For statistical inference on an infinite-dimensional Hilbert space $\H $ with no moment conditions we introduce a new class of energy distances on the space of probability measures on $\H$. The proposed distances consist of the integrated squared modulus of the corresponding difference of the characteristic functionals with respect to a reference probability measure on the Hilbert space. Necessary and sufficient conditions are established for the reference probability measure to be {\em characteristic}, the property that guarantees that the distance defines a metric on the space of probability measures on $\H$. We also use these results to define new distance covariances, which can be used to measure the dependence between the marginals of a two dimensional distribution of $\H^2$ without existing moments. On the basis of the new distances we develop statistical inference for Hilbert space valued data, which does not require any moment assumptions. As a consequence, our methods are robust with respect to heavy tails in finite dimensional data. In particular, we consider the problem of comparing the distributions of two samples and the problem of testing for independence and construct new minimax optimal tests for the corresponding hypotheses. We also develop aggregated (with respect to the reference measure) procedures for power enhancement and investigate the finite-sample properties by means of a simulation study.

翻译：针对无矩条件的无穷维希尔伯特空间 $\H$ 上的统计推断问题，我们在 $\H$ 上的概率测度空间中引入了一类新的能量距离。所提出的距离由特征泛函差值的积分平方模相对于希尔伯特空间上的参考概率测度构成。我们建立了参考概率测度具备特征性（即保证该距离在 $\H$ 上概率测度空间中定义度量性质）的充分必要条件。同时，我们利用这些结果定义了新的距离协方差，可在无需矩存在性的情况下度量 $\H^2$ 上二维分布边际分量间的依赖性。基于这些新距离，我们发展了无需任何矩假设的希尔伯特空间值数据的统计推断方法，因此所提方法对有限维数据中的重尾分布具有稳健性。特别地，我们考虑了两样本分布比较问题与独立性检验问题，构建了相应假设的极小极大最优检验。此外，我们还开发了基于参考测度的聚合方法以增强检验功效，并通过模拟研究考察了有限样本性质。