Abstract
This paper is devoted to the offline multiple changes detection for long-range dependence processes. The observations are supposed to satisfy a semi-parametric long-range dependence assumption with distinct memory parameters on each stage. A penalized local Whittle contrast is considered for estimating all the parameters, notably the number of changes. The consistency as well as convergence rates are obtained. Monte-Carlo experiments exhibit the accuracy of the estimators. They also show that the estimation of the number of breaks is improved by using a data-driven slope heuristic procedure of choice of the penalization parameter.
Highlights
There exists a very large literature devoted to long-range dependent processes
The aim of this paper is to present a method for estimating from (X1, . . . , Xn) the number K∗ of abrupt changes, the K∗ change-times (t∗1, . . . , t∗K∗ ) and the K∗ + 1 different long-memory parameters (d∗1, . . . , d∗K∗+1), which are unknown
For our offline framework, following the previous purposes, we have chosen to set to build a penalized contrast based on a sum of successive local Whittle contrasts and to minimize it
Summary
There exists a very large literature devoted to long-range dependent processes. Their most commonly used definition requires a second order stationary process X = (Xn)n∈Z with spectral density f such that:. For our offline framework, following the previous purposes, we have chosen to set to build a penalized contrast based on a sum of successive local Whittle contrasts and to minimize it The principle of this method, minimizing a penalized contrast, provides very convincing results in many frameworks: in case of mean changes with least squares contrast (see Bai, 1998), in case of linear models changes with least squares contrast (see Bai and Perron, 1998, generalized by Lavielle, 1999, and Lavielle and Moulines, 2000) or least absolute deviations (see Bai, 1998), in case of spectral densities changes with usual Whittle contrasts (see Lavielle and Ludena, 2000), in case of time series changes with quasi-maximum likelihood (see Bardet et al, 2012), . When the number of changes is known, the theoretical results concerning the consistencies of the estimator are satisfying and n = 5000 provides very convincing results while they are still mediocre for n = 2000 and bad for n = 500 This is not surprising since we considered a semi-parametric statistical framework.
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