Modeling and Optimal Inference in change-point models with ultra-high dimensional data

关于多元或张量设置中的变化点的建模和推理出现在统计应用的许多领域，例如金融市场、神经成像、计量经济学、社交网络分析和反恐。该提案涉及在随机微分方程 (SDE)、分数随机微分方程 (fSDE)、常微分方程 (ODE) 中开发此类方法。它由三个主要部分组成：受变化影响的 SDE 和 fSDE 模型中的鲁棒推理、具有变化点的高维张量回归中的改进推理以及通过 ODE 在动态系统中建模和推理。 在第 1 部分中，我将研究一些具有未知多个变化点的多元 SDE 中的推理问题，其中漂移参数可能满足一些限制。我将在 fSDE 背景下研究类似的推理问题。在这两种情况下，我都会考虑漂移系数不一定是线性的一般情况。因此，这些方法将适用于数据集由广义 Cox-Ingersoll-Ross 过程或 Ornstein-Uhlenbeck 过程生成的特殊情况。上述问题也将在政权转换的背景下进行研究。除了处理因变化点的存在和参数先验知识的不确定性而给这些模型带来的复杂性之外，我还计划针对估计量本身的维度是随机的情况开发渐近结果。 在第 2 部分中，当张量参数被怀疑满足某些限制时，我将考虑具有多个变化点的张量回归模型中的推理问题。我还将研究高维数据背景下的类似问题。与最近文献中的类似模型相比，我将放宽误差项的条件，使其不需要独立同分布。依赖结构最多是 mixale 的依赖结构。我还将考虑误差项是长记忆平稳过程的情况。  在第 3 部分中，我将考虑动态系统的随机版本的建模，例如生态或生物医学系统中出现的系统，通常由 ODE 建模。  我将首先获得 ODE 轨迹的初步非参数估计量，然后使用它们构建估计函数来推断 ODE 的参数。然后，这些推理工具将用于检测此类系统中的多个变化点。同样在这种情况下，我将把常用的误差独立性强假设放宽为较弱的混合依赖性。 例如，在生态系统中，所提出的模型预计会考虑一些现实因素，例如动物适应、迁徙和/或隐藏策略。

Modeling and inference about change-points in multivariate or tensor setups appear in many areas of statistical applications such as financial markets, neuroimaging, econometrics, social network analysis and counter-terrorism. This proposal is concerned with developing such methodologies in stochastic differential equations (SDE), fractional stochastic differential equations (fSDE), ordinary differential equations (ODEs). It is composed of three main parts: robust inference in SDE and fSDE models subject to changes, improved inference in high dimensional tensor regression with change-points, and modeling and inference in dynamical systems via ODEs. In Part 1, I will study inference problems in some multivariate SDE with unknown multiple change-points for which the drift parameters may satisfy some restrictions. I will study similar inference problems in context of  fSDE. In both cases, I will consider the general case where the drift coefficient is not necessarily linear. Thus, the methods will be applicable to special cases where the datasets are generated by the generalized Cox-Ingersoll-Ross processes or Ornstein-Uhlenbeck processes. The above problems will also be studied in the context of regime switching. In addition to dealing with complexities brought into these models by the presence of change-points and by the uncertainty in the prior knowledge about the parameters, I plan to develop asymptotic results for the situation where the dimensions of the estimators themselves are random. In Part 2, I will consider inference problems in tensor regression models with multiple change-points when the tensor parameter is suspected to satisfy some restriction. I will also study similar problems in the context of high dimensional data. As compared to similar models in recent literature, I will relax the conditions on the error term so that it does not need to be independent and identically distributed. The dependence structure will be at most that of mixingale. I will also consider the case where the error term is a long memory stationary process.  In Part 3, I will consider the modeling of stochastic versions of dynamical systems such as those appearing in ecological or biomedical systems which are commonly modeled by ODEs.  I will first obtain preliminary nonparametric estimators of the trajectory of the ODEs and then use them to construct estimating functions to make inferences about the parameters of the ODEs. These inferential tools will then be used to detect multiple change-points in such systems. Also in this case, I will relax the commonly used strong assumptions of independence of the errors to the weaker mixingale dependence. For instance, in ecological systems, the proposed models are expected to account for some realistic factors such as animal adaptation, migration and/or hiding strategies.