BROUSTE Alexandre

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.

The subject of the thesis is parametric and non-parametric estimation in jump process models. The thesis is composed of 3 parts which regroup 4 works. The first part, which is composed of two chapters, deals with the estimation of drift and volatility parameters by contrast methods from discrete observations, with the main objective of minimizing the conditions on the observation step, so that it can for example go arbitrarily slowly towards 0. The second part of the thesis concerns asymptotic developments, and bias correction, for the estimation of the integrated volatility. The third part of the thesis, concerns the adaptive estimation of the stationary measure for jump processes.

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.

Mean times to failure are fundamental indicators in reliability and related fields. Here we focus on the conditional mean time to failure defined in a semi-Markov context. A discrete time semi-Markov model with discrete state space is employed, which allows for realistic description of systems under risk. Our main objective is to estimate the conditional mean time to failure and provide asymptotic properties of its nonparametric estimator. Consistency and asymptotic normality results are provided. Our methodology is tested in a real wind dataset and indicators associated with the wind energy production are estimated.

A new sequence of estimators is introduced to estimate the parameters of the fractional Gaussian noise at high-frequency. This sequence is defined by an initial sequence of quadratic generalized variations based estimators (QGV) and a single Fisher scoring step. It presents certain advantages over the sequence of maximum likelihood estima-tors (MLE) which is rate and variance efficient and the QGV which is only rate efficient. Indeed, it is much less computationally demanding than the MLE while keeping the efficient asymptotic variance. The local weak efficiency of crude oil markets is analyzed through likelihood-ratio hypothesis tests based on the joint estimation of the volatility and the Hurst exponent in the high-frequency scheme.

Generalized Linear Models with categorical explanatory variables are considered and parameters of the model are estimated with an original exact maximum likelihood method. The existence of a sequence of maximum likelihood estimators is discussed and considerations on possible link functions are proposed. A focus is then given on two particular positive distributions: the Pareto 1 distribution and the shifted log-normal distributions. Finally, the approach is illustrated on a actuarial dataset to model insurance losses.

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We propose a dynamical model for the wind speed which is a Markov diffusion process with Weibull marginal distribution. It presents several advantages, namely nice modeling features both in terms of marginal probability density function and temporal correlation. The characteristics can be interpreted in terms of shape and scale parameters of a Weibull law which is convenient for practitioners to analyze the results. We calibrate the parameters with the maximum quasi-likelihood method and use the model to generate and forecast the wind speed process. We have tested the model with wind speed dataset provided by the National Renewable Energy Laboratory. The model fits very well with the data. Besides, we obtain a very good performance in point and probabilistic forecasting in the short-term in comparison to benchmarks.

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A new methodology to test the accuracy and to assess the category of a bridge weigh-in-motion (BWIM) system is described in this paper. It is based on a statistical composite hypothesis test. This statistical test can be build either in the unbiased setting or in the asymptotical setting. This procedure is compared to the benchmark detailed by the COST 323 action in terms of maximal admissible empirical standard deviation to be in a fixed class of accuracy. This methodology gives also simpler formulae to practitioners still implying the costumer risk and the supplier risk but also the observation sample size whereas the observation sample size is tabulated in the benchmark.

A parametric framework is proposed to model both attritional and atypical claims for insurance pricing. This model relies on a classical Generalized Linear Model for attritional claims and a non-standard Generalized Pareto distribution regression model for atypical claims. Maximum likelihood estimators (closed-form for the Generalized Linear Model part and computed with Iterated Weighted Least Square procedure for the Generalized Pareto distribution regression part) are proposed to calibrate the model. Two premium principles (expected value principle and standard deviation principle) are computed on a real data set of fire warranty of a corporate line-of-business. In our methodology, the tuning of the safety loading in the two premium principles is performed to meet a solvency constraint so that the premium caps a high-level quantile of the aggregate annual claim distribution over a reference portfolio.

We propose a dynamical model for the wind speed which is a Markov diffusion process with Weibull marginal distribution. It presents several advantages, namely nice modeling features both in terms of marginal probability density function and temporal correlation. The characteristics can be interpreted in terms of shape and scale parameters of a Weibull law which is convenient for practitioners to analyze the results. We calibrate the parameters with the maximum quasi-likelihood method and use the model to generate and forecast the wind speed process. We have tested the model with wind speed dataset provided by the National Renewable Energy Laboratory. The model fits very well with the data. Besides, we obtain a very good performance in point and probabilistic forecasting in the short-term in comparison to benchmarks.

Mean times to failure are fundamental indicators in reliability and related fields. Here we focus on the conditional mean time to failure defined in a semi-Markov context. A discrete time semi-Markov model with discrete state space is employed, which allows for realistic description of systems under risk. Our main objective is to estimate the conditional mean time to failure and provide asymptotic properties of its nonparametric estimator. Consistency and asymptotic normality results are provided. Our methodology is tested in a real wind dataset and indicators associated with the wind energy production are estimated.

Probabilistic modeling of climate and environmental events must take into account their spatial nature. This thesis focuses on the study of risk measures for spatial processes. In a first part, we introduce risk measures able to take into account the dependency structure of the underlying spatial processes when dealing with environmental data. A second part is devoted to the estimation of parameters of max-mix processes. The first part of the thesis is dedicated to risk measures. We extend the work done in [44] on the one hand to Gaussian processes, on the other hand to other max-stable processes and to max-mix processes, other dependence structures are thus considered. The risk measures considered are based on the average L(A,D) of losses or damages D over a region of interest A. We then consider the expectation and variance of these normalized damages. First, we focus on the axiomatic properties of the risk measures, their computation and their asymptotic behavior (when the size of the region A tends to infinity). We compute the risk measures in different cases. For a Gaussian process, X, we consider the excess function: D+ X,u = (X-u)+ where u is a fixed threshold. For max-stable and max-mixed processes X, we consider the power function: DνX = Xν. In some cases, semi-explicit formulas for the corresponding risk measures are given. A simulation study tests the behavior of the risk measures with respect to the many parameters involved and the different forms of the correlation kernel. We also evaluate the computational performance of the different proposed methods. This one is satisfactory. Finally, we have used a previous study on pollution data in the Italian Piedmont, which can be considered as Gaussian. We study the risk measure associated to the legal pollution threshold given by the European directive 2008/50/EC. In a second part, we propose a procedure for estimating the parameters of a max-mix process, as an alternative to the composite maximum likelihood estimation method. This more classical method of estimation by composite maximum likelihood is especially efficient to estimate the parameters of the max-stable part of the mixture (and less efficient to estimate the parameters of the asymptotically independent part). We propose a least squares method based on the F-madogram: minimization of the quadratic difference between the theoretical F-madogram and the empirical F-madogram. This method is evaluated by simulation and compared to the composite maximum likelihood method. The simulations indicate that the F-madogram least squares method performs better in estimating the parameters of the asymptotically independent part.

We propose an effective and fast method to simulate multidimensional conditional fractional Gaussian fields with the package FieldSim. Our method is valid not only for conditional simulations associated to fractional Brownian fields, but to any Gaussian field and on any (non regular) grid of points.

Insurance reserving is a key topic for both actuaries and academics. In the present paper, we present an efficient way to compute all the key indicators in a unified approach of the ruin theory and claim reserving methods. The proposed framework allows to derive closed-form formulas for both ruin theory and claim reserves indicators. A numerical illustration of these indicators is carried out on a real dataset from a private insurer.

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Wind power is an intermittent resource due to wind speed intermittency. However wind speed can be described as a stochastic process with short memory. This allows us to derive a central limit theorem for the annual or pluri-annual wind power production and then get quantiles of the wind power production for one, ten or twenty years future periods. On the one hand, the interquantile spread offers a measurement of the intrinsic uncertainties of wind power production. On the other hand, different quantiles with different periods of time are used by financial institutions to quantify the financial risk of the wind turbine. Our method is then applied to real datasets corresponding to a French wind turbine. Since confidence intervals can be enhanced by taking into account seasonality, we present some tools for change point analysis on wind series.

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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.

In Chapter 1, we study the maximum likelihood estimation (MLE) problem of the parameters of a p-order autoregressive process (AR(p)) driven by a stationary Gaussian noise, which can be long-memory like the fractional Gaussian noise. We give an explicit formula for the MLE and analyze its asymptotic properties. In fact, in our model the covariance function of the noise is assumed to be known, but the asymptotic behavior of the estimator (speed of convergence, Fisher information) does not depend on it.Chapter 2 is devoted to the determination of the optimal input (from an asymptotic point of view) for the estimation of the drift parameter in a partially observed but controlled fractional Ornstein-Uhlenbeck process. We present a separation principle that allows us to achieve this goal. The asymptotic properties of the MLE are demonstrated using the Ibragimov-Khasminskii program and the computation of Laplace transforms of a quadratic functional of the process.In Chapter 3, we present a new approach to study the properties of the mixed fractional Brownian motion and related models, based on the theory of filtering Gaussian processes. The results highlight the semimartingale structure and lead to a number of useful continuity absolute properties. We establish the equivalence of measures induced by mixed fractional Brownian motion with a stochastic drift, and derive the corresponding expression for the Radon-Nikodym derivative. For a Hurst index H > 3=4, we obtain a representation of the mixed fractional Brownian motion as a diffusion-like process in its natural filtration and deduce a formula for the Radon-Nikodym derivative with respect to the Wiener measure. For H < 1=4, we show the equivalence of the measure with that of the fractional component and obtain a formula for the corresponding density. A potential field of application is the statistical analysis of models governed by mixed fractional noises. As an example, we consider the basic linear regression model and show how to define the MLE and study its asymptotic behavior.

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Since the use of fractional Brownian motion for statistical applications by Mandelbrot and Van Ness in 1968, a vast literature has been built up around the estimation of self-similarity and Hölderian regularity. In the framework of the multifractal analysis of Fourier series and wavelet series (full and lacunar), random wavelet series based on a branching process are presented. Their analytical properties (bifractality, non-integer Hausdorff dimension of the graph) and the simulation of their trajectories show their ability to model intermittent processes. A method for estimating the two parameters of the model (estimation of the regularity parameter and the gap parameter) by filtering a trajectory is developed for this model. It will find a natural application for the classification, in geophysics, of stylolitic morphologies (sedimentary rocks of limestone subjected to stress dissolution processes). These series generalize the properties of fractional Brownian motion and can explain the phenomena of persistence and leptokurticity.