SOLTANE Marius

This paper considers the joint estimation of the parameters of a first-order fractional autoregressive model by constructing an initial estimator with convergence speed lower than √ n and singular asymptotic joint distribution. The one-step procedure is then used in order to obtain an asymptoticallyefficient estimator. This estimator is computed faster than the maximum likelihood or Whittle estimator and therefore allows for faster inference on large samples. The paper illustrates the performance of this method on finite-size samples via Monte Carlo simulations.

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The present paper concerns the parametric estimation for the fractional Gaussian noise in a high-frequency observation scheme. The sequence of Le Cam’s one-step maximum likelihood estimators (OSMLE) is studied. This sequence is defined by an initial sequence of quadratic generalized variations-based estimators (QGV) and a single Fisher scoring step. The sequence of OSMLE is proved to be asymptotically efficient as the sequence of maximum likelihood estimators but is much less computationally demanding. It is also advantageous with respect to the QGV which is not variance efficient. Performances of the estimators on finite size observation samples are illustrated by means of Monte-Carlo simulations.

This thesis is devoted to the asymptotic inferential statistics of different noise-driven time series models with memory. In these models, the least squares estimator is not consistent and we consider alternative estimators. We start by studying the asymptotic properties of the maximum likelihood estimator of the autoregression coefficient in an autoregressive process driven by stationary Gaussian noise. We then present a statistical procedure to detect a regime shift in this model based on the classical case driven by strong white noise. We then discuss an autoregressive model where the coefficients are random and have a short memory. Here again the least squares estimator is not consistent and we correct the estimation in order to correctly estimate the model parameters. Finally we study a new joint estimator of the Hurst exponent and the variance in a high frequency fractional Gaussian noise whose qualities are comparable to maximum likelihood.

This thesis is devoted to the asymptotic inferential statistics of different noise-driven time series models with memory. In these models, the least squares estimator is not consistent and we consider alternative estimators. We start by studying the asymptotic properties of the maximum likelihood estimator of the autoregression coefficient in an autoregressive process driven by stationary Gaussian noise. We then present a statistical procedure to detect a regime shift in this model based on the classical case driven by strong white noise. We then discuss an autoregressive model where the coefficients are random and have a short memory. Here again the least squares estimator is not consistent and we correct the estimation in order to correctly estimate the model parameters. Finally we study a new joint estimator of the Hurst exponent and the variance in a high frequency fractional Gaussian noise whose qualities are comparable to maximum likelihood.