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Parametric estimation and spectral analysis of piecewise linear maps of the interval

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Artur Lopes  and
Sílvia Lopes
Artur Lopes*
Affiliation:
Universidade Federal do Rio Grande do Sul
Sílvia Lopes*
Affiliation:
Universidade Federal do Rio Grande do Sul
*
∗ Postal address: Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, Porto Alegre, RS-91540-000, Brasil.
∗ Postal address: Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, Porto Alegre, RS-91540-000, Brasil.
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Abstract

We present an estimation procedure and analyse spectral properties of stochastic processes of the kind Zt = Xt + ξt = ϕ(Tt(ψ)) + ξt, for t ∊ Z, where T is a deterministic map, ϕ is a given function and ξt is a noise process. The examples considered in this paper generalize the classical harmonic model Zt = Acos(ω0t + ψ) + ξt, for t ∊ Z. Two examples are developed at length. In the first one, the spectral measure is discrete and in the second it is continuous. In the second example, the time series is obtained from a chaotic map. These two examples exhibit the extremal cases of different possibilities for the spectral measure of time series and they are both associated with ergodic deterministic transformations with noise. We present a method for obtaining explicitly the spectral density function (second example) and the autocorrelation coefficients (first example). In the first example the rotation number plays an important role. We also consider large deviation properties of the estimated parameters of the model.

Keywords

Spectral analysischaotic time seriesrotation number

MSC classification

Primary: 60G10: Stationary processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 757 - 776
Copyright
Copyright © Applied Probability Trust 1998 

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