Estimation of value-at-risk for conduct risk losses using pseudo-marginal Markov chain Monte Carlo

We propose a model for conduct risk losses, in which conduct risk losses are characterized by having a small number of extremely large losses (perhaps only one) with more numerous smaller losses. It is assumed that the largest loss is actually a provision from which payments to customers are made periodically as required. We use the pseudo-marginal (PM) Markov chain Monte Carlo method to decompose the largest loss into smaller partitions in order to estimate 99.9% value-at-risk. The partitioning is done in a way that makes no assumption about the size of the partitions. The advantages and problems of using this method are discussed. The PM procedures were run on several representative data sets. The results indicate that, in cases where using approaches such as calculating a Monte Carlo-derived loss distribution yields a result that is not consistent with the risk profile expressed by the data, using the PM method yields results that have the required consistency

Quantification of uncertainty in probabilistic safety analysis

This thesis develops methods for quantification and interpretation of uncertainty in probabilistic safety analysis, focussing on fault trees. The output of a fault tree analysis is, usually, the probability of occurrence of an undesirable event (top event) calculated using the failure probabilities of identified basic events. The standard method for evaluating the uncertainty distribution is by Monte Carlo simulation, but this is a computationally intensive approach to uncertainty estimation and does not, readily, reveal the dominant reasons for the uncertainty. A closed form approximation for the fault tree top event uncertainty distribution, for models using only lognormal distributions for model inputs, is developed in this thesis. Its output is compared with the output from two sampling based approximation methods; standard Monte Carlo analysis, and Wilks’ method, which is based on order statistics using small sample sizes. Wilks’ method can be used to provide an upper bound for the percentiles of top event distribution, and is computationally cheap. The combination of the lognormal approximation and Wilks’ Method can be used to give, respectively, the overall shape and high confidence on particular percentiles of interest. This is an attractive, practical option for evaluation of uncertainty in fault trees and, more generally, uncertainty in certain multilinear models. A new practical method of ranking uncertainty contributors in lognormal models is developed which can be evaluated in closed form, based on cutset uncertainty. The method is demonstrated via examples, including a simple fault tree model and a model which is the size of a commercial PSA model for a nuclear power plant. Finally, quantification of “hidden uncertainties” is considered; hidden uncertainties are those which are not typically considered in PSA models, but may contribute considerable uncertainty to the overall results if included. A specific example of the inclusion of a missing uncertainty is explained in detail, and the effects on PSA quantification are considered. It is demonstrated that the effect on the PSA results can be significant, potentially permuting the order of the most important cutsets, which is of practical concern for the interpretation of PSA models. Finally, suggestions are made for the identification and inclusion of further hidden uncertainties.Open Acces

Statistical issues in the assessment of undiscovered oil and gas resources

Prior to his untimely death, my friend Dave Wood gave me wise counsel about how best to organize a paper describing uses of statistics in oil and gas exploration. A preliminary reconnaissance of the literature alerted me to the enormous range of topics that might be covered. Geology, geophysics with particular attention to seismology, geochemistry, petroleum engineering and petroleum economics--each of these disciplines plays an important role in petroleum exploration and each weaves statistical thinking into its fabric in a distinctive way. An exhaustive review would be book length. Dave and I agreed that a timely review paper of reasonable length would: (1) Illustrate the range of statistical thinking of oil and gas exploratists. (2) Concentrate on topics with statistical novelty, show how statistical thinking can lead to better decision making and let the reader now about important controversies that might be resolved by better use of statistical methods. (3) Focus on topics that are directly relevant to exploration decision making and resource estimation. In response to Dave's sensible suggestions, the Department of Interior's 1989 assessment of U.S. undiscovered oil and gas will be a tour map for a short trip through a large territory of statistical methods and applications. Were he here to review this review, I know that it would be better than it is.Supported in part by the MIT Center for Energy and Environmental Policy Research

A central limit theorem formulation for empirical bootstrap value-at-risk

ABSTRACT In this paper, the importance of the empirical bootstrap (EB) in assessing minimal operational risk capital is discussed, and an alternative way of estimating minimal operational risk capital using a central limit theorem (CLT) formulation is presented. The results compare favorably with risk capital obtained by fitting appropriate distributions to the same data. The CLT formulation is significant in validation because it provides an alternative approach to the calculation that is independent of both the empirical severity distribution and any dependent fitted distributio

Demand Distribution Dynamics in Creative Industries: the Market for Books in Italy

We studied the distribution dynamics of the demand for books in Italy. We found that for each of the three broad sub-markets into which the book publishing industry can be classified - Italian novels, foreign novels and non-fiction - sales over a three-year sample can be adequately fitted by a power law distribution. Our results can be plausibly interpreted in terms of a model of interactions among buyers exchanging information on the books they buy.Book publishing industry; Information transmission; Power law distribution

Some non-standard approaches to the study of sums of heavy-tailed distributions

Heavy-tailed phenomena arise whenever events with very low probability have sufficiently large consequences that these events cannot be treated as negligible. These are sometimes described as low intensity, high impact events. Sums of heavy-tailed random variables play a major role in many areas of applied probability, for instance in risk theory, insurance mathematics, financial mathematics, queueing theory, telecommunications and computing, to name but a few areas. The theory of the asymptotic behaviour of a sum of independent heavy-tailed random variables is well-understood. We give a review of known results in this area, stressing the importance of some insensitivity properties of the class of long-tailed distributions. We introduce the new concept of the Boundary Class for a long-tailed distribution, and describe some of its properties and uses. We give examples of calculating the boundary class. Geometric sums of random variables are a useful model in their own right, for instance in reliability theory, but are also useful because they model the maximum of a random walk, which is itself a model that occurs in many applications. When the summands are heavy-tailed and independent then the asymptotic behaviour has been known since the 1970s. The asymptotic expression for the geometric sum is often used as an approximation to the actual distribution, owing to the (usually) analytically intractable form of the exact distribution. However the accuracy of this asymptotic approximation can be very poor, as we demonstrate. Following and further developing work by Kalashnikov and Tsitsiashvili we construct an upper bound for the relative accuracy of this approximation. We then develop new techniques for the application of our analytical results, and apply these in practice to several examples. Source code viii for the computer algorithms used in these calculations is given. As we have said, the asymptotic behaviour of a sum of heavy-tailed random variables is well-understood when the random variables are independent, the main characteristic being the principle of the single big jump. However, the case when the random variables are dependent is much less clear. We study this case for both deterministic and random sums using a novel approach, by considering conditional independence structures on the random variables. We seek sufficient conditions for the results of the theory with independent random variables still to hold. We give several examples to show how to apply and check our conditions, and the examples demonstrate a variety of effects owing to the dependence, and are also interesting in their own right. All the results we develop on this topic are entirely new. Some of the examples also include results that are new and have not been obtainable through previously existing techniques. For some examples we study the asymptotic behaviour is known, and this allows us to contrast our approach with previous approaches

A note on maximum likelihood estimation of a Pareto mixture

In this paper we study Maximum Likelihood Estimation of the parameters of a Pareto mixture. Application of standard techniques to a mixture of Pareto is problematic. For this reason we develop two alternative algorithms. The first one is the Simulated Annealing and the second one is based on Cross-Entropy minimization. The Pareto distribution is a commonly used model for heavy-tailed data. It is a two-parameter distribution whose shape parameter determines the degree of heaviness of the tail, so that it can be adapted to data with different features. This work is motivated by an application in the operational risk measurement field: we fit a Pareto mixture to operational losses recorded by a bank in two different business lines. Losses below an unknown threshold are discarded, so that the observed data are truncated. The thresholds used in the two business lines are unknown. Thus, under the assumption that each population follows a Pareto distribution, the appropriate model is a mixture of Pareto where all the parameters have to be estimated.