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Thomas Mikosch - One of the best experts on this subject based on the ideXlab platform.
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quasi maximum likelihood estimation in conditionally heteroscedastic time series a stochastic Recurrence Equations approach
arXiv: Statistics Theory, 2007Co-Authors: Daniel Straumann, Thomas MikoschAbstract:This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form $X_t=\sigma_tZ_t$, where the unobservable volatility $\sigma_t$ is a parametric function of $(X_{t-1},...,X_{t-p},\sigma_{t-1},... ,\sigma_{t-q})$ for some $p,q\ge0$, and $(Z_t)$ is standardized i.i.d. noise. We assume that these models are solutions to stochastic Recurrence Equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution $(X_t)$ to the stochastic Recurrence Equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models.
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quasi maximum likelihood estimation in conditionally heteroscedastic time series a stochastic Recurrence Equations approach
Annals of Statistics, 2006Co-Authors: Daniel Straumann, Thomas MikoschAbstract:This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form Xt = σt Zt , where the unobservable volatility σt is a parametric function of (Xt−1 ,...,X t−p ,σ t−1 ,...,σ t−q ) for some p, q ≥ 0, and (Zt ) is standardized i.i.d. noise. We assume that these models are solutions to stochastic Recurrence Equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution (Xt ) to the stochastic Recurrence Equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models. 1. Introduction. Gaussian quasi-maximum-likelihood estimation, that is, likelihood estimation under the hypothesis of Gaussian innovations, is a popular method which is widely used for inference in time series models. However, it is often a nontrivial task to establish the consistency and asymptotic normality of the quasi-maximum-likelihood estimator (QMLE) applied to specific models and, therefore, an in-depth analysis of the probabilistic structure generated by the model is called for. A classical example of this kind is the seminal paper by Hannan [18] on estimation in linear ARMA time series. In this paper we study the QMLE for a general class of conditionally heteroscedastic time series models, which includes GARCH, asymmetric GARCH and exponential GARCH. Recall that a GARCH(p, q) [generalized autoregressive conditionally heteroscedastic of order (p, q)] process [4 ]i s def ined by
Daniel Straumann - One of the best experts on this subject based on the ideXlab platform.
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quasi maximum likelihood estimation in conditionally heteroscedastic time series a stochastic Recurrence Equations approach
arXiv: Statistics Theory, 2007Co-Authors: Daniel Straumann, Thomas MikoschAbstract:This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form $X_t=\sigma_tZ_t$, where the unobservable volatility $\sigma_t$ is a parametric function of $(X_{t-1},...,X_{t-p},\sigma_{t-1},... ,\sigma_{t-q})$ for some $p,q\ge0$, and $(Z_t)$ is standardized i.i.d. noise. We assume that these models are solutions to stochastic Recurrence Equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution $(X_t)$ to the stochastic Recurrence Equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models.
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quasi maximum likelihood estimation in conditionally heteroscedastic time series a stochastic Recurrence Equations approach
Annals of Statistics, 2006Co-Authors: Daniel Straumann, Thomas MikoschAbstract:This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form Xt = σt Zt , where the unobservable volatility σt is a parametric function of (Xt−1 ,...,X t−p ,σ t−1 ,...,σ t−q ) for some p, q ≥ 0, and (Zt ) is standardized i.i.d. noise. We assume that these models are solutions to stochastic Recurrence Equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution (Xt ) to the stochastic Recurrence Equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models. 1. Introduction. Gaussian quasi-maximum-likelihood estimation, that is, likelihood estimation under the hypothesis of Gaussian innovations, is a popular method which is widely used for inference in time series models. However, it is often a nontrivial task to establish the consistency and asymptotic normality of the quasi-maximum-likelihood estimator (QMLE) applied to specific models and, therefore, an in-depth analysis of the probabilistic structure generated by the model is called for. A classical example of this kind is the seminal paper by Hannan [18] on estimation in linear ARMA time series. In this paper we study the QMLE for a general class of conditionally heteroscedastic time series models, which includes GARCH, asymmetric GARCH and exponential GARCH. Recall that a GARCH(p, q) [generalized autoregressive conditionally heteroscedastic of order (p, q)] process [4 ]i s def ined by
Pedro Tirado - One of the best experts on this subject based on the ideXlab platform.
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a characterization of smyth complete quasi metric spaces via caristi s fixed point theorem
Fixed Point Theory and Applications, 2015Co-Authors: Salvador Romaguera, Pedro TiradoAbstract:We obtain a quasi-metric generalization of Caristi’s fixed point theorem for a kind of complete quasi-metric spaces. With the help of a suitable modification of its proof, we deduce a characterization of Smyth complete quasi-metric spaces which provides a quasi-metric generalization of the well-known characterization of metric completeness due to Kirk. Some illustrative examples are also given. As an application, we deduce a procedure which allows to easily show the existence of solution for the Recurrence Equation of certain algorithms.
Salvador Romaguera - One of the best experts on this subject based on the ideXlab platform.
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a characterization of smyth complete quasi metric spaces via caristi s fixed point theorem
Fixed Point Theory and Applications, 2015Co-Authors: Salvador Romaguera, Pedro TiradoAbstract:We obtain a quasi-metric generalization of Caristi’s fixed point theorem for a kind of complete quasi-metric spaces. With the help of a suitable modification of its proof, we deduce a characterization of Smyth complete quasi-metric spaces which provides a quasi-metric generalization of the well-known characterization of metric completeness due to Kirk. Some illustrative examples are also given. As an application, we deduce a procedure which allows to easily show the existence of solution for the Recurrence Equation of certain algorithms.
Tran Christophe - One of the best experts on this subject based on the ideXlab platform.
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Addition formulae on Jacobians of hyperelliptic curves : application to cryptography
2014Co-Authors: Tran ChristopheAbstract:Dans cette thèse, j'étudie deux aspects distincts de la cryptographie basée sur les courbes elliptiques et hyperelliptiques. Dans une première partie, je confronte deux méthodes de calcul de couplages, originales car ne reposant pas sur le traditionnel algorithme de Miller. Ainsi, dans [42], K. Stange calcula le couplage de Tate sur une courbe elliptique à partir d'un nouvel outil, les elliptic nets. Y. Uchida et S. Uchiyama généralisèrent ces objets au cas hyperelliptique ([47]), mais ne donnèrent un algorithme pour le calcul de couplages que dans le cas des courbes de genre 2. Mon premier travail dans cette thèse fut de donner cet algorithme pour le cas général. De leur côté, D. Lubicz et D. Robert donnèrent dans [28] une autre méthode de calcul de couplage, basée sur les fonctions thêta. Le second résultat de ma thèse est de réunifier ces deux méthodes : je montre que la formule de récurrence à la base des nets est une conséquence des formules d'addition des fonctions thêta utilisées dans l'algorithme de Lubicz et Robert. Dans la seconde partie de ma thèse, je me suis intéressé à l'algorithme de calcul d'index attaquant le problème du logarithme discret sur les courbes elliptiques et hyperelliptiques. Dans le cas elliptique, une des étapes principales de cette attaque repose sur les polynômes de Semaev. Je donne une nouvelle construction ces polynômes en utilisant la fonction sigma de Weierstrass, pour pouvoir ensuite les généraliser pour la première fois au cas hyperelliptique.In this thesis, I study two different aspects of elliptic and hyperelliptic curves based cryptography.In the first part, I confront two methods of pairings computation, whose original feature is that they are not based the traditional Miller algorithm. Therefore, in [42], K. Stange computed Tate pairings on elliptic curves using a new tool, the elliptic nets. Y. Uchida and S. Uchiyama generalized these objects to hyperelliptic case ([47]), but they gave an algorithm for pairing computation only for the genus 2 case. My first work in this thesis was to give this algorithm for the general case. Meanwhile, D. Lubicz and D. Robert gave in [28] an other pairing computation method, based on theta functions. The second result of my thesis is the reunification of these two methods : I show that the Recurrence Equation which is the basis of nets theory is a consequence of the addition law of theta functions used in the Lubicz and Robert’s algorithm. In the second part, I study the index calculus algorithm attacking the elliptic and hyperelliptic curve discrete logarithm problem. In the elliptic case, one of the main steps of this attack requires the Semaev polynomials. I reconstruct these polynomials using Weierstrass sigma function, with the purpose of giving their first hyperelliptic generalization
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Addition formulae on hyperelliptic curves : applications to cryptography
HAL CCSD, 2014Co-Authors: Tran ChristopheAbstract:In this thesis, I study two different aspects of elliptic and hyperelliptic curves based cryptography.In the first part, I confront two methods of pairings computation, whose original feature is that they are not based the traditional Miller algorithm. Therefore, K. Stange computed Tate pairings on elliptic curves using a new tool, the elliptic nets. Y. Uchida and S. Uchiyama generalized these objects to hyperelliptic case, but they gave an algorithm for pairing computation only for the genus 2 case. My first work in this thesis was to give this algorithm for the general case. Meanwhile, D. Lubicz and D. Robert gave an other pairing computation method, based on theta functions. The second result of my thesis is the reunification of these two methods : I show that the Recurrence Equation which is the basis of nets theory is a consequence of the addition law of theta functions used in the Lubicz and Robert’s algorithm.In the second part, I study the index calculus algorithm attacking the elliptic and hyperelliptic curve discrete logarithm problem. In the elliptic case, one of the main steps of this attack requires the Semaev polynomials. I reconstruct these polynomials using Weierstrass sigma function, with the purpose of giving their first hyperelliptic generalization.Dans cette thèse, j’étudie deux aspects distincts de la cryptographie basée sur les courbes elliptiques et hyperelliptiques.Dans une première partie, je confronte deux méthodes de calcul de couplages, originales car ne reposant pas sur le traditionnel algorithme de Miller. Ainsi, K. Stange calcula le couplage de Tate sur une courbe elliptique à partir d’un nouvel outil, les elliptic nets. Y. Uchida et S. Uchiyama généralisèrent ces objets au cas hyperelliptique, mais ne donnèrent un algorithme pour le calcul de couplages que dans le cas des courbes de genre 2. Mon premier travail dans cette thèse fut de donner cet algorithme pour le cas général. De leur côté, D. Lubicz et D. Robert donnèrent une autre méthode de calcul de couplage, basée sur les fonctions thêta. Le second résultat de ma thèse est de réunifier ces deux méthodes : je montre que la formule de récurrence à la base des nets est une conséquence des formules d’addition des fonctions thêta utilisées dans l’algorithme de Lubicz et Robert.Dans la seconde partie de ma thèse, je me suis intéressé à l’algorithme de calcul d’index attaquant le problème du logarithme discret sur les courbes elliptiques et hyperellip- tiques. Dans le cas elliptique, une des étapes principales de cette attaque repose sur les polynômes de Semaev. Je donne une nouvelle construction ces polynômes en utilisant la fonction sigma de Weierstrass, pour pouvoir ensuite les généraliser pour la première fois au cas hyperelliptique
