A new unimodal distribution family indexed by the mode and three other parameters is derived from a mixture of a Gumbel distribution for the maximum and a Gumbel distribution for the minimum. Properties of the proposed distribution are explored, including model identifiability and flexibility in capturing heavy-tailed data that exhibit different directions of skewness over a wide range. Both frequentist and Bayesian methods are developed to infer parameters in the new distribution. Simulation studies are conducted to demonstrate satisfactory performance of both methods. By fitting the proposed model to simulated data and data from an application in hydrology, it is shown that the proposed flexible distribution is especially suitable for data containing extreme values in either direction, with the mode being a location parameter of interest. A regression model concerning the mode of a response given covariates based on the proposed unimodal distribution can be easily formulated, which we apply to data from an application in criminology to reveal interesting data features that are obscured by outliers.

翻译：基于最大值Gumbel分布与最小值Gumbel分布的混合，本文推导出一个以众数及其他三个参数为索引的新单峰分布族。研究探讨了该分布的特性，包括模型可识别性以及在不同偏度方向范围内对厚尾数据的灵活捕捉能力。基于频率学派和贝叶斯方法分别发展了该新分布的参数推断技术。仿真研究表明两种方法均具有令人满意的表现。通过将所提模型拟合至模拟数据和水文领域实际应用数据，证实该灵活分布尤其适用于包含双向极端值的数据，其中众数作为具有实际意义的定位参数。基于该单峰分布可简便构建关于响应变量条件众数的回归模型，并将其应用于犯罪学领域数据，揭示出被异常值掩盖的有趣数据特征。