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Adaptive estimation of the fractional differencing coefficient. (English) Zbl 1006.62082

Summary: Semi-parametric estimation of the fractional differencing coefficient \(d\) of a long-range dependent stationary time series has received substantial attention in recent years. Some of the so-called local estimators introduced earlier were proved rate-optimal over relevant classes of spectral densities. The rates of convergence of these estimators are limited to \(n^{2/5}\), where \(n\) is the sample size.
This paper focuses on the fractional exponential (FEXP) or broadband estimator of \(d\). Minimax-rates of convergence over classes of spectral densities which are smooth outside the zero frequency are obtained, and the FEXP estimator is proved rate-optimal over these classes. On a certain functional class which contains the spectral densities of FARIMA processes, the rate of convergence of the FEXP estimator is \((n/ \log(n))^{1/2}\), thus making it a reasonable alternative to parametric estimators.
As usual in semi-parametric estimation problems, these rate-optimal estimators are infeasible, since they depend on an unknown smoothness parameter defining the functional class. A feasible adaptive version of the broadband estimator is constructed. It is shown that this estimator is minimax rate-optimal up to a factor proportional to the logarithm of the sample size.

MSC:

62M15 Inference from stochastic processes and spectral analysis
62C20 Minimax procedures in statistical decision theory
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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