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 $q_0$, popular in the engineering community, are considered. These are: $q_1$, the quantile of the distribution of the annual maximum event with non-exceedance probability $1-1/N$, estimated using mean model parameters; $q_2$, the mean of different quantile estimates of the annual maximum event with non-exceedance probability $1-1/N$; $q_3$, the quantile of the predictive distribution of the annual maximum event with non-exceedance probability $1-1/N$; and $q_4$, the quantile of the predictive distribution of the N-year maximum event with non-exceedance probability $\text{exp}\left[\text{−}1\right]$. Using theoretical arguments, and simulation of samples of GP-distributed peaks over threshold (with $\xi \in [\text{−}0.4,0.1]$) and different GP parameter estimation schemes, we show that the rank order of estimators q and true value $q_0$ can be predicted, and that differences between estimators q and $q_0$ 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 $q_2$ is probably the best overall. The estimation scheme of Zhang (2010) provides low bias for $q_1$.

我们考虑使用最大似然法和其他估计方案，在存在不确定的极值模型参数的情况下对返回值进行估计。当忽略不确定性参数时，返回值的估计值将产生相同的值，而当考虑不确定性时，将给出不同的值。鉴于不确定形状ξ一个广义帕累托（GP）分布和尺度参数，四个采样估计q为Ñ -年返回值$q_0$在工程界很流行。这些是：$q_{\mathrm{1个}}$，具有不超过概率的年度最大事件的分布的分位数 $\mathrm{1个}-\mathrm{1个}/ñ$，使用均值模型参数估算； $q_2$，不超过概率的年度最大事件的不同分位数估计的均值 $\mathrm{1个}-\mathrm{1个}/ñ$; $q_3$，具有非超标概率的年度最大事件的预测分布的分位数 $\mathrm{1个}-\mathrm{1个}/ñ$; 和$q_4$，具有非超标概率的N年最大事件的预测分布的分位数$\text{经验值}\left[\text{-}\mathrm{1个}\right]$。使用理论参数并模拟GP分布超过阈值的峰的样本（具有$\xi \in [\text{-}0.4，0.1]$）和不同的GP参数估计方案，我们证明了估计器q和真实值的等级顺序$q_0$可以预测，并且估计值q和$q_0$可以很大。有关估计量相对性能的判断取决于用来评估它们的效用函数的选择。我们考虑返回值的偏差，超出概率的偏差和对数超出概率的偏差。在最大似然估计下，四个估计器对于所有三个效用都没有表现良好，但是平均分位数$q_2$可能是最好的整体。Zhang（2010）的估计方案为$q_{\mathrm{1个}}$。