates to the use of Bayesian methodology inorder to obtain rational, admissible decisions. Particular advancesover previous research allow for informative prior distributions,better approximations which lead to improved accuracy, non-negligibledowntimes, and general lifetime distributions. General analyticsolutions are sought for the simpler models, in order to achieveaccuracy and insight. The resulting integrals can only be solvedto give an infinite series and one approximation to the requiredsolution is obtained by truncating this series. Two other approximationsare developed, based on expansions of the prior predictive andlog-posterior distributions. A simulation approach is also developed to include prior informationand hence provide alternative approximations to these optimaldecisions. With exponential lifetime distributions, the relevantposterior lifetime distributions are non-central Pareto. Thissimulation is simple to program, compared to the approximations,but requires more computing time. It is accurate and extendseasily to situations involving greater complexity. We considertwo such extensions, the inclusion of downtimes and Weibulllifetime distributions.     

Maximum entropy analysis to the N policy M/G/1 queueing system with a removable server

We study a single removable server in an M/G/1 queueing system operating under the N policy in steady-state. The server may be turned on at arrival epochs or off at departure epochs. Using the maximum entropy principle with several well-known constraints, we develop the approximate formulae for the probability distributions of the number of customers and the expected waiting time in the queue. We perform a comparative analysis between the approximate results with exact analytic results for three different service time distributions, exponential, 2-stage Erlang, and 2-stage hyper-exponential. The maximum entropy approximation approach is accurate enough for practical purposes. We demonstrate, through the maximum entropy principle results, that the N policy M/G/1 queueing system is sufficiently robust to the variations of service time distribution functions.     

A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets

We introduce and analyze a novel scalarization technique and an associated algorithm for generating an approximation of the Pareto front (i.e., the efficient set) of nonlinear multiobjective optimization problems. Our approach is applicable to nonconvex problems, in particular to those with disconnected Pareto fronts and disconnected domains (i.e., disconnected feasible sets). We establish the theoretical properties of our new scalarization technique and present an algorithm for its implementation. By means of test problems, we illustrate the strengths and advantages of our approach over existing scalarization techniques such as those derived from the Pascoletti–Serafini method, as well as the popular weighted-sum method.     

Analysis of the M/G/1 processor-sharing queue with bulk arrivals

We analyze the single server processor-sharing queue for the case of bulk arrivals. We obtain an expression for the expected response time of a job as a function of its size, when the service times of jobs have a generalized hyperexponential distribution and more generally for distributions with rational Laplace transforms. Our analysis significantly extends the class of distributions for which processor-sharing queues with bulk arrivals were previously analyzed.     

Stochastic comparison of aggregate claim amounts between two heterogeneous portfolios and its applications

The aggregate claim amount in a particular time period is a quantity of fundamental importance for proper management of an insurance company and also for pricing of insurance coverages. In this paper, we show that the proportional hazard rates (PHR) model, which includes some well-known distributions such as exponential, Weibull and Pareto distributions, can be used as the aggregate claim amount distribution. We also present some conditions for the use of exponentiated Weibull distribution as the claim amount distribution. The results established here complete and extend the well-known result of Khaledi and Ahmadi (2008).     

Discrete Approximation of a Mixture Distribution via Restricted Divergence

Mixture distributions arise in many application areas, for example, as marginal distributions or convolutions of distributions. We present a method of constructing an easily tractable discrete mixture distribution as an approximation to a mixture distribution with a large to infinite number, discrete or continuous, of components. The proposed DIRECT (divergence restricting conditional tesselation) algorithm is set up such that a prespecified precision, defined in terms of Kullback–Leibler divergence between true distribution and approximation, is guaranteed. Application of the algorithm is demonstrated in two examples. Supplementary materials for this article are available online.     

Finding optimal hyperpaths in large transit networks with realistic headway distributions

This paper implements and tests a label-setting algorithm for finding optimal hyperpaths in large transit networks with realistic headway distributions. It has been commonly assumed in the literature that headway is exponentially distributed. To validate this assumption, the empirical headway data archived by Chicago Transit Agency are fitted into various probabilistic distributions. The results suggest that the headway data fit much better with Loglogistic, Gamma and Erlang distributions than with the exponential distribution. Accordingly, we propose to model headway using the Erlang distribution in the proposed algorithm, because it best balances realism and tractability. When headway is not exponentially distributed, finding optimal hyperpaths may require enumerating all possible line combinations at each transfer stop, which is tractable only for a small number of alternative lines. To overcome this difficulty, a greedy method is implemented as a heuristic and compared to the brute-force enumeration method. The proposed algorithm is tested on a large scale CTA bus network that has over 10,000 stops. The results show that (1) the assumption of exponentially distributed headway may lead to sub-optimal route choices and (2) the heuristic greedy method provides near optimal solutions in all tested cases.     

Computing the Stationary Distribution of an SRBM in an Orthant with Applications to Queueing Networks

In [15], a BNAfm (Brownian network analyzer with finite element method) algorithm was developed for computing the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in a hypercube. In this companion paper, that BNAfm algorithm is extended to computing the stationary distribution of an SRBM in an orthant, which is achieved by constructing a converging sequence of SRBMs in hypercubes. The SRBM in the orthant serves as an approximation model of queueing networks with infinite buffers. We show that the constructed sequence of SRBMs in the hypercubes converges weakly to the SRBM in the orthant as the hypercubes approach the orthant. Under the conjecture that the set of the stationary distributions of the SRBMs in the hypercubes is relatively compact, we prove that the sequence of the stationary distributions of the SRBMs in the hypercubes converges weakly to the stationary distribution of the SRBM in the orthant. A three-machine job shop example is presented to illustrate the effectiveness of the SRBM approximation model and our BNAfm algorithm. The BNAfm algorithm is shown to produce good estimates for stationary probabilities of queueing networks.     

Using mixture density functions for modelling of wage distributions

This article explores the possibility of modeling the wage distribution using a mixture of density functions. We deal with this issue for a long time and we build on our earlier work. Classical models use the probability distribution such as normal, lognormal, Pareto, etc., but the results are not very good in the last years. Changing the parameters of a probability density over time has led to a degradation of such models and it was necessary to choose a different probability distribution. We were using the idea of mixtures of distributions (instead of using one classical density) in previous articles. We tried using a mixture of probability distributions (normal, lognormal and a mixture of Johnson’s distribution densities) in our models. The achieved results were very good. We used data from Czech Statistical Office covering the wages of the last 18 years in Czech Republic.     

Local polynomial maximum likelihood estimation for Pareto-type distributions

We discuss the estimation of the tail index of a heavy-tailed distribution when covariate information is available. The approach followed here is based on the technique of local polynomial maximum likelihood estimation. The generalized Pareto distribution is fitted locally to exceedances over a high specified threshold. The method provides nonparametric estimates of the parameter functions and their derivatives up to the degree of the chosen polynomial. Consistency and asymptotic normality of the proposed estimators will be proven under suitable regularity conditions. This approach is motivated by the fact that in some applications the threshold should be allowed to change with the covariates due to significant effects on scale and location of the conditional distributions. Using the asymptotic results we are able to derive an expression for the asymptotic mean squared error, which can be used to guide the selection of the bandwidth and the threshold. The applicability of the method will be demonstrated with a few practical examples.     

Performance analysis approximation in a queueing system of type M/G/1

In this work, we apply the strong stability method to obtain an estimate for the proximity of the performance measures in the M/G/1 queueing system to the same performance measures in the M/M/1 system under the assumption that the distributions of the service time are close and the arrival flows coincide. In addition to the proof of the stability fact for the perturbed M/M/1 queueing system, we obtain the inequalities of the stability. These results give with precision the error, on the queue size stationary distribution, due to the approximation. For this, we elaborate from the obtained theoretical results, the STR-STAB algorithm which we execute for a determined queueing system: M/Coxian − 2/1. The accuracy of the approach is evaluated by comparison with simulation results.     

Distributed localized bi-objective search

We propose a new distributed heuristic for approximating the Pareto set of bi-objective optimization problems. Our approach is at the crossroads of parallel cooperative computation, objective space decomposition, and adaptive search. Given a number of computing nodes, we self-coordinate them locally, in order to cooperatively search different regions of the Pareto front. This offers a trade-off between a fully independent approach, where each node would operate independently of the others, and a fully centralized approach, where a global knowledge of the entire population is required at every step. More specifically, the population of solutions is structured and mapped into computing nodes. As local information, every node uses only the positions of its neighbors in the objective space and evolves its local solution based on what we term a ‘localized fitness function  ’. This has the effect of making the distributed search evolve, over all nodes, to a high quality approximation set, with minimum communications. We deploy our distributed algorithm using a computer cluster of hundreds of cores and study its properties and performance on ρ$\rho$MNK-landscapes. Through extensive large-scale experiments, our approach is shown to be very effective in terms of approximation quality, computational time and scalability.     

Efficient simulation and conditional functional limit theorems for ruinous heavy-tailed random walks

The contribution of this paper is to introduce change of measure based techniques for the rare-event analysis of heavy-tailed random walks. Our changes of measures are parameterized by a family of distributions admitting a mixture form. We exploit our methodology to achieve two types of results. First, we construct Monte Carlo estimators that are strongly efficient (i.e. have bounded relative mean squared error as the event of interest becomes rare). These estimators are used to estimate both rare-event probabilities of interest and associated conditional expectations. We emphasize that our techniques allow us to control the expected termination time of the Monte Carlo algorithm even if the conditional expected stopping time (under the original distribution) given the event of interest is infinity–a situation that sometimes occurs in heavy-tailed settings. Second, the mixture family serves as a good Markovian approximation (in total variation) of the conditional distribution of the whole process given the rare event of interest. The convenient form of the mixture family allows us to obtain functional conditional central limit theorems that extend classical results in the literature.     

Risk aggregation in multivariate dependent Pareto distributions

In this paper we obtain closed expressions for the probability distribution function of aggregated risks with multivariate dependent Pareto distributions. We work with the dependent multivariate Pareto type II proposed by Arnold (1983, 2015), which is widely used in insurance and risk analysis. We begin with an individual risk model, where the probability density function corresponds to a second kind beta distribution, obtaining the VaR, TVaR and several other tail risk measures. Then, we consider a collective risk model based on dependence, where several general properties are studied. We study in detail some relevant collective models with Poisson, negative binomial and logarithmic distributions as primary distributions. In the collective Pareto–Poisson model, the probability density function is a function of the Kummer confluent hypergeometric function, and the density of the Pareto–negative binomial is a function of the Gauss hypergeometric function. Using data based on one-year vehicle insurance policies taken out in 2004–2005 (Jong and Heller, 2008) we conclude that our collective dependent models outperform other collective models considered in the actuarial literature in terms of AIC and CAIC statistics.