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Nonlinear positive AR(2) processes. (English) Zbl 0714.62087

Summary: Let \(e_ t\) be a strict white noise such that \(e_ t>0\). Consider a process \(X_ t\) given by \[ X_ t=b_ 1h_ 1(X_{t-1})+b_ 2h_ 2(X_{t-2})+e_ t, \] where \(b_ 1\geq 0\), \(b_ 2\geq 0\) and \(h_ 1,h_ 2\) are nondecreasing positive functions. Let a realization \(X_ 1,...,X_ n\) be given. Denote \(b^*_ 1\), \(b^*_ 2\) the solution of the linear program \(\max (\beta_ 1+\beta_ 2)\) under the conditions \(\beta_ 1\geq 0\), \(\beta_ 2\geq 0\), \(\beta_ 1h_ 1(X_{t- 1})+\beta_ 2h_ 2(X_{t-2})\leq X_ t\) \((t=3,...,n)\). It is proved that \(b^*_ 1\) and \(b^*_ 2\) are strongly consistent estimators for \(b_ 1\) and \(b_ 2\), respectively.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G12 General second-order stochastic processes
90C05 Linear programming
90C90 Applications of mathematical programming
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References:

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