Uncertainties in return values from extreme value analysis of peaks over threshold using the generalised Pareto distribution

Jonathan, P. and Randell, D. and Wadsworth, J. and Tawn, J. (2021) Uncertainties in return values from extreme value analysis of peaks over threshold using the generalised Pareto distribution. Ocean Engineering, 220: 107725. ISSN 0029-8018

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Abstract

We consider the estimation of return values in the presence of uncertain extreme value model parameters, using maximum likelihood and other estimation schemes. Estimators for return value, which yield identical values when parameter uncertainty is ignored, give different values when uncertainty is taken into account. Given uncertain shape ξ and scale parameters of a generalised Pareto (GP) distribution, four sample estimators q for the N-year return value q0, popular in the engineering community, are considered. These are: q1, the quantile of the distribution of the annual maximum event with non-exceedance probability 1−1/N, estimated using mean model parameters; q2, the mean of different quantile estimates of the annual maximum event with non-exceedance probability 1−1/N; q3, the quantile of the predictive distribution of the annual maximum event with non-exceedance probability 1−1/N; and q4, the quantile of the predictive distribution of the N-year maximum event with non-exceedance probability exp[−1]. Using theoretical arguments, and simulation of samples of GP-distributed peaks over threshold (with ξ∈[−0.4,0.1]) and different GP parameter estimation schemes, we show that the rank order of estimators q and true value q0 can be predicted, and that differences between estimators q and q0 can be large. Judgements concerning the relative performance of estimators depend on the choice of utility function adopted to assess them. We consider bias in return value, bias in exceedance probability and bias in log exceedance probability. None of the four estimators performs well with respect to all three utilities under maximum likelihood estimation, but the mean quantile q2 is probably the best overall. The estimation scheme of Zhang (2010) provides low bias for q1.