Volume 4/Number 3, Fall 2009
Bayesian analysis of extreme operational losses
Algorithmics (UK) Limited, Eldon House, 2 Eldon Street, London EC2M 7LS, UK; email: firstname.lastname@example.org
Bayesian techniques offer an alternative to parameter estimation methods, such as maximum likelihood estimation, for extreme value models. Thesetechniques treat the parameters to be estimated as random variables, instead of some ﬁxed, possibly unknown, constants. We investigate, with simulated examples, how Bayesian analysis can be used to estimate the parameters of extreme value models, for the case where we have no prior knowledge at all and the case where we have prior knowledge in the form of expert opinion. In addition, Bayesian analysisprovides a framework for the incorporation of information from external data into a loss model based on internal data; this is again illustrated using simulation.
Maximum likelihood estimation (MLE) techniques are not the only way to draw inferences from the likelihood function; Bayesian inference offers an alternative methodology, as well as viewpoint. There is some debateconcerning the viability of these methods, which we will only brieﬂy touch upon. We will be concentrating instead on how these methods might be applied in practice. The Bayesian techniques we will consider here have wide applicability, but we are particularly interested in how these techniques may give us greater insight into the behavior of loss processes at extreme levels. There has been a greatdeal of interest in the use of Bayesian methods in operational risk; recent papers on the subject include those by Shevchenko and Wüthrich (2006) and Peters and Sisson (2006). We will not be restricting ourselves to conjugate priors (see Shevchenko and Wüthrich (2006) for classes of frequency and severity models admitting conjugate forms) and will instead be concentrating on the ﬁtting of extremelosses, using the distributions suggested by extreme value theory. In Section 2, we introduce the concepts behind Bayesian analysis; in Section 3 we brieﬂy describe how simulation-based techniques, and in particular Markov chain Monte Carlo (MCMC) techniques, can help us overcome the difﬁculty of computation in Bayesian analysis. In Section 4, we consider parameter estimation for the extreme valuemodel for annual maxima, through a small simulation study. For simulated data, we compare the Bayesian estimates for the model parameters when we have no prior information (Section 4.1) with those obtained when we have
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prior expertopinion (Section 4.2). In the latter case, there is no obvious methodology for transforming expert opinions into prior distributions for the parameters. We will be following the method for eliciting prior information proposed by Coles and Tawn (1996). We consider using external data for prior speciﬁcation in Section 5. Typically there are concerns about the incorporation of external data into themodeling process – questions about scaling, applicability and loss severity thresholds. We illustrate a possible use of Bayesian analysis with an example showing where the external data can provide us with information concerning the shape of the distribution.
2 BAYESIAN INFERENCE
Suppose we have data x = (x1 , . . . , xn ), constituting independent identically distributed (iid) realizationsof a random variable, X, whose density belongs to a parametric family parametrized by θ ∈ . The likelihood for θ may then be given by P (x | θ) = n P (xi | θ), since the observations xi , i = 1, . . . , n, are independent. i=1 In the classical framework, θ is a constant – an unknown constant to be estimated in many cases. In the Bayesian setting, θ itself is a random variable, with an a priori...