Testing simple or composite hypothesis on a functional parameter has attracted considerable attention in time series analysis. To accommodate for the unknown temporal dependence, classical nonparametric approaches such as block bootstrapping and subsampling all involve a bandwidth parameter, the choice of which can substantially affect the finite sample performance. The self normalization (SN) method is tuning parameter free when applied to the inference of a finite-dimensional parameter but its applicability to a functional parameter is unknown. In this paper, we propose a sample splitting based approach to generalize the SN method to hypothesis testing of a functional parameter. Our SS-SN (sample splitting plus self-normalization) idea is broadly applicable to many testing problems for functional parameters, including testing for simple/composite hypothesis on marginal cumulative distribution function, testing for time-reversibility and testing for a change point on the spectral distribution of a multivariate time series. Specifically, we derive the pivotal limiting distributions of our SS-SN test statistics under the null for both simple and composite null hypothesis, and derive the limiting power function under the local alternatives. Numerical simulations show that our new tests tend to yield accurate size with competitive power performance as compared to many existing ones.

翻译：对函数参数进行简单或复合假设检验在时间序列分析中引起了广泛关注。为适应未知的时间依赖性，经典的半参数方法（如块自助法和子抽样法）均涉及带宽参数，其选择会显著影响有限样本表现。自正则化方法在应用于有限维参数推断时无需调参，但其对函数参数的适用性尚不明确。本文提出一种基于样本分割的方法，将自正则化方法推广至函数参数的假设检验。我们的SS-SN（样本分割+自正则化）思想广泛适用于函数参数的多种检验问题，包括对边际累积分布函数的简单/复合假设检验、时间可逆性检验以及多元时间序列谱分布变点检验。具体而言，我们推导了在简单与复合原假设下SS-SN检验统计量的渐近枢轴分布，并获得了局部备择假设下的渐近势函数。数值模拟表明，与现有多种检验方法相比，我们的新方法在保持可控检验水平的同时，具有具有竞争力的检验功效。