Master's Thesis Defense: Change-Point Detection for Conditional Distributions via Kernel and Neural Network Methods

We study the problem of detecting change-points in time-varying conditional distributions from observations consisting of covariates and responses collected over time. The goal is to identify time points at which the conditional distribution of the response given the covariates undergoes structural changes. A natural approach is to estimate the conditional distribution at each time and combine these estimates with a cumulative sum (CUSUM) procedure to detect changes. We observe that kernel estimators based on local normalization admit an interpretation as solutions to localized least-squares projection problems. This provides a structural perspective for conditional distribution estimation. Motivated by this observation, we propose a unified projection-based framework that extends localized smoothing to empirical projection onto general function classes. This formulation encompasses kernel methods as a special case while allowing for more flexible estimators, including those based on feedforward neural networks. Building on this framework, we construct a CUSUM-type statistic and develop a seeded binary segmentation procedure for multiple change-point detection. We further consider a stabilized kernel estimator based on global normalization and establish its consistency under suitable conditions. Simulation studies and real data applications demonstrate that the proposed framework achieves strong performance, particularly in settings with complex and nonlinear dependence on covariates.

Thesis Advisor: Carlos Misael Madrid Padilla