We frame novelty detection on path space as a hypothesis testing problem with signature-based test statistics. Using transportation-cost inequalities of Gasteratos and Jacquier (2023), we obtain tail bounds for false positive rates that extend beyond Gaussian measures to laws of RDE solutions with smooth bounded vector fields, yielding estimates of quantiles and p-values. Exploiting the shuffle product, we derive exact formulae for smooth surrogates of conditional value-at-risk (CVaR) in terms of expected signatures, leading to new one-class SVM algorithms optimising smooth CVaR objectives. We then establish lower bounds on type-$\mathrm{II}$ error for alternatives with finite first moment, giving general power bounds when the reference measure and the alternative are absolutely continuous with respect to each other. Finally, we evaluate numerically the type-$\mathrm{I}$ error and statistical power of signature-based test statistic, using synthetic anomalous diffusion data and real-world molecular biology data.

翻译：我们将路径空间上的新颖性检测构建为基于签名检验统计量的假设检验问题。利用Gasteratos和Jacquier（2023）的传输成本不等式，获得了超出高斯测度范围的误报率尾部界限，该界限可扩展至具有光滑有界向量场的随机微分方程解律，从而得到分位数与p值的估计。通过利用洗牌积，我们推导出条件风险价值光滑代理量关于期望签名的精确公式，进而提出优化光滑条件风险价值目标的新型单类支持向量机算法。随后，针对具有有限一阶矩的备择假设建立了第二类错误下界，当参考测度与备择假设相互绝对连续时给出普适的功效界限。最后，我们使用合成异常扩散数据和真实世界分子生物学数据，对基于签名的检验统计量的第一类错误与统计功效进行了数值评估。