Parameter estimation in Manneville–Pomeau processes

Abstract

In this paper, we study a class of stochastic processes [...], where[...] is obtained from the iterations of the transformation , invariant for an ergodic probability[...] on [...] and a certain constant by partial function [...]. We consider here the family of transformations[...] indexed by a parameter , known as the Manneville–Pomeau family of transformations. The autocorrelation function of the resulting process decays hyperbolically (or polynomially) and we obtain efficient methods to estimate the parameter[...] from a finite time series. As a consequence, we also estimate the rate of convergence of the autocorrelation decay of these processes. We compare different estimation methods based on the periodogram function, the smoothed periodogram function, the variance of the partial sum, and the wavelet theory. To obtain our results we analyzed the properties of the spectral density function and the associated Fourier series.