Exact finite-sample geometry of the Jarque-Bera statistic: support, discriminants and elliptic integrals
Abstract
Under Gaussian sampling, removing location and scale turns the sample into a direction that is uniformly distributed on a sphere of dimension n−2, and sample skewness and kurtosis become a cubic-quartic polynomial image of spherical measure. We use this geometry to obtain exact finite-sample results for the Jarque-Bera statistic JBn. First, for every n≥ 3 the largest attainable value of JBn is n{(n−2)4+4(n−1)2}/{24(n−1)2}, attained by a single-outlier sample, so the asymptotic 10% and 5% tests have size zero for n≤ 6 and the 1% test for n≤ 7; near this maximum the density of JBn behaves like an explicit multiple of (Jn+−x)(n−4)/2. Second, passing from residual coordinates to power sums writes the joint density of skewness and kurtosis as an integral of the reciprocal square root of a polynomial discriminant; for every n≥ 5 the innermost integral runs over a single interval and is a Lauricella FD period. Third, this yields explicit laws: an arcsine law for n=3, an algebraic joint density for n=4, and a single complete elliptic integral, equivalently 2F1(1/2,1/2;1;·), for n=5. Residual-coordinate collisions generate the discriminant singularities of the joint law, while stationary points of JBn on the residual sphere govern the singularities of its one-dimensional density.
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BibTeX
@misc{lawford2026exact,
title = {{Exact finite-sample geometry of the Jarque-Bera statistic: support, discriminants and elliptic integrals}},
author = {Steve Lawford},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29976},
eprint = {2609.29976},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29976},
}
RIS
TY - UNPB
TI - Exact finite-sample geometry of the Jarque-Bera statistic: support, discriminants and elliptic integrals
AU - Steve Lawford
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29976
UR - https://arxiv.org/abs/2609.29976
AB - Under Gaussian sampling, removing location and scale turns the sample into a direction that is uniformly distributed on a sphere of dimension n−2, and sample skewness and kurtosis become a cubic-quartic polynomial image of spherical measure. We use this geometry to obtain exact finite-sample results for the Jarque-Bera statistic JB_n. First, for every n≥ 3 the largest attainable value of JB_n is n{(n−2)^4+4(n−1)^2}/{24(n−1)^2}, attained by a single-outlier sample, so the asymptotic 10% and 5% tests have size zero for n≤ 6 and the 1% test for n≤ 7; near this maximum the density of JB_n behaves like an explicit multiple of (J_n^+−x)^(n−4)/2. Second, passing from residual coordinates to power sums writes the joint density of skewness and kurtosis as an integral of the reciprocal square root of a polynomial discriminant; for every n≥ 5 the innermost integral runs over a single interval and is a Lauricella F_D period. Third, this yields explicit laws: an arcsine law for n=3, an algebraic joint density for n=4, and a single complete elliptic integral, equivalently _2F_1(1/2,1/2;1;·), for n=5. Residual-coordinate collisions generate the discriminant singularities of the joint law, while stationary points of JB_n on the residual sphere govern the singularities of its one-dimensional density.
ER -