逐渐变化的平均值的偏差是否相关？一种基于超范数估计的检验方法
Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators
摘要：
经典变点分析的目的是（1）检测可能非平稳时间序列的均值的突变，以及（2）识别均值呈现分段恒定行为的区域。然而，在许多应用中，更合理的假设是平均值以平滑的方式逐渐变化。这些逐渐变化可能是不相关的（即，小），或与手头的具体问题有关，本文件提出了统计方法来检测后者。更确切地说，我们认为常见的非参数回归模型$X_{i} = μ(i/n) + \varepsilon_{i}$可能具有非平稳误差，并提出了零假设的最大绝对偏差的回归函数$μ$从功能$g (μ)$（如值$μ(0)$或积分$\int_{0}^{1} μ(t) dt$）小于给定的阈值在给定的间隔$[x_{0},x_{1}] \subseteq [0,1]$。使用适当的估计量（例如$\hat d_{\infty, n}$）对最大偏差$ d_{\infty}= \sup_{t \in [x_{0},x_{1}]} |μ(t) - g( μ) |$进行检验。我们推导出一个适当的标准化版本的$\hat d_{\infty,n}$，其中的标准化取决于勒贝格措施的极值点的功能$μ(\cdot)-g(μ)$的集合的极限分布。一个完善的过程的基础上估计这一套，并证明其一致性。通过仿真研究和数据实例说明了结果。
英文摘要：
Classical change point analysis aims at (1) detecting abrupt changes in the mean of a possibly non-stationary time series and at (2) identifying regions where the mean exhibits a piecewise constant behavior. In many applications however, it is more reasonable to assume that the mean changes gradually in a smooth way. Those gradual changes may either be non-relevant (i.e., small), or relevant for a specific problem at hand, and the present paper presents statistical methodology to detect the latter. More precisely, we consider the common nonparametric regression model $X_{i} = μ(i/n) + \varepsilon_{i}$ with possibly non-stationary errors and propose a test for the null hypothesis that the maximum absolute deviation of the regression function $μ$ from a functional $g (μ)$ (such as the value $μ(0)$ or the integral $\int_{0}^{1} μ(t) dt$) is smaller than a given threshold on a given interval $[x_{0},x_{1}] \subseteq [0,1]$. A test for this type of hypotheses is developed using an appropriate estimator, say $\hat d_{\infty, n}$, for the maximum deviation $ d_{\infty}= \sup_{t \in [x_{0},x_{1}]} |μ(t) - g( μ) |$. We derive the limiting distribution of an appropriately standardized version of $\hat d_{\infty,n}$, where the standardization depends on the Lebesgue measure of the set of extremal points of the function $μ(\cdot)-g(μ)$. A refined procedure based on an estimate of this set is developed and its consistency is proved. The results are illustrated by means of a simulation study and a data example.
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