发表机构
ENAC (University of Toulouse)（国立民航工程学院（图卢兹大学））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用球面几何推导Jarque-Bera统计量的精确有限样本分布，给出最大可达值、显式密度及椭圆积分表示，并揭示检验水平为零的样本量界限。
AI 中文摘要
在高斯抽样下，去除位置和尺度参数后，样本转化为一个在 $n-2$ 维球面上均匀分布的方向，样本偏度和峰度成为球面测度的三次-四次多项式映射。我们利用这一几何结构获得了 Jarque-Bera 统计量 $\mathrm{JB}_n$ 的精确有限样本结果。首先，对于每个 $n\ge 3$，$\mathrm{JB}_n$ 的最大可达值为 $n\{(n-2)^4+4(n-1)^2\}/\{24(n-1)^2\}$，该值由单离群样本达到，因此当 $n\le 6$ 时渐近 $10\\%$ 和 $5\\%$ 检验的检验水平为零，当 $n\le 7$ 时 $1\\%$ 检验的检验水平为零；在该最大值附近，$\mathrm{JB}_n$ 的密度表现为 $(J_n^+-x)^{(n-4)/2}$ 的显式倍数。其次，从残差坐标转换到幂和，将偏度和峰度的联合密度写为多项式判别式平方根倒数的积分；对于每个 $n\ge 5$，最内层积分在单个区间上进行，且是 Lauricella $F_D$ 周期。第三，这产生了显式分布：$n=3$ 时的反正弦分布，$n=4$ 时的代数联合密度，以及 $n=5$ 时的单个完全椭圆积分，等价于 ${}_2F_1(\tfrac12,\tfrac12;1;\cdot)$。残差坐标的碰撞产生了联合分布的判别式奇异性，而 $\mathrm{JB}_n$ 在残差球面上的驻点控制其一维密度的奇异性。
英文摘要
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 $\mathrm{JB}_n$. First, for every $n\ge 3$ the largest attainable value of $\mathrm{JB}_n$ 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 $n\le 6$ and the $1\%$ test for $n\le 7$; near this maximum the density of $\mathrm{JB}_n$ 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 $n\ge 5$ 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 ${}_2F_1(\tfrac12,\tfrac12;1;\cdot)$, for $n=5$. Residual-coordinate collisions generate the discriminant singularities of the joint law, while stationary points of $\mathrm{JB}_n$ on the residual sphere govern the singularities of its one-dimensional density.
Comments14 pages, 2 figures, 2 tables
