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Bayesian semi-parametric estimation of the long-memory parameter under FEXP-priors. (English) Zbl 1349.62100
Summary: In this paper we study the semi-parametric problem of the estimation of the long-memory parameter \(d\) in a Gaussian long-memory model. Considering a family of priors based on FEXP models, called FEXP priors in [the second author et al., Ann. Stat. 40, No. 2, 964–995 (2012; Zbl 1274.62340)], we derive concentration rates together with a Bernstein-von Mises theorem for the posterior distribution of \(d\), under Sobolev regularity conditions on the short-memory part of the spectral density. Three different variations on the FEXP priors are studied. We prove that one of them leads to the minimax (up to a log \(n\) term) posterior concentration rate for \(d\), under Sobolev conditions on the short memory part of the spectral density, while the other two lead to sub-optimal posterior concentration rates in \(d\). Interestingly these results are contrary to those obtained in [loc. cit.] for the global estimation of the spectral density.

62G05 Nonparametric estimation
62F15 Bayesian inference
62G20 Asymptotic properties of nonparametric inference
62M15 Inference from stochastic processes and spectral analysis
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