×

zbMATH — the first resource for mathematics

The stationary marginal distribution of a threshold AR(1) process. (English) Zbl 1051.62080
The author considers the threshold AR(1) process defined by \(X_{s}=\tau(X_{s-1})+e_{s}\), where \[ \tau(x)=\begin{cases} \alpha x, & \text{if \(x\geq0\)},\\ \beta x, & \text{if \(x<0\),}\end{cases} \] and \(0<\alpha<1\) and \(0<\beta<1\). It is known that under these conditions the process is ergodic. Hence, the stationary marginal distribution exists and is unique. The invariant distribution of this process is computed in the case of the Laplace distributed noise process \(e_{s}\). The distributions of the threshold AR(1) process for the cases \(0<\alpha<1\), \(\beta<0\) and \(0<\alpha<1\), \(\beta=-\alpha\) are obtained. The derivation is based on the use of the Frobenius-Perron operator.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E15 Exact distribution theory in statistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 1J. ANDEL(1983 ) Marginal distributions of autoregressive processes . InTrans. 9th Prague Conference Information TheoryVol. A. Prague: Academia, 127-35.
[2] ANDEL J., Kybernetika 20 pp 89– (1984)
[3] ANDEL J., J. Time Ser. Anal. 7 pp 1– (1986)
[4] 4A. BOYARSKI, and P. GORA(1997 )Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension. Boston: Birkhauser.
[5] CHAN K. S., Probab. Theory Related Fields 73 pp 153– (1986)
[6] 6H. DORRIE(1951 )Unendliche Reihen. Munchen: Oldenbourg.
[7] PETRUCCELLI J. D., J. Appl. Prob. 21 pp 270– (1984)
[8] 8H. TONG(1983 )Threshold Models in Non-linear Time Series Analysis. Lecture Notes in Statistics. Vol. 21. New York: Springer.
[9] 9H. TONG(1990 )Non-linear Time Series. Oxford Statistical Science Series, Vol. 6. Oxford: Clarendon Press. Reprinted 1999.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.