Testing epidemic change in nearly nonstationary process with statistics based on residuals

Abstract

We study an epidemic type change in innovations of a first order autoregressive process $y_{n,k}={\phi }_n{y}_{n,k-1}+{\epsilon }_k+a_{n,k}$, where ${\varphi }_n$ is either a constant in $(-1,1)$ or a sequence in $(0,1)$, converging to 1. For $k$ inside some unknown interval ${\mathbb{I}}_n^{\ast }=(k^{\ast },k^{\ast }+{\ell }^{\ast }]$, $a_{n,k}=a_n$ while $a_{n,k}=0$ for $k$ outside ${\mathbb{I}}_n^{\ast }$. When $a_n\ne 0$, we have an epidemic deviation from the usual (zero) mean of innovations. Since innovations are not observed, we build uniform increments statistics on residuals $({\hat{\epsilon }}_k)$ of the process $y_{n,k}$. We assume that innovations $({\epsilon }_k)$ are regularly varying with index $p\ge 2$ or satisfies integrability condition ${\lim }_{t\to \infty }{t}^{p}P(|{\epsilon }_1|>t)=0$ for $p>2$ and $E{\epsilon }_k^2<\infty$ for $p=2$. We find the limit distributions of the tests under no change and prove consistency under short epidemics that is ${\ell }^{\ast }=O(n^{\beta })$ for some $0<\beta \le 1/2$.