Theoretical guarantees for change localization using conformal p-values

TL;DR

This work develops a unified distribution-free framework for changepoint inference by analyzing and extending conformal p-values to distribution-change settings. It provides rigorous finite-sample guarantees for changepoint localization via the MCP algorithm, along with a consistent, parametric-free estimator of the changepoint and distribution-free tests for both changepoint presence and exchangeability. A Neyman–Pearson-type principle yields near-optimal score functions for conformal p-values, implemented via kernel density estimates on data split by the estimated changepoint. Theoretical results are complemented by simulation studies that validate finite-sample validity, shrinking confidence-set length, and robust performance across mean and non-mean changes. Collectively, the paper broadens the applicability and credibility of conformal inference in modern changepoint analysis with distribution-free guarantees.

Abstract

Changepoint localization aims to provide confidence sets for a changepoint (if one exists). Existing methods either relying on strong parametric assumptions or providing only asymptotic guarantees or focusing on a particular kind of change(e.g., change in the mean) rather than the entire distributional change. A method (possibly the first) to achieve distribution-free changepoint localization with finite-sample validity was recently introduced by \cite{dandapanthula2025conformal}. However, while they proved finite sample coverage, there was no analysis of set size. In this work, we provide rigorous theoretical guarantees for their algorithm. We also show the consistency of a point estimator for change, and derive its convergence rate without distributional assumptions. Along that line, we also construct a distribution-free consistent test to assess whether a particular time point is a changepoint or not. Thus, our work provides unified distribution-free guarantees for changepoint detection, localization, and testing. In addition, we present various finite sample and asymptotic properties of the conformal $p$-value in the distribution change setup, which provides a theoretical foundation for many applications of the conformal $p$-value. As an application of these properties, we construct distribution-free consistent tests for exchangeability against distribution-change alternatives and a new, computationally tractable method of optimizing the powers of conformal tests. We run detailed simulation studies to corroborate the performance of our methods and theoretical results. Together, our contributions offer a comprehensive and theoretically principled approach to distribution-free changepoint inference, broadening both the scope and credibility of conformal methods in modern changepoint analysis.

Figures (8)

• Figure 1: $p$-values produced by the MCP Algorithm (with the score function $s(z) = z$) for $N(0,1)$ pre-change and $N(0,5)$ post-change distribution on a sample of size $n = 4000$ and true changepoint at $\tau_n = 0.5n$(marked by red line). In the presence of a distribution change, the $p$-values exhibit a sharp spike-type pattern around the true changepoint.
• Figure 2: $p$-values produced by the MCP Algorithm (with the score function $s(z) = z$) for a sample of size $200$ exchangeably drawn from $N(0,1)$(left) and Exp$(1)$(right). This figure demonstrates \ref{['prop:pval exchan']} and illustrates the existence of atoms in the marginal distribution of the $p$-values under exchangeability of the whole sample.
• Figure 3: Relative length of the CI of MCP Algorithm for mentioned pre and post-change distributions, sample size $n$, change-point at $\tau_n = \frac{n}{2}$ i.e. $c = 0.5$, significance level $\alpha = 0.05$ and score function $s(z) = z$. For all the cases, the relative length is seen to converge to $0$, as guaranteed in \ref{['th: rel len CONCH']}.
• Figure 4: Ratio of estimated change-point ($\widehat{\tau}_n$) and true change-point ($\tau_n = cn, c = 0.5$) for mentioned pre and post-change distributions with $n$ to be the total sample size. As the sample size increases, the ratio $\frac{\widehat{\tau}_n}{\tau_n}$ is seen to converge to $1$, as guaranteed in \ref{['th: rel len CONCH']}.
• Figure 5: $p$-values of the MCP Algorithm for sample of size $n = 400$(above), $n = 4000$(below), changepoint at $\tau_n = \frac{n}{2}$ i.e. $c = 0.5$, pre and post-change distributions $\mathcal{N}(0,1)$ and $\mathcal{N}(5,1)$ respectively, for identity and oracle score functions. The oracle score function yields a narrower confidence set for $\tau_n$ as compared to the identity score function. As the sample size increases, the confidence set size decreases.

Theorems & Definitions (22)

• Definition 1
• Definition 2: consistency of changepoint estimator
• Definition 3: Consistency of a sequence of test function
• Definition 4: Score Functions
• Proposition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5