超几何重尾族中的高斯边界推断
Gaussian Boundary Inference in a Hypergeometric Heavy-Tailed Family
- ENAC (University of Toulouse)（国立民用航空学院（图卢兹大学））
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对嵌套高斯的超几何重尾分布族提出推断方法，建立可容许性、推导最小距离估计量及边界渐近，模拟验证功效，并应用于S&P 500收益率，发现GARCH滤波减少但未消除重尾偏离。
AI中文摘要：
本文针对一个嵌套高斯的超几何分布函数族发展了推断方法。该族定义为 \\[ G_c(z) = \frac12 + z\\,\frac{\Gamma(c-1/2)}{2\sqrt2\\,\Gamma(c)}\\,{}_1F_1\\!\left(\frac12;c;-\frac{z^2}{2}\right),\quad c\ge\frac32, \\] 在边界 $c=3/2$ 处包含标准正态分布。在远离边界处，密度具有代数尾部行为 $g_c(z)\sim(c-3/2)|z|^{-3}$，因此参数 $c$ 索引了高斯分布的一种有向重尾变形。该分布还等价于一个 beta 精度正态尺度混合表示，其中高斯边界对应于退化的单位精度。我建立了该超几何族的可容许性，推导了最小距离估计量，并获得了在正态原假设下的正则内部渐近和非标准边界渐近。标准化后的拟合改进统计量收敛于混合分布 $\tfrac12\delta_0+\tfrac12\chi_1^2$。我将该理论扩展到插入位置-尺度程序，包括稳健的中位数/IQR 版本。模拟实验记录了准确的零假设尺寸和对重尾备择假设的有向功效。对每日 S&P 500 收益率的应用（无条件及 GARCH(1,1) 滤波后）展示了在尾部拟合和风险分位数方面的实证意义，并记录了 GARCH 滤波显著减少但并未消除检验所检测到的对称重尾偏离。
英文摘要：
This paper develops inference for a Gaussian-nested hypergeometric family of distribution functions. The family \[ G_c(z) = \frac12 + z\,\frac{Γ(c-1/2)}{2\sqrt2\,Γ(c)}\,{}_1F_1\!\left(\frac12;c;-\frac{z^2}{2}\right),\quad c\ge\frac32, \] contains the standard normal distribution at the boundary $c=3/2$. Away from the boundary, the density has algebraic tail behaviour $g_c(z)\sim(c-3/2)|z|^{-3}$, so the parameter $c$ indexes a directed heavy-tailed deformation of the Gaussian law. The distribution also admits an equivalent beta-precision normal scale-mixture representation, in which the Gaussian boundary corresponds to degenerate unit precision. I establish the admissibility of the hypergeometric family, derive minimum-distance estimators, and obtain both regular interior asymptotics and nonstandard boundary asymptotics under the normal null. The standardised fit-improvement statistic converges to the mixture distribution $\tfrac12δ_0+\tfrac12χ_1^2$. I extend the theory to plug-in location-scale procedures, including a robust median/IQR version. Simulations document accurate null size and directed power against heavy-tailed alternatives. An application to daily S&P 500 returns, both unconditionally and after GARCH(1,1) filtering, illustrates the empirical implications for tail fitting and risk quantiles, and documents that GARCH filtering substantially reduces but does not eliminate the symmetric heavy-tailed departure detected by the test.
