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

An, Hong-Zhi; Cheng, Bing

Int. Stat. Rev. 59, No. 3, 287-307 (1991).

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)

MSC:

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

Keywords:

Brownian motion; stochastic weighting empirical processes; strong mixing sequence; time series; stationary ergodic sequence; conditional expectation; stationary martingale difference; testing nonlinearity; Kolmogorov-Smirnov type statistic; autoregressive model; simulation studies