精确逼近极值统计量
Accurately approximating extreme value statistics
摘要：
考虑了概率论中的经典问题--$N$独立同分布随机变量的极值统计问题。当$N\to\infty$时，变量最大值附近的波动由Fisher-Tippett-Gnedenko定理描述，该定理指出最大值的分布收敛于三种极限形式之一。其中包括Gumbel分布，对于该分布，使用$N$的收敛速率具有对数性质。在这里，我们提出了一个理论，允许使用Gumbel极限精确地近似精确的极值分布。我们通过将尺度和宽度参数表示为幂级数，并通过底层分布的变换来实现这一点。我们认为功能的修正，以及Gumbel极限，表明他们可以通过泰勒展开。我们的方法也改善了对偏离均值极值的大偏差的描述。此外，它有助于在基础分布未知时（例如拟合实验数据时）表征极值统计。
英文摘要：
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.
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