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Large sample inference based on multiple observations from nonlinear autoregressive processes. (English) Zbl 0796.62074

Let \(\{Y_ t(i) \mid t = 1,\dots,n\}\) be a time series for individual \(i\), \(i = 1,\dots,m\). The nonlinear autoregressive model \(Y_ t(i) = H(Y_{t - 1}(i); \theta_ i) + \varepsilon_ t(i)\) is considered, where the \(\theta_ i\)’s are \(p \times 1\) vectors of unknown parameters, \(H(y;\theta)\) is a known function and \(\varepsilon_ t(i)\) is a sequence of iid random variables for each \(i\).
In section 2 of the paper the case “\(n \to \infty\), \(m\) fixed” is dealt with. The Wald statistic based on an asymptotically optimal one-step maximum likelihood estimator is proposed for testing the hypothesis \(H_ 0 : \theta_ 1 = \dots \theta_ m\). The null and non-null limiting distributions are deduced from the LAN property.
Section 3 studies a special threshold autoregressive model of order 1 for the case “\(n\) fixed, \(m \to \infty\)”. This section is mainly concerned with the derivation of the least squares estimator of the parameter vector and its limiting distribution. Section 4 discusses briefly some related topics.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F05 Asymptotic properties of parametric tests
62E20 Asymptotic distribution theory in statistics
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