Consistency of the posterior distribution and MLE for piecewise linear regression

TL;DR

This paper establishes the weak consistency of the posterior distribution and Bayes estimator in a two-phase piecewise linear regression model with an unknown break-point, overcoming technical challenges due to non-differentiability.

Contribution

It introduces a regularised approach to handle non-differentiability and proves that the regularised and original models share the same asymptotic properties, ensuring consistency.

Findings

Proves weak consistency of the posterior distribution.

Demonstrates strong consistency for the regularised model.

Shows asymptotic equivalence between regularised and original models.

Abstract

We prove the weak consistency of the posterior distribution and that of the Bayes estimator for a two-phase piecewise linear regression mdoel where the break-point is unknown. The non-differentiability of the likelihood of the model with regard to the break- point parameter induces technical difficulties that we overcome by creating a regularised version of the problem at hand. We first recover the strong consistency of the quantities of interest for the regularised version, using results about the MLE, and we then prove that the regularised version and the original version of the problem share the same asymptotic properties.