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Abstract：
We propose a construction of simultaneous confidence bands (SCBs) for functional parameters over arbitrary dimensional compact domains using the Gaussian Kinematic formula of t-processes (tGKF). Although the tGKF relies on Gaussianity, we show that a central limit theorem (CLT) for the parameter of interest is enough to obtain asymptotically precise covering even if the observations are non-Gaussian processes. As a proof of concept we study the functional signal-plus-noise model and derive a CLT for an estimator of the Lipshitz-Killing curvatures, the only data-dependent quantities in the tGKF. We further discuss extensions to discrete sampling with additive observation noise using scale space ideas from regression analysis. Our theoretical work is accompanied by a simulation study comparing different methods to construct SCBs for the population mean. We show that the tGKF outperforms state-of-the-art methods with precise covering for small sample sizes, and only a Rademacher multiplier-t bootstrap performs similarly well. A further benefit is that our SCBs are computational fast even for domains of dimension greater than one. Applications of SCBs to diffusion tensor imaging (DTI) fibers (1D) and spatio-temporal temperature data (2D) are discussed.
摘要：
我们建议使用t过程的高斯运动学公式（tGKF）为任意维紧凑域上的功能参数构造同时置信带（SCB）。尽管tGKF依赖于高斯性，我们证明，即使观测值是非高斯过程，感兴趣参数的中心极限定理（CLT）也足以获得渐近精确的覆盖。作为概念证明，我们研究了功能信号加噪声模型，并得出了Lipshitz-Killing曲率估计器的CLT，tGKF中唯一的数据相关量。我们使用回归分析的尺度空间思想进一步讨论了具有加性观测噪声的离散采样的扩展。我们的理论工作伴随着一项模拟研究，比较了为人口均值构建SCB的不同方法。我们证明了tGKF优于最先进的方法，对小样本量具有精确的覆盖，并且只有Rademacher乘数-tbootstrap表现类似。另一个好处是，即使对于维度大于1的域，我们的SCB也可以快速计算。讨论了SCB在扩散张量成像（DTI）纤维（1D）和时空温度数据（2D）中的应用。
