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

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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.