Diffusions with measurement errors. I. Local asymptotic normality

Arnaud Gloter; Jean Jacod

Displaying similar documents to “Diffusions with measurement errors. I. Local asymptotic normality”

Diffusions with measurement errors. II. Optimal estimators

Arnaud Gloter, Jean Jacod (2001)

ESAIM: Probability and Statistics

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We consider a diffusion process $X$ which is observed at times $i / n$ for $i = 0, 1, ..., n$ , each observation being subject to a measurement error. All errors are independent and centered gaussian with known variance $ρ_{n}$ . There is an unknown parameter to estimate within the diffusion coefficient. In this second paper we construct estimators which are asymptotically optimal when the process $X$ is a gaussian martingale, and we conjecture that they are also optimal in the general case.

On the asymptotic properties of a simple estimate of the Mode

Christophe Abraham, Gérard Biau, Benoît Cadre (2004)

ESAIM: Probability and Statistics

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We consider an estimate of the mode $θ$ of a multivariate probability density $f$ with support in $ℝ^{d}$ using a kernel estimate $f_{n}$ drawn from a sample $X_{1}, \dots, X_{n}$ . The estimate $θ_{n}$ is defined as any $x$ in ${X_{1}, \dots, X_{n}}$ such that $f_{n} (x) = {max}_{i = 1, \dots, n} f_{n} (X_{i})$ . It is shown that $θ_{n}$ behaves asymptotically as any maximizer ${\hat{θ}}_{n}$ of $f_{n}$ . More precisely, we prove that for any sequence ${(r_{n})}_{n \geq 1}$ of positive real numbers such that $r_{n} \to \infty$ and $r_{n}^{d} log n / n \to 0$ , one has $r_{n} ∥ θ_{n} - {\hat{θ}}_{n} ∥ \to 0$ in probability. The asymptotic normality of $θ_{n}$ follows without further work.

The efficiency of approximating real numbers by Lüroth expansion

Chunyun Cao, Jun Wu, Zhenliang Zhang (2013)

Czechoslovak Mathematical Journal

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For any $x \in (0, 1]$ , let $x = \frac{1}{d_{1}} + \frac{1}{d_{1} (d_{1} - 1) d_{2}} + \dots + \frac{1}{d_{1} (d_{1} - 1) \dots d_{n - 1} (d_{n - 1} - 1) d_{n}} + \dots$ be its Lüroth expansion. Denote by $P_{n} (x) / Q_{n} (x)$ the partial sum of the first $n$ terms in the above series and call it the $n$ th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents ${P_{n} (x) / Q_{n} (x)}_{n \geq 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify...

On the approximation of functions on a Hodge manifold

Alessandro Ghigi (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

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If $(M, ω)$ is a Hodge manifold and $f \in C^{\infty} (M, ℝ)$ we construct a canonical sequence of functions $f_{N}$ such that $f_{N} \to f$ in the $C^{\infty}$ topology. These functions have a simple geometric interpretation in terms of the moment map and they are real algebraic, in the sense that they are regular functions when $M$ is regarded as a real algebraic variety. The definition of $f_{N}$ is inspired by Berezin-Toeplitz quantization and by ideas of Donaldson. The proof follows quickly from known results of Fine, Liu and Ma.

Adaptive estimation of the stationary density of discrete and continuous time mixing processes

Fabienne Comte, Florence Merlevède (2002)

ESAIM: Probability and Statistics

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In this paper, we study the problem of non parametric estimation of the stationary marginal density $f$ of an $α$ or a $β$ -mixing process, observed either in continuous time or in discrete time. We present an unified framework allowing to deal with many different cases. We consider a collection of finite dimensional linear regular spaces. We estimate $f$ using a projection estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the...

Constraints on distributions imposed by properties of linear forms

Denis Belomestny (2003)

ESAIM: Probability and Statistics

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Let $(X_{1}, Y_{1}), ..., (X_{m}, Y_{m})$ be $m$ independent identically distributed bivariate vectors and $L_{1} = β_{1} X_{1} + ... + β_{m} X_{m}$ , $L_{2} = β_{1} Y_{1} + ... + β_{m} Y_{m}$ are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of $L_{1}$ and $L_{2}$ imply the same property for $X_{1}$ and $Y_{1}$ , and under what conditions does the independence of $L_{1}$ and $L_{2}$ entail independence of $X_{1}$ and $Y_{1}$ ? Some analytical sufficient conditions are obtained and it is shown that in general they can not be weakened.

Mittag-Leffler type expansions of $\bar{\partial}$ -closed $(0, n - 1)$ -forms in certain domains in $ℂ^{n}$

Telemachos Hatziafratis (2003)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we will prove a Mittag-Leffler type theorem for $\bar{\partial}$ -closed $(0, n - 1)$ -forms in $ℂ^{n}$ by addressing the question of constructing such differential forms with prescribed periods in certain domains.

Extreme values and kernel estimates of point processes boundaries

Stéphane Girard, Pierre Jacob (2004)

ESAIM: Probability and Statistics

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We present a method for estimating the edge of a two-dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on a Parzen-Rosenblatt kernel and extreme values of point processes. We give conditions for various kinds of convergence and asymptotic normality. We propose a method of reducing the negative bias and edge effects, illustrated by some simulations.