Posterior analysis in assessing risk of extreme events: a conjugate family approach

The motivation of this study is the need to update constantly the assessment of risk of extreme events. This paper investigates the use of a posterior analysis using a conjugate family approach. Risk of extreme events is characterized by the Gumbel type of the extreme-value distribution and two associated parameters, the characteristic largest value, u_t; and the inverse measure of dispersion, α_t. The conjugate family approach derived in this paper facilitates the calculation of the posterior expected value and variance for both u_t and α_t. The Gumbel type and parameters u_t and α_t can be combined to estimate a conditional expected value of extreme events, f₄, that measures risks with low exceedance probability and high consequence. For each generating process that the authors study, it can be shown that the variances of u_t, α_t, and f ₄ decrease as additional samples are taken. This indicates that the uncertainty in assessing risk of extreme events decreases in light of additional information. The theoretical and methodological developments are illustrated in a case study involving coastal storm wave heights