Estimating a gradual parameter change in an AR(1)-process. (English) Zbl 07575564
Summary: We discuss the estimation of a change-point \(t_0\) at which the parameter of a (non-stationary) AR(1)-process possibly changes in a gradual way. Making use of the observations \(X_1,\ldots, X_n\), we shall study the least squares estimator \(\widehat{t_0}\) for \(t_0\), which is obtained by minimizing the sum of squares of residuals with respect to the given parameters. As a first result it can be shown that, under certain regularity and moment assumptions, \(\widehat{t_0} /n\) is a consistent estimator for \(\tau_0\), where \(t_0 =\lfloor n\tau_0 \rfloor\), with \(0<\tau_0 <1\), i.e., \(\widehat{t_0} /n \,\overset{P}{\rightarrow}\,\tau_0 (n\rightarrow \infty)\). Based on the rates obtained in the proof of the consistency result, a first, but rough, convergence rate statement can immediately be given. Under somewhat stronger assumptions, a precise rate can be derived via the asymptotic normality of our estimator. Some results from a small simulation study are included to give an idea of the finite sample behaviour of the proposed estimator.
MSC:
| 62F10 | Point estimation |
| 62F12 | Asymptotic properties of parametric estimators |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
Keywords:
AR(1)-process; gradual change; change-point estimator; consistency; convergence rate; asymptotic normalityReferences:
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