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

We present an estimation procedure and analyse spectral properties of stochastic processes of the kind Z t = X t + ξ t = ϕ(T t (ψ)) + ξ t , for t ∊ Z, where T is a deterministic map, ϕ is a given function and ξ t is a noise process. The examples considered in this paper generalize the classical harmonic model Z t = Acos(ω0 t + ψ) + ξ t , 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.