Hermite谱与核分解用于高斯加权正态性检验
Hermite Spectra and Kernel Factorization for Gaussian-Weighted Tests of Normality
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- National and Kapodistrian University of Athens（雅典国立卡波迪斯特里安大学）
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中文总结 AI 辅助
本文研究高斯加权正态性检验的渐近零谱，证明特征值简单性、交替性及极点归属，并通过核分解和数值计算验证理论结果。
中文摘要 AI 辅助
我们研究了在位置和尺度被估计的情况下，单变量正态性的高斯加权检验的渐近零谱。闭合的Hermite系数给出了在未扰动极点之外和极点处的方程。完全标准化协方差的每个正特征值都是简单的，且奇偶特征值严格交替。对于除最大极点外的每个未扰动偶极点，恰好存在一个权重参数使得该极点属于扰动后的偶谱。这些参数严格有序并收敛到参数区间的边界。一个有符号的全正性论证更一般地证明了连续高斯协方差修正和正偶可积权重的简单性。我们还直接从标准化观测处的退化核推导出极限二次型。一个导数可和性条件控制了完整的谱尾，包括有符号核，并在具有有限四阶权重矩的特征函数核上得到验证。一个实Hilbert空间分解识别了观测谱和Fourier谱。数值计算评估了渐近校准。
英文摘要
We study the asymptotic null spectrum of Gaussian-weighted tests of univariate normality with estimated location and scale. Closed Hermite coefficients give equations both away from and at the unperturbed poles. Every positive eigenvalue of the fully standardized covariance is simple, and the odd and even eigenvalues alternate strictly. For each unperturbed even pole except the largest, there is exactly one weight parameter at which it belongs to the perturbed even spectrum. These parameters are strictly ordered and converge to the boundary of the parameter interval. A signed total-positivity argument proves simplicity more generally for consecutive Gaussian covariance corrections and positive even integrable weights. We also derive the limiting quadratic forms directly from degenerate kernels at standardized observations. A derivative summability condition controls the complete spectral tail, including signed kernels, and is verified for characteristic-function kernels with a finite fourth weight moment. A real Hilbert-space factorization identifies the observation and Fourier spectra. Numerical calculations assess asymptotic calibration.
