Statistical Inference for Multiple Change Points and Implicit Network Structures

This dissertation contains two research projects: Multiple Change Point Detection in Linear Models and Statistical Inference for Implicit Network Structures. In the first project, a new method to detect the number and locations of change points in piecewise linear models under stationary Gaussian noise is proposed. The method transforms the problem of detecting change points to the detection of local extrema by kernel smoothing and differentiating the data sequence. The change points are detected by computing the p-values for all local extrema using the derived peak height distributions of smooth Gaussian processes, and then applying the Benjamini-Hochberg procedure to identify significant local extrema. Theoretical results show that the method can guarantee asymptotic control of the False Discover Rate (FDR) and power consistency, as the length of the sequence, and the size of slope changes and jumps get large. In addition, compared to traditional methods for change point detection based on recursive segmentation, The proposed method tests the candidate local extrema only one time, achieving the smallest computational complexity. Numerical studies show that the properties on FDR control and power consistency are maintained in non-asymptotic cases. In the second project, identifiability and estimation consistency under mild conditions in hub model are proved. Hub Model is a model-based approach, introduced by Zhao and Weko (2019), to infer implicit network structuress from grouping behavior. The hub model assumes that each member of the group is brought together by a member of the group called the hub. This paper generalize the hub model by introducing a model component that allows hubless groups in which individual nodes spontaneously appear independent of any other individual. The new model bridges the gap between the hub model and the degenerate case of the mixture model -- the Bernoulli product. Furthermore, a penalized likelihood approach is proposed to estimate the set of hubs when it is unknown.