Alexandre Brouste

Variety of stylolites' morphologies and statistical characterization of the amount of heterogeneities in the rock

The surface roughness of several stylolites in limestones was measured using high-resolution laser profilometry. The 1D signals obtained were statistically analyzed to determine the scaling behavior and calculate a roughness exponent, also called Hurst exponent. Statistical methods based on the characterization of a single Hurst exponent imply strong assumptions on the mathematical characteristics of the signal: the derivative of the signal (or local increments) should be stationary and has finite variance. The analysis of the measured stylolites shows that these properties are not always verified simultaneously. The stylolite profiles show persistence and jumps and several stylolites are not regular, with alternating regular and irregular portions. A new statistical method is proposed here, based on a non-stationary but Gaussian model, to estimate the roughness of the profiles and quantify the heterogeneity of stylolites. This statistical method is based on two parameters: the local roughness (H ), which describes the local amplitude of the stylolite, and the amount of irregularities on the signal (m), which can be linked to the heterogeneities initially present in the rock before the stylolite formed. Using this technique, a classification of the stylolites in two families is proposed: those for which the morphology is homogeneous everywhere and those with alternating regular and irregular portions. � 2006 Elsevier Ltd. All rights reserved.

A generalized linear model approach to seasonal aspects of wind speed modeling

The aim of the article is to identify the intraday seasonality in a wind speed time series. Following the traditional approach, the marginal probability law is Weibull and, consequently, we consider seasonal Weibull law. A new estimation and decision procedure to estimate the seasonal Weibull law intraday scale parameter is presented. We will also give statistical decision-making tools to discard or not the trend parameter and to validate the seasonal model.

Local asymptotic normality property for fractional Gaussian noise under high-frequency observations

Local Asymptotic Normality (LAN) property for fractional Gaussian noise under high-frequency observations is proved with a non-diagonal rate matrix depending on the parameter to be estimated. In contrast to the LAN families in the literature, non-diagonal rate matrices are inevitable.

Fractional Diffusion with Partial Observations

This article is devoted to the large sample asymptotic properties of the Maximum Likelihood Estimator (MLE) for the signal drift parameter in a partially observed and possibly controlled fractional diffusion system, perturbed by independent normalized fBms with the same Hurst parameter.

CONTROLLED DRIFT ESTIMATION IN FRACTIONAL DIFFUSION LINEAR SYSTEMS

This paper is devoted to the determination of the asymptotical optimal input for the estimation of the drift parameter in a partially observed but controlled fractional Ornstein–Uhlenbeck process. Large sample asymptotical properties of the Maximum Likelihood Estimator are deduced using Ibragimov–Khasminskii program and Laplace transform computations.

Efficient estimation of stable Lévy process with symmetric jumps

Efficient estimation of a non-Gaussian stable Lévy process with drift and symmetric jumps observed at high frequency is considered. For this statistical experiment, the local asymptotic normality of the likelihood is proved with a non-singular Fisher information matrix through the use of a non-diagonal norming matrix. The asymptotic normality and efficiency of a sequence of roots of the associated likelihood equation are shown as well. Moreover, we show that a simple preliminary method of moments can be used as an initial estimator of a scoring procedure, thereby conveniently enabling us to bypass numerically demanding likelihood optimization. Our simulation results show that the one-step estimator can exhibit quite similar finite-sample performance as the maximum likelihood estimator.

Estimation Theory for Generalized Linear Models

Generalized Linear Models have been introduced by (Nelder and Wedderburn 1972). See also the book (McCullagh and Nelder 1983). They describe random observations depending on unobservable variables of interest, generalizing the standard gaussian error model. Many estimation results can be obtained in this context, which generalize with some approximation procedures the gaussian case. We revisit and extend the results. In particular, we prove the cental limit theorem for the MLE, maximum likelihood estimator, in a general setting. We also provide a recursive estimator, similar to the Kalman filter. We also consider dynamic models and develop several methods, including that of (West et al. 1985).

Asymptotic Properties of the MLE for the Autoregressive Process Coefficients under Stationary Gaussian Noise

In this paper we study the Maximum Likelihood Estimator (MLE) of the vector parameter of an autoregressive process of order p with regular stationary Gaussian noise. We prove the large sample asymptotic properties of the MLE under very mild conditions. We do simulations for fractional Gaussian noise (fGn), autoregressive noise (AR(1)) and moving average noise (MA(1)).

Kalman type filter under stationary noises

Abstract In this paper, we are interested in finding an explicit solution to the filtering problem for a d -dimensional autoregressive signal observed through a linear channel when the noises are stationary Gaussian with the same covariance. We represent the signal–observation pair in terms of a 2 × d -dimensional autoregressive process driven by a white Gaussian noise. Simulations are given for fractional Gaussian noises (fGn), autoregressive noises (AR(1)) and moving average noises (MA) in order to analyze the performance of the filtering algorithm compared to other approaches in the literature.