The Asymptotic Distribution for a Single Joinpoint Changepoint Model

TL;DR

This paper derives the exact asymptotic distribution of a test statistic for detecting a single changepoint in a time series, providing theoretical insights and quantiles for practical use.

Contribution

It provides the first derivation of the asymptotic distribution for the changepoint existence test statistic in a single joinpoint model.

Findings

Derived the asymptotic distribution as a supremum of a Gaussian process.

Provided quantiles of the distribution for practical application.

Addressed a subtle gap in changepoint statistical theory.

Abstract

A single joinpoint changepoint model partitions a time series into two segments, joined at the changepoint time by constraining the estimated piecewise linear regression responses to be continuous. This manuscript derives the exact asymptotic distribution of the changepoint existence test statistic gauging whether or not a second segment is necessary. The identified asymptotic distribution, a supremum of a Gaussian process over the unit interval, is rather unwieldy. The work presented here provides the result and its derivation; quantiles of the asymptotic distribution are presented for the user. This addresses a subtle gap in the changepoint literature.