Gaussian Boundary Inference in a Hypergeometric Heavy-Tailed Family
Abstract
This paper develops inference for a Gaussian-nested hypergeometric family of distribution functions. The family Gc(z) = 1/2 + z (Γ(c−1/2))/(2√2 Γ(c)) 1F1 (1/2;c;−z2/2), c≥3/2, contains the standard normal distribution at the boundary c=3/2. Away from the boundary, the density has algebraic tail behaviour gc(z)∼(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 1/2δ0+1/2χ12. 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.
Related
- Sub-Gaussian Confidence Intervals for Heavy-Tailed Data: Characterizing the Limits of Inference
- The first type of power mean bounds for the Gaussian hypergeometric function
- An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron
- Causal Inference for Heterogeneous Extreme Quantiles with Heavy-Tailed Outcomes
- Conventional Inference under Nonstationary Heavy Tailed Fractional Processes & Arrays
BibTeX
@misc{lawford2026gaussian,
title = {{Gaussian Boundary Inference in a Hypergeometric Heavy-Tailed Family}},
author = {Steve Lawford},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20393},
eprint = {2609.20393},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20393},
}
RIS
TY - UNPB
TI - Gaussian Boundary Inference in a Hypergeometric Heavy-Tailed Family
AU - Steve Lawford
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20393
UR - https://arxiv.org/abs/2609.20393
AB - This paper develops inference for a Gaussian-nested hypergeometric family of distribution functions. The family G_c(z) = 1/2 + z (Γ(c−1/2))/(2√2 Γ(c)) _1F_1 (1/2;c;−z^2/2), c≥3/2, contains the standard normal distribution at the boundary c=3/2. Away from the boundary, the density has algebraic tail behaviour g_c(z)∼(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 1/2δ_0+1/2χ_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.
ER -