通过非中心偏态 t 分布实现多元偏态和重尾数据的灵活建模：在肿瘤形状数据中的应用
Flexible Modeling of Multivariate Skewed and Heavy-Tailed Data via a Non-Central Skew t Distribution: Application to Tumor Shape Data
AI总结：
提出多元非中心偏态 t 分布，通过卡方变量缩放偏态正态向量实现灵活建模，推导理论性质并用蒙特卡洛近似进行估计，在乳腺癌肿瘤形状数据上表现优于标准替代模型。
摘要：
我们提出了一种多元非中心偏态 t（NCST）分布的灵活公式，通过用独立的卡方变量缩放偏态正态随机向量来定义。该构造通过同时允许非对称性和非中心性，扩展了经典的多元 t 族，为现有的偏态 t 模型提供了一种替代方案，这些现有模型通常依赖限制性假设以保持可处理性。我们推导了 NCST 分布的关键理论性质，包括其矩结构、仿射变换行为以及二次型的分布。由于缺乏闭式密度，我们实现了蒙特卡洛似然近似以进行最大似然估计，并通过模拟研究评估其性能。为展示实用性，我们将 NCST 模型应用于乳腺癌诊断数据，对肿瘤形状的多个特征进行建模。基于信息准则和视觉诊断，NCST 模型实现了更优的拟合，特别是在存在偏态和重尾的情况下，相比标准替代方案，包括多元正态、偏态正态和 Azzalini 的偏态 t 分布。我们的研究结果表明，NCST 分布为建模复杂多元数据提供了一个有用且可解释的选择，突显了似然推断、贝叶斯计算以及涉及非对称和非高斯依赖性应用的未来发展方向。
英文摘要：
We propose a flexible formulation of the multivariate non-central skew t (NCST) distribution, defined by scaling skew-normal random vectors with independent chi-squared variables. This construction extends the classical multivariate t family by allowing both asymmetry and non-centrality, which provides an alternative to existing skew t models that often rely on restrictive assumptions for tractability. We derive key theoretical properties of the NCST distribution, which includes its moment structure, affine transformation behavior, and the distribution of quadratic forms. Due to the lack of a closed-form density, we implement a Monte Carlo likelihood approximation to enable maximum likelihood estimation and evaluate its performance through simulation studies. To demonstrate practical utility, we apply the NCST model to breast cancer diagnostic data, modeling multiple features of tumor shape. The NCST model achieves a superior fit based on information criteria and visual diagnostics, particularly in the presence of skewness and heavy tails compared to standard alternatives, including the multivariate normal, skew normal, and Azzalini's skew $t$ distribution. Our findings suggest that the NCST distribution offers a useful and interpretable choice for modeling complex multivariate data, which highlights promising directions for future development in likelihood inference, Bayesian computation, and applications involving asymmetry and non-Gaussian dependence.
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