An outlier test for linear processes (Q685769)

Let \(Y_ 1,\ldots,Y_ n\) be an undisturbed linear process. An observed process \(X_ 1,\ldots,X_ n\) coincides with \(Y_ 1,\ldots,Y_ n\) except for some outliers. Let \(s_ n\) be an upper bound for the total number of outliers. The authors consider an outlier test based on the statistic \[ T_ n = \max_{1 \leq \mu(1)<\cdots< \mu(s_ n) \leq n} \sum^{s_ n}_{i=1} \bigl( X_{\mu(i)} -\hat Y_{\mu(i)} \bigr)^ 2 \] where \(\hat Y_ t\) is a predictor of \(Y_ t\). It is assumed that \(s_ n \to \infty\), \(s_ n/n \to 0\) and that the distribution of the squared standardized white noise belongs to a subset of the domain of attraction of the Gumbel distribution; the Gaussian white noise is included. The asymptotic distribution of \(T_ n\) under the null hypothesis `no outlier' is derived.