A Kolmogorov-Smirnov type statistic with application to test for nonlinearity in time series. (English) Zbl 0748.62049

The authors consider the model \(w_ t=\psi(w_{t-1})+\varepsilon_ t\) for a stationary ergodic sequence \(\{w_ t\}\), where \(\psi(x)\) is the conditional expectation of \(w_ t\) given \(x\), and \(\varepsilon_ t\) is a stationary martingale difference, and they discuss the problem of testing nonlinearity of \(\psi(x)\) by using samples.
For finding the significance of nonlinearity of \(\psi(x)\), they introduce a Kolmogorov-Smirnov type statistic \(K_ n\) and show some asymptotic statistical properties of \(K_ n\) when \(\psi(x)\) is linear or nonlinear. Furthermore they consider a more general autoregressive model \(w_ t=\psi(w_{t-1},\dots,w_{t-p})+\varepsilon_ t\), propose Kolmogorov- Smirnov type statistics and mention some asymptotic properties of the statistics. Finally they show some simulation studies and comparisons with other methods.
Reviewer: M.Huzii (Tokyo)


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
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