We develop a nonparametric two-sample test for distributions supported on the cone of symmetric positive definite matrices. The procedure relies on the Wishart kernel density estimator (KDE) introduced by Belzile et al. (2025), whose support-adaptive kernel alleviates boundary bias by remaining confined to the cone. Our test statistic is the rescaled integrated squared difference between two Wishart KDEs and can be expressed as a two-sample $V$-statistic via an explicit closed-form overlap of Wishart kernels, avoiding numerical integration. Under the null hypothesis of equal densities, we derive the asymptotic distribution in both the common shrinking-bandwidth and fixed-bandwidth regimes. The proposed method provides a kernel-based competitor to the empirical Laplace-transform two-sample test of Lukić (2024). Unlike the orthogonally invariant Hankel-transform test of Lukić and Milošević (2024), our statistic can detect alternatives that differ only through eigenvector structure, for instance, Wishart models with the same shape parameter and the same scale eigenvalues but different orientations.

翻译：本文针对支撑在对称正定矩阵锥上的分布，提出了一种非参数双样本检验方法。该过程依赖于Belzile等人（2025）提出的Wishart核密度估计器，其支撑自适应的核函数通过始终限制在该锥体内，有效缓解了边界偏差。我们的检验统计量是两个Wishart核密度估计器之间经重标定的积分平方差，并可通过Wishart核显式闭式重叠公式表示为双样本$V$统计量，从而避免了数值积分。在密度相等的原假设下，我们推导了其分别在公共收缩带宽与固定带宽机制下的渐近分布。所提出的方法为Lukić（2024）的经验拉普拉斯变换双样本检验提供了基于核函数的竞争方案。与Lukić和Milošević（2024）的正交不变Hankel变换检验不同，我们的统计量能够检测仅通过特征向量结构产生差异的备择假设，例如具有相同形状参数、相同尺度特征值但不同取向的Wishart模型。