The generalized extreme value (GEV) distribution is the only possible limiting distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. As such, it has been widely applied to approximate the distribution of maxima over blocks. In these applications, GEV properties such as finite lower endpoint when the shape parameter $\xi$ is positive or the loss of moments due to the magnitude of $\xi$ are inherited by the finite-sample maxima distribution. The extent to which these properties are realistic for the data at hand has been widely ignored. Motivated by these overlooked consequences in a regression setting, we here make three contributions. First, we propose a blended GEV (bGEV) distribution, which smoothly combines the left tail of a Gumbel distribution (GEV with $\xi =0$) with the right tail of a Fréchet distribution (GEV with $\xi >0$). Our resulting distribution has, therefore, unbounded support. Second, we proposed a principled method called property-preserving penalized complexity (P${}^3$C) prior to decide on the existence of the GEV distribution first and second moments a priori. Third, we propose a reparametrization of the GEV distribution that provides a more natural interpretation of the (possibly covariate-dependent) model parameters, which in turn helps define meaningful priors. We implement the bGEV distribution with the new parameterization and the P${}^3$C prior approach in the R-INLA package to make it readily available to users. We illustrate our methods with a simulation study that reveals that the GEV and bGEV distributions are comparable when estimating the right tail under large-sample settings. Moreover, some small-sample settings show that the bGEV fit slightly outperforms the GEV fit. Finally, we conclude with an application to NO${}_2$ pollution levels in California that illustrates the suitability of the new parameterization and the P${}^3$C prior approach in the Bayesian framework.

中文翻译：

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基于广义极值回归模型的实用策略

广义极值 (GEV) 分布是一系列独立且同分布的随机变量的正确归一化最大值的唯一可能限制分布。因此，它已被广泛应用于近似块上的最大值分布。在这些应用中，GEV 属性如有限下端点时的形状参数$\xi$是正数或由于$\xi$由有限样本最大值分布继承。这些属性对于手头数据的现实程度已被广泛忽略。受回归设置中这些被忽视的后果的启发，我们在这里做出了三个贡献。首先，我们提出了一个混合 GEV（bGEV）分布，它平滑地结合了 Gumbel 分布的左尾（GEV 和$\xi =0$) 具有 Fréchet 分布的右尾 (GEV 与$\xi >0$）。因此，我们得到的分布具有无限的支持。其次，我们提出了一种原则性方法，称为保属性惩罚复杂度（P${}^3$C) 先验地决定 GEV 分布的第一和第二矩的存在。第三，我们提出了 GEV 分布的重新参数化，它提供了对（可能与协变量相关的）模型参数的更自然的解释，这反过来又有助于定义有意义的先验。我们使用新的参数化和 P${}^3$R-INLA 包中的 C 先前方法使其易于用户使用。我们通过一项模拟研究来说明我们的方法，该研究表明，在大样本设置下估计右尾时，GEV 和 bGEV 分布具有可比性。此外，一些小样本设置表明 bGEV 拟合略优于 GEV 拟合。最后，我们以申请 NO 结束${}_2$加利福尼亚州的污染水平，说明了新参数化的适用性和 P${}^3$贝叶斯框架中的 C 先验方法。