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A multiple-threshold AR(1) model

Published online by Cambridge University Press:  14 July 2016

K. S. Chan ,
Joseph D. Petruccelli ,
H. Tong  and
Samuel W. Woolford
K. S. Chan*
Affiliation:
Chinese University of Hong Kong
Joseph D. Petruccelli*
Affiliation:
Worcester Polytechnic Institute
H. Tong*
Affiliation:
Chinese University of Hong Kong
Samuel W. Woolford*
Affiliation:
Worcester Polytechnic Institute
*
∗Postal address: Department of Statistics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong.
∗∗Postal address: Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA.
∗Postal address: Department of Statistics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong.
∗∗Postal address: Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA.
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Abstract

We consider the model Zt = φ (0, k)+ φ(1, k)Zt–1 + at (k) whenever rk−1<Zt−1≦rk, 1≦k≦l, with r0 = –∞ and rl =∞. Here {φ (i, k); i = 0, 1; 1≦k≦l} is a sequence of real constants, not necessarily equal, and, for 1≦k≦l, {at(k), t≧1} is a sequence of i.i.d. random variables with mean 0 and with {at(k), t≧1} independent of {at(j), t≧1} for j ≠ k. Necessary and sufficient conditions on the constants {φ (i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly consistent and asymptotically normal.

Keywords

NON-LINEAR TIME SERIESSETAR MODELSAUTOREGRESSIVE MODELSMARKOV CHAINS
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 2 , June 1985 , pp. 267 - 279
Copyright
Copyright © Applied Probability Trust 1985 

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References

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