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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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