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Theses

Estimation dans des modèles à variables cachées

Abstract : This thesis is devoted to the inference in hidden-variables models. The Chapter 1 considers the properties of the maximum likelihood estimator (MLE) for a possibly not stationary hidden Markov model, where the hidden state space is a metric compact space, and both the transition kernel of the hidden chain and the conditional distribution of the observations depend on a parameter . For identifiable models, consistency, asymptotic normality and efficiency of the MLE is shown to follow from exponential memorylessness properties of the state prediction filter and geometric ergodicity of suitably extended Markov chains. Chapter 2 deals with a semiparametric deconvolution model. The observations come from a signal of i.i.d. random variables with common unknown density g, and a white noise sequence Gaussian centered with unknown variance \sigma^2. When \sigma is unknown, we prove that the rate of convergence for the estimation of g is seriously deteriorated: this rate is slower than (log n)^(-1/2) in regular cases. We propose an estimator of \sigma that is nearly minimax when g has a support included in some fixed compact set. We also construct a universal estimator of \sigma (i.e. without any constraint on g except the one that ensures the identifiability of the model). Chapter 3 still deals with the convolution model but assuming that the Gaussian noise has a known variance (fixed to 1). We study the estimation properties of linear functionals of g given by \int f(x)\Phi_1(y-x) g(x)dx where \Phi_1 is the noise density and f is an entirely known function. We extend the results of Taupin [2,1] when f is a polynomial or a trigonometric function, proving lower bounds for the pointwise minimax quadratic risk and for the minimax risk with respect to the L_\infinity norm, and establishing lower and upper bounds for the mimimax risk with respect to the Lp-norm when . We prove that the estimator given by Taupin [2] reaches the optimal rates when f is a polynomial function and is nearly minimax when f is a trigonometric function.
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Submitted on : Monday, February 7, 2005 - 4:48:05 PM
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Catherine Matias. Estimation dans des modèles à variables cachées. Mathématiques [math]. Université Paris Sud - Paris XI, 2001. Français. ⟨tel-00008383⟩

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