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A note on the threshold AR(1) model with Cauchy innovations. (English) Zbl 0587.60033
The paper investigates the autoregressive process \(\{X_ s\}\) of first order with one threshold \(r=0\) given by the following formulae \[ (1)\quad X_ s=\alpha X_{s-1}+\ell_ s\quad if\quad X_{s-1}\leq r,\quad X_ s=\beta X_{s-1}+\ell_ s\quad if\quad X_{s-1}>r \] where \(\alpha =-\beta\), \(\alpha\in (0,1)\) and \(\ell_ 1,...,\ell_ n,..\). are independent C(0,1) Cauchy variables with density \(g(x)=\pi^{-1}\{1+x^ 2\}^{-1}\), \(-\infty <x<+\infty\). The authors have found an exact solution for the stationary density h(x) of the distribution of \(X_ s\) of the threshold autoregressive process (1) with Cauchy innovations.
Reviewer: I.G.Zhurbenko

60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
Full Text: DOI
[1] Andel J., Kybernetika 20 pp 89– (1984)
[2] DOI: 10.2307/3213639 · Zbl 0541.62073 · doi:10.2307/3213639
[3] DOI: 10.2307/3213771 · Zbl 0579.62074 · doi:10.2307/3213771
[4] Tong H., Pattern Recognition and Signal Processing pp 575– (1978) · doi:10.1007/978-94-009-9941-1_24
[5] Tong H., Threshold Models in Non-Linear Time Series Analysis. Lecture Notes in Statistics 21 (1983) · Zbl 0527.62083 · doi:10.1007/978-1-4684-7888-4
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