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**Gaussian semiparametric estimation of long range dependence.**
*(English)*
Zbl 0843.62092

Summary: Assuming the model \(f(\lambda) \sim G \lambda^{1 - 2H}\), as \(\lambda \to 0+\), for the spectral density of a covariance stationary process, we consider an estimate of \(H \in (0,1)\) which maximizes an approximate form of frequency domain Gaussian likelihood, where discrete averaging is carried out over a neighbourhood of zero frequency which degenerates slowly to zero as sample size tends to infinity.

The estimate has several advantages. It is shown to be consistent under mild conditions. Under conditions which are not greatly stronger, it is shown to be asymptotically normal and more efficient than previous estimates. Gaussianity is nowhere assumed in the asymptotic theory, the limiting normal distribution is of very simple form, involving a variance which is not dependent on unknown parameters, and the theory covers simultaneously the cases \(f(\lambda) \to \infty\), \(f(\lambda) \to 0\), and \(f(\lambda) \to C \in (0,\infty)\), as \(\lambda\to 0\). Monte Carlo evidence on finite-sample performance is reported, along with an application to a historical series of minimum levels of the River Nile.

The estimate has several advantages. It is shown to be consistent under mild conditions. Under conditions which are not greatly stronger, it is shown to be asymptotically normal and more efficient than previous estimates. Gaussianity is nowhere assumed in the asymptotic theory, the limiting normal distribution is of very simple form, involving a variance which is not dependent on unknown parameters, and the theory covers simultaneously the cases \(f(\lambda) \to \infty\), \(f(\lambda) \to 0\), and \(f(\lambda) \to C \in (0,\infty)\), as \(\lambda\to 0\). Monte Carlo evidence on finite-sample performance is reported, along with an application to a historical series of minimum levels of the River Nile.

### MSC:

62M15 | Inference from stochastic processes and spectral analysis |

62G05 | Nonparametric estimation |

65C99 | Probabilistic methods, stochastic differential equations |

62G20 | Asymptotic properties of nonparametric inference |