论文标题
逐渐变化的偏差是否相关？基于SUP-NORM估计器的测试方法
Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators
论文作者
论文摘要
经典变更点分析的目的是（1）检测可能非平稳时间序列的平均值的突然变化，并在（2）识别平均值表现出分段恒定行为的区域。但是，在许多应用中，假设平均值以平稳的方式逐渐变化是更合理的。这些逐渐变化可能是非相关的（即小），也可以与当前特定问题相关，并且本文提出了统计方法来检测后者。更准确地说，我们考虑常见的非参数回归模型$ x_ {i} =μ（i/n） + \ varepsilon_ {i} $，可能是非平稳错误的，并提出了一个无效假设的测试，即回归$ g（例如ande $ g（your）$ g（例如，$ g（例如，$ g）的最大绝对偏差，例如，$ g（例如，$ g（例如）$（例如，$ g（例如，$ g）（$）$（$ g（use）$ g（useve $ g（use）$ g（useve $ g（use）$ g（例如，$）$（$ g（n） $ \ int_ {0}^{1}μ（t）dt $）小于给定间隔$ [x_ {0}，x_ {1}] \ subseteq [0,1] $上给定阈值。使用适当的估计量（例如$ \ hat d _ {\ infty，n} $）开发了对这种假设的测试，对于最大偏差$ d _ {\ infty} = \ sup_ {t \ in [x__ {0}，x_ {1}，x_ {1}}}} |μ（t） - t） - g（t） - g（t） - g（us） - $ $ $。我们得出了$ \ hat d _ {\ infty，n} $的适当标准化版本的限制分布，其中标准化取决于函数$μ（\ cdot）-g（μ）$的极端点的lebesgue度量。开发了基于该集合的估计的精制程序，并证明了其一致性。结果通过仿真研究和数据示例说明了结果。
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.
