Asymptotics for the Sequential Empirical Process and Testing for Distributional Change for Stationary Linear Models

Abstract

Detecting a change in the structure of a time series is a classical statistical problem. Here we consider a short memory causal linear process $X_i=\sum_{j=0}^\infty a_j\xi_{i-j}$, $i=1,\cdots,n$, where the innovations $\xi_i$ are independent and identically distributed and the coefficients $a_j$ are summable. The goal is to detect the existence of an unobserved time at which there is a change in the marginal distribution of the $X_i$'s. Our model allows us to simultaneously detect changes in the coefficients and changes in location and/or scale of the innovations. Under very simple moment and summability conditions, we investigate the asymptotic behaviour of the sequential empirical process based on the $X_i$'s both with and without a change-point, and show that two proposed test statistics are consistent. In order to find appropriate critical values for the test statistics, we then prove the validity of the moving block bootstrap for the sequential empirical process under both the hypothesis and the alternative, again under simple conditions. Finally, the performance of the proposed test statistics is demonstrated through Monte Carlo simulations.