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Linear and quadratic serial rank tests for randomness against serial dependence. (English) Zbl 0661.62091
The \(ARMA(p_ 1,p_ 2)\) model \(X_ t-n^{-1/2}\sum^{p_ 1}_{i=1}a_ i X_{t-i}=\epsilon_ t+n^{-1/2}\sum^{p_ 2}_{i=1}b_ i \epsilon_{t-i}\) is considered, where \(\underset \tilde{} a=(a_ 1,...,a_{p_ 1})'\) and \(\underset \tilde{} b=(b_ 1,...,b_{p_ 2})'\) are parameters and \(\epsilon_ t\) is white noise with density f. \(\underset \tilde{} X^{(n)}=(X_ 1^{(n)},...,X_ n^{(n)})'\) is the vector of \(observations.\)
H\({}_ 0^{(n)}\) is the family of distributions of all i.i.d. n-tuples \(\underset \tilde{} X^{(n)}\), and \(H^{(n)}_{\underset \tilde{} d,f}\) are the ARMA alternatives specified by a density f and a vector \(\underset \tilde{} d=(d_ 1,...,d_ p)'\) such that \(d_ i=a_ i+b_ i\), \(p=\max (p_ 1,p_ 2)\). \(H_ 0^{(n)}\) and \(H^{(n)}_{d,f}\) are contiguous, and the paper considers testing the null hypothesis of randomness against the ARMA alternatives. Let \(\underset \tilde{} R^{(n)}=(R_ 1^{(n)},...,R_ n^{(n)})'\) be the vector of ranks of \(\underset \tilde{} X^{(n)}\), and \[ S^{(n)}=(n-p)^{-1}\sum^{n}_{t=p+1}a_ n(R_ t^{(n)},...,R^{(n)}_{t-p}) \] a linear serial rank statistic defined in terms of a score function \(a_ n\). If \(\underset \tilde{} S^{(n)}\) is a vector of such statistics and \(\underset \tilde{} m^{(n)}\) its mean under \(H_ 0^{(n)}\), \(n^{1/2}(\underset \tilde{} S^{(n)}-\underset \tilde{} m^{(n)})\) could be used as a test statistic, and would be asymptotically N(\(\underset \tilde{} O,\underset \tilde{} V)\). Due to the difficulties with this procedure, the quadratic test statistic \[ n(\underset \tilde{} S^{(n)}-\underset \tilde{} m^{(n)})' V^{-2}(\underset \tilde{} S^{(n)}- \underset \tilde{} m^{(n)}) \] is considered: it is asymptotically \(\chi^ 2\), resp. noncentral \(\chi^ 2\), under the two hypotheses. A version of this procedure is proved to be asymptotically most powerful at level \(\alpha\) when \(\underset \tilde{} d\) and f are specified, and asymptotically maximum most powerful with \(\underset \tilde{} d\) unspecified, f specified.
For various choices of f these results are related to rank “portmanteau” statistics: van der Warden, Wilcoxon, Laplace, Spearman and Box-Pierce. A final section includes questions unanswered by this approach.
Reviewer: R.Mentz

MSC:
62G10 Nonparametric hypothesis testing
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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