论文标题
准确近似极值统计
Accurately approximating extreme value statistics
论文作者
论文摘要
我们考虑$ n $独立变量的极端价值统计数据，这是概率理论中的经典问题。当$ n \ to \ infty $时，Fisher-Tippett-gendenko定理描述了变量最大变量的波动，该定理指出，最大值的分布在三种限制形式中收敛到三种。其中包括牙龈分布，$ n $的收敛速率具有对数性质。在这里，我们提出了一种理论，该理论允许一个人使用牙龈胶合限制准确近似确切的极值分布。我们通过将量表和宽度参数表示为功率序列，以及通过基础分布的转换来做到这一点。我们也考虑了对牙龈极限的功能校正，这表明它们是通过泰勒扩展获得的。我们的方法还改善了对平均极值的大偏差的描述。此外，当拟合实验数据时，当基础分布未知时，它有助于表征极值统计。
We consider the extreme value statistics of $N$ independent and identically distributed random variables, which is a classic problem in probability theory. When $N\to\infty$, fluctuations around the maximum of the variables are described by the Fisher-Tippett-Gnedenko theorem, which states that the distribution of maxima converges to one out of three limiting forms. Among these is the Gumbel distribution, for which the convergence rate with $N$ is of a logarithmic nature. Here, we present a theory that allows one to use the Gumbel limit to accurately approximate the exact extreme value distribution. We do so by representing the scale and width parameters as power series, and by a transformation of the underlying distribution. We consider functional corrections to the Gumbel limit as well, showing they are obtainable via Taylor expansion. Our method also improves the description of large deviations from the mean extreme value. Additionally, it helps to characterize the extreme value statistics when the underlying distribution is unknown, for example when fitting experimental data.
