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Theses

Long-memory time series models with dependent innovations

Abstract : We first consider, in this thesis, the problem of statistical analysis of FARIMA (Fractionally AutoRegressive Integrated Moving-Average) models endowed with uncorrelated but non-independent error terms. These models are called weak FARIMA and can be used to fit long-memory processes with general nonlinear dynamics. Relaxing the independence assumption on the noise, which is a standard assumption usually imposed in the literature, allows weak FARIMA models to cover a large class of nonlinear long-memory processes. The weak FARIMA models are dense in the set of purely non-deterministic stationary processes, the class of these models encompasses that of FARIMA processes with an independent and identically distributed noise (iid). We call thereafter strong FARIMA models the models in which the error term is assumed to be an iid innovations.We establish procedures for estimating and validating weak FARIMA models. We show, under weak assumptions on the noise, that the least squares estimator of the parameters of weak FARIMA(p,d,q) models is strongly consistent and asymptotically normal. The asymptotic variance matrix of the least squares estimator of weak FARIMA(p,d,q) models has the "sandwich" form. This matrix can be very different from the asymptotic variance obtained in the strong case (i.e. in the case where the noise is assumed to be iid). We propose, by two different methods, a convergent estimator of this matrix. An alternative method based on a self-normalization approach is also proposed to construct confidence intervals for the parameters of weak FARIMA(p,d,q) models.We then pay particular attention to the problem of validation of weak FARIMA(p,d,q) models. We show that the residual autocorrelations have a normal asymptotic distribution with a covariance matrix different from that one obtained in the strong FARIMA case. This allows us to deduce the exact asymptotic distribution of portmanteau statistics and thus to propose modified versions of portmanteau tests. It is well known that the asymptotic distribution of portmanteau tests is correctly approximated by a chi-squared distribution when the error term is assumed to be iid. In the general case, we show that this asymptotic distribution is a mixture of chi-squared distributions. It can be very different from the usual chi-squared approximation of the strong case. We adopt the same self-normalization approach used for constructing the confidence intervals of weak FARIMA model parameters to test the adequacy of weak FARIMA(p,d,q) models. This method has the advantage of avoiding the problem of estimating the asymptotic variance matrix of the joint vector of the least squares estimator and the empirical autocovariances of the noise.Secondly, we deal in this thesis with the problem of estimating autoregressive models of order 1 endowed with fractional Gaussian noise when the Hurst parameter H is assumed to be known. We study, more precisely, the convergence and the asymptotic normality of the generalized least squares estimator of the autoregressive parameter of these models.
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Youssef Esstafa. Long-memory time series models with dependent innovations. Statistics [math.ST]. Université Bourgogne Franche-Comté, 2019. English. ⟨NNT : 2019UBFCD021⟩. ⟨tel-02548451⟩

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