Jump or kink: on super-efficiency in segmented linear regression breakpoint estimation

Yining Chen

doi:10.1093/biomet/asaa049

Abstract

Summary We consider the problem of segmented linear regression with a single breakpoint, with the focus on estimating the location of the breakpoint. If $n$ is the sample size, we show that the global minimax convergence rate for this problem in terms of the mean absolute error is $O(n^{-1/3})$. On the other hand, we demonstrate the construction of a super-efficient estimator that achieves the pointwise convergence rate of either $O(n^{-1})$ or $O(n^{-1/2})$ for every fixed parameter value, depending on whether the structural change is a jump or a kink. The implications of this example and a potential remedy are discussed.

Highlights

Asymptotic analysis is commonly used to facilitate comparison between different statistical estimators from a frequentist’s perspective
The pointwise rate, the uniform rate and the minimax rate are the same, in which case the corresponding estimator is usually regarded as rate optimal
By focusing on estimating the location of the single breakpoint and taking the loss function to be the Euclidean distance between the true location and estimated location of the breakpoint, we show that the global minimax convergence rate of the risk is at least O(n−1/3), i.e., lim inf inf sup n1/3R(θ, θ) > 0. n→∞ θθ ∈

Summary

We consider the problem of segmented linear regression with a single breakpoint, with the focus on estimating the location of the breakpoint. If n is the sample size, we show that the global minimax convergence rate for this problem in terms of the mean absolute error is O(n−1/3). We demonstrate the construction of a super-efficient estimator that achieves the pointwise convergence rate of either O(n−1) or O(n−1/2) for every fixed parameter value, depending on whether the structural change is a jump or a kink. The implications of this example and a potential remedy are discussed. Some key words: Changepoint; Minimax rate; Pointwise rate; Structural break

Introduction

Minimax rate of convergence

A super-efficient estimator

Numerical experiment

Discussion