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Theses

Tests optimaux pour détecter les signaux faibles dans les séries chronologiques

Abstract : This thesis focuses on the construction of locally asymptotically optimal tests to detect breaks in the mean of Conditional Heteroskedastic AutoRegressive Nonlinear (CHARN) models described by the following stochastic equation: $X_t=T(Z_{t-1})+gamma^{top}omega(t)+V(Z_{t-1})varepsilon_t,quad tinzz,$ where $gamma=(gamma_1,ldots,gamma_k,gamma_{k+1})^{top} inrr^{k+1}$ and for $t_1,ldots,t_k,$ $1< t_10,forall xinrr^p$ and $n$ the number of observations. The model (2) contains a large class of time series models like AR, MA, ARMA, ARIMA, ARCH etc. Attention is paid to small breaks. Those which are difficult to observe with the naked eye, unlike those considered in the literature. Such a study does not appear to have already been carried out in the context of time series. The test studied is the likelihood ratio test to test $$H_0:gamma=gamma_0text{ against } H^{(n)}_beta:gamma=gamma_0+dfrac{beta}{sqrt{n}}=gamma_{n},quad n>1,$$ for $gamma_0inrr^{k+1}$ and $betainrr^{k+1}$ characterizing respectively the situation where there is no break, and that where there is at least one break to be found. This document is organized as follows: Chapter 1 constitutes the general introduction to the thesis. There, some useful basic concepts and tools are recalled. Chapter 2 reviews the state of the art on the detection of breaks in time series. This chapter is divided into two parts. The first concerns the estimation of breaks and their locations. The second concerns the tests for the existence of break-points. Chapter 3 deals with the case where the functions $ T $ and $ V $ are known, and when they are known but depend on unknown parameters. In the latter case, the situation where the parameter $ gamma_0 $ is known and the one where it is unknown are studied. When it is unknown, it is estimated by the maximum likelihood method. The study of the test is based essentially on the asymptotic local property (LAN) stated for example in cite{droesbeke1996}. Chapter 4 is a generalization of chapter 3. Here, the magnitude of the jump is arbitrary and unknown. Therefore, one has to test $$H_0:gamma=gamma_0text{ against }H^{(n)}=displaystylebigcup_{betainrr^{k+1}}^{}{ H^{(n)}_beta}.$$ A Cramer-Von-Mises type test is constructed. The techniques of cite{ngatchou2009} are used to find the asymptotic distribution of the test under the alternative hypothesis. Chapter 5 presents numerical results obtained using software R. The results obtained for simulated data are first presented and commented. This is followed by those for applications with several real datasets. Chapter 6 concludes the thesis and sets out some prospects.
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Submitted on : Wednesday, February 2, 2022 - 3:14:33 PM
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Marwa Ltaifa. Tests optimaux pour détecter les signaux faibles dans les séries chronologiques. Mathématiques [math]. Université de Lorraine; Université de Sousse (Tunisie), 2021. Français. ⟨NNT : 2021LORR0189⟩. ⟨tel-03552957⟩

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