Estimation, inference and model selection for jump regression models

2026-02-25

AI Summary

This study addresses the challenge of simultaneously unknown jump locations and segment levels in piecewise constant regression. The authors propose an integrated estimation framework that combines least squares with Bayesian inference and introduce two novel information criteria—AJIC and BJIC—specifically designed for selecting the number of change points in jump models. Theoretical analysis establishes a convergence rate of order $n$ for jump locations and $\sqrt{n}$ for level parameters. The proposed Bayesian approach demonstrates superior estimation accuracy compared to conventional methods, while the new criteria effectively identify the optimal jump configuration, enabling both efficient parameter estimation and reliable statistical inference.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationMachine Learning: Bayesian LearningSearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We consider regression models with data of the type $y_i=m(x_i)+\varepsilon_i$, where the $m(x)$ curve is taken locally constant, with unknown levels and jump points. We investigate the large-sample properties of the minimum least squares estimators, finding in particular that jump point parameters and level parameters are estimated with respectively $n$-rate precision and $\sqrt{n}$-rate precision, where $n$ is sample size. Bayes solutions are investigated as well and found to be superior. We then construct jump information criteria, respectively AJIC and BJIC, for selecting the right number of jump points from data. This is done by following the line of arguments that lead to the Akaike and Bayesian information criteria AIC and BIC, but which here lead to different formulae due to the different type of large-sample approximations involved.

Problem

• jump regression
• model selection
• least squares estimation
• Bayesian inference
• information criteria

Innovation

• jump regression
• least squares estimation
• Bayesian inference
• model selection
• information criteria