Modelling and structural break detection using nonparametric methods

Abstract

Understanding the behavior of time-evolving systems is central to contemporary data science, with wide-ranging applications in domains such as finance, biology and environmental studies. These systems are often high-dimensional, nonlinear, and governed by complex internal dynamics as well as external shocks. Time series data offer a natural lens through which to understand and model such systems. However, conventional modeling approaches, especially parametric methods, are often inadequate in capturing the nuanced and structurally evolving nature of real-world data. This thesis develops a set of novel nonparametric tools aimed at advancing statistical inference in such dynamic and complex settings.

A primary focus of the thesis is the detection of structural breaks- abrupt changes in the underlying relationship between a pair of time-dependent variables. Classical break detection methods typically rely on strong parametric assumptions and are often limited to changes in mean or variance. In contrast, this work introduces a fully nonparametric, data-driven methodology capable of identifying changes in more intricate functional relationships, including nonlinear and time-varying patterns. The proposed approach employs adaptive smoothing techniques within a minimax optimal framework, ensuring robust performance even under model misspecification and irregular error structures. Simulation studies and real-world applications in economics and finance demonstrate the method’s ability to uncover structural changes that remain undetected by conventional techniques. The second component of the thesis explores the use of shape-constrained nonparametric regression, motivated by the observation that many real-world processes exhibit known structural properties, such as monotonicity, convexity, or quasiconvexity. Incorporating these constraints into a nonparametric framework allows for more interpretable and stable estimation while preserving model flexibility. The proposed estimators are shown to be statistically consistent and particularly effective in settings with limited data or high noise levels.

The thesis also addresses a critical but often overlooked problem: testing the adequacy of ordinary differential equation (ODE) models used to describe dynamic systems. Existing lack-of-fit tests are typically not well suited for ODEs, as they fail to consider the structural constraints imposed by the differential formulation. This work proposes a novel adaptive goodness-of-fit test for both fully and partially observed ODE systems. The test is constructed to directly assess deviations in the trajectories and derivative structures implied by the ODE, offering a rigorous and sensitive diagnostic tool. The methodology is supported by theoretical results guaranteeing optimal detection rates, and its practical utility is confirmed through simulations and applications in biomedical and agricultural data.

Collectively, the contributions of this thesis advance the field of nonparametric inference for dynamic systems. By developing methodologies that are robust, adaptive, and theoretically grounded, this work addresses key limitations of existing approaches and provides practical tools for analyzing complex, real-world time series. The results have broad applicability across disciplines where understanding structural change, validating mechanistic models, and incorporating qualitative prior knowledge are essential for reliable statistical analysis.