Change-Points in Nonparametric Regression Analysis

Abstract

an otherwise smooth regression model are proposed. The assumptions needed are much weaker than those made in parametric models. The proposed estimators apply as well to the detection of discontinuities in derivatives and therefore to the detection of change-points of slope and of higher order curvature. The proposed estimators are based on a comparison of left and right one-sided kernel smoothers. Weak convergence of a stochastic process in local differences to a Gaussian process is established for properly scaled versions of estimators of the location of a change-point. The continuous mapping theorem can then be invoked to obtain asymptotic distributions and corresponding rates of convergence for change-point estimators. These rates are typically faster than n- 1/2. Rates of global LP convergence of curve estimates with appropriate kernel modifications adapting to estimated change-points are derived as a consequence. It is shown that these rates of convergence are the same as if the location of the change-point was known. The methods are illustrated by means of the well known data on the annual flow volume of the Nile river between 1871 and 1970. 1. Introduction. Nonparametric regression methods are usually applied in order to obtain a smooth fit of a regression curve without having to specify a parametric class of regression functions. Sometimes a generally smooth curve might contain an isolated discontinuity or change-point in the curve or in a (possibly higher order) derivative, and in many cases interest focuses on the occurrence of such change-points. In parametric approaches to the regression change-point problem, simple linear regressions before and after a possible change-point are assumed, and then the possibility of a discontinuity in the form of a jump or of a jump in the first derivative, or, equivalently, a slope change, is incorporated into the model; see for instance Hinkley (1969) and Brown, Durbin and Evans (1975). The analysis of change-points which describe sudden, localized changes typically occurring in economics, medicine and the physical sciences has recently found increasing interest. General smoothness assumptions, allowing for a large class of regression functions to be considered, seem to be more appropriate in a variety of applied problems than parametric modelling. An