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Estimating functions for diffusiontype processes
"... In this chapter we consider parametric inference based on discrete time observations X0, Xt1,...,Xtn from a ddimensional stochastic process. In most of the chapter the statistical model for the data will be a diffusion model given by a stochastic differential equation. We shall, however, also consi ..."
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Cited by 7 (4 self)
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In this chapter we consider parametric inference based on discrete time observations X0, Xt1,...,Xtn from a ddimensional stochastic process. In most of the chapter the statistical model for the data will be a diffusion model given by a stochastic differential equation. We shall, however, also consider some examples of nonMarkovian models, where we typically assume
Mighty convergence of the Gaussian quasilikelihood random fields for ergodic Lévy driven SDE observed at high frequency
 Kyushu University
, 2012
"... Mighty convergence of the ..."
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Simulation of multivariate diffusion bridges
, 2014
"... We propose simple methods for multivariate diffusion bridge simulation, which plays a fundamental role in simulationbased likelihood and Bayesian inference for stochastic differential equations. By a novel application of classical coupling methods, the new approach generalizes a previously proposed ..."
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Cited by 1 (1 self)
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We propose simple methods for multivariate diffusion bridge simulation, which plays a fundamental role in simulationbased likelihood and Bayesian inference for stochastic differential equations. By a novel application of classical coupling methods, the new approach generalizes a previously proposed simulation method for onedimensional bridges to the multivariate setting. First a method of simulating approximate, but often very accurate, diffusion bridges is proposed. These approximate bridges are used as proposal for easily implementable MCMC algorithms that produce exact diffusion bridges. The new method is much more generally applicable than previous methods. Another advantage is that the new method works well for diffusion bridges in long intervals because the computational complexity of the method is linear in the length of the interval. In a simulation study the new method performs well, and its usefulness is illustrated by an application to Bayesian estimation for the multivariate hyperbolic diffusion model. Key words: Bayesian inference; coupling; discretely sampled diffusions; likelihood inference; stochastic differential equation; timereversal.
observations of a diffusion process
, 2012
"... We consider the estimation of unknown parameters in the drift and diffusion coefficients of a onedimensional ergodic diffusion X when the observation Y is a discrete sampling of X with an additive noise, at times iδ, i = 1... N. Assuming that the sampling interval tends to 0 while the total length ..."
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We consider the estimation of unknown parameters in the drift and diffusion coefficients of a onedimensional ergodic diffusion X when the observation Y is a discrete sampling of X with an additive noise, at times iδ, i = 1... N. Assuming that the sampling interval tends to 0 while the total length time interval tends to infinity, we prove limit theorems for functionals associated with the observations, based on local means of the sample. We apply these results to obtain a contrast function. The associated minimum contrast estimators are shown to be consistent. Some examples are discussed with numerical simulations.
Estimating functions for noisy observations of ergodic diffusions
, 2010
"... In this article, general estimating functions for ergodic diffusions sampled at high frequency with noisy observations are presented. The theory is formulated in term of approximate martingale estimating functions based on local means of the observations, and simple conditions are given for rate ..."
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In this article, general estimating functions for ergodic diffusions sampled at high frequency with noisy observations are presented. The theory is formulated in term of approximate martingale estimating functions based on local means of the observations, and simple conditions are given for rate optimality. The estimation of diffusion parameter is faster that the estimation of drift parameter, and the rate of convergence in the Central Limit Theorem is classical for the drift parameter but not classical for the diffusion parameter. The link with specific minimum contrast estimators is established, as an example. Key Words: estimating functions, diffusion process, parametric inference, discrete time noisy observations, central limit theorem
vorgelegt von
"... I would like to thank a number of people who have accompanied me during the writing of this thesis. First and foremost, my sincere gratitude goes to my supervisors Ludwig Fahrmeir and Gareth Roberts, who enriched my work through their advice, ideas and encouragement. I also thank Leonhard Held for h ..."
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I would like to thank a number of people who have accompanied me during the writing of this thesis. First and foremost, my sincere gratitude goes to my supervisors Ludwig Fahrmeir and Gareth Roberts, who enriched my work through their advice, ideas and encouragement. I also thank Leonhard Held for his directions during the first stage of my thesis. My research has financially been supported by the German Research Foundation (DFG), the German Academic Exchange Service (DAAD) and the LMU Mentoring programme, in which Francesca Biagini has been a dedicated mentor to me. I deeply appreciate the careful proofreading and helpful comments by Michael Höhle. Furthermore, I am grateful to my former and present colleagues for their interest in my research and their friendship, in particular to the members of the Semwiso group, my FRAP collaborators, the advocates of good teaching, my fellow women’s representatives, the Cozi Family and my office mates. My family has been a constant source of support, and I greatly acknowledge their personal way of understanding my work. I owe my heartful gratitude to Florian Fuchs, who has been a strong and close partner during all stages of my thesis and who stayed awake until the last sentence was written.
Submitted to the Annals of Statistics ON THE APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATION FOR DIFFUSION PROCESSES
"... The transition density of a diffusion process does not admit an explicit expression in general, which prevents the full maximum likelihood estimation (MLE) based on discretely observed sample paths. Aı̈tSahalia (1999, 2002) proposed asymptotic expansions to the transition densities of diffusion p ..."
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The transition density of a diffusion process does not admit an explicit expression in general, which prevents the full maximum likelihood estimation (MLE) based on discretely observed sample paths. Aı̈tSahalia (1999, 2002) proposed asymptotic expansions to the transition densities of diffusion processes, which lead to an approximate maximum likelihood estimation (AMLE) for parameters. Built on Aı̈tSahalia (2002, 2008)’s proposal and analysis on the AMLE, we establish the consistency and convergence rate of the AMLE, which reveal the roles played by the number of terms used in the asymptotic density expansions and the sampling interval between successive observations. We find conditions under which the AMLE has the same asymptotic distribution as that of the full MLE. A first order approximation to the Fisher information matrix is proposed. 1. Introduction. Continuoustime
Approximate quadratic
, 2010
"... estimating function for discretely observed Levy driven SDEs with application to a noise normality test ..."
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estimating function for discretely observed Levy driven SDEs with application to a noise normality test
Efficient Estimation for Diffusions Sampled at High Frequency Over a Fixed Time Interval
, 2015
"... Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimat ..."
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Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimating functions under which estimators are consistent, rate optimal, and efficient under high frequency (infill) asymptotics. The asymptotic distributions of the estimators are shown to be normal variancemixtures, where the mixing distribution generally depends on the full sample path of the diffusion process over the observation time interval. Utilising the concept of stable convergence, we also obtain the more easily applicable result that for a suitable data dependent normalisation, the estimators converge in distribution to a standard normal distribution. The theory is illustrated by a small simulation study comparing an efficient and a nonefficient estimating function. Key words: Approximate martingale estimating functions, discrete time sampling of diffusions, infill asymptotics, normal variancemixtures, optimal rate, random Fisher information, stable convergence, stochastic differential equation.