56 research outputs found
Estimation for L\'{e}vy processes from high frequency data within a long time interval
In this paper, we study nonparametric estimation of the L\'{e}vy density for
L\'{e}vy processes, with and without Brownian component. For this, we consider
discrete time observations with step . The asymptotic framework is:
tends to infinity, tends to zero while tends
to infinity. We use a Fourier approach to construct an adaptive nonparametric
estimator of the L\'{e}vy density and to provide a bound for the global
-risk. Estimators of the drift and of the variance of the
Gaussian component are also studied. We discuss rates of convergence and give
examples and simulation results for processes fitting in our framework.Comment: Published in at http://dx.doi.org/10.1214/10-AOS856 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Asymptotic equivalence of nonparametric diffusion and Euler scheme experiments
We prove a global asymptotic equivalence of experiments in the sense of Le
Cam's theory. The experiments are a continuously observed diffusion with
nonparametric drift and its Euler scheme. We focus on diffusions with
nonconstant-known diffusion coefficient. The asymptotic equivalence is proved
by constructing explicit equivalence mappings based on random time changes. The
equivalence of the discretized observation of the diffusion and the
corresponding Euler scheme experiment is then derived. The impact of these
equivalence results is that it justifies the use of the Euler scheme instead of
the discretized diffusion process for inference purposes.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1216 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Computable infinite dimensional filters with applications to discretized diffusion processes
Let us consider a pair signal-observation ((xn,yn),n 0) where the unobserved
signal (xn) is a Markov chain and the observed component is such that, given
the whole sequence (xn), the random variables (yn) are independent and the
conditional distribution of yn only depends on the corresponding state variable
xn. The main problems raised by these observations are the prediction and
filtering of (xn). We introduce sufficient conditions allowing to obtain
computable filters using mixtures of distributions. The filter system may be
finite or infinite dimensional. The method is applied to the case where the
signal xn = Xn is a discrete sampling of a one dimensional diffusion process:
Concrete models are proved to fit in our conditions. Moreover, for these
models, exact likelihood inference based on the observation (y0,...,yn) is
feasable
Filtering the Wright-Fisher diffusion
We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be
observed directly. Instead, at times t1 < t2 < . . ., the observations y(ti)
are such that, given the process (x(t)), the random variables (y(ti)) are
independent and the conditional distribution of y(ti) only depends on x(ti).
When this conditional distribution has a specific form, we prove that the model
((x(ti), y(ti)), i 1) is a computable filter in the sense that all
distributions involved in filtering, prediction and smoothing are exactly
computable. These distributions are expressed as finite mixtures of parametric
distributions. Thus, the number of statistics to compute at each iteration is
finite, but this number may vary along iterations.Comment: 24 page
Penalized nonparametric mean square estimation of the coefficients of diffusion processes
We consider a one-dimensional diffusion process which is observed at
discrete times with regular sampling interval . Assuming that
is strictly stationary, we propose nonparametric estimators of the
drift and diffusion coefficients obtained by a penalized least squares
approach. Our estimators belong to a finite-dimensional function space whose
dimension is selected by a data-driven method. We provide non-asymptotic risk
bounds for the estimators. When the sampling interval tends to zero while the
number of observations and the length of the observation time interval tend to
infinity, we show that our estimators reach the minimax optimal rates of
convergence. Numerical results based on exact simulations of diffusion
processes are given for several examples of models and illustrate the qualities
of our estimation algorithms.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5173 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Filtering the Wright-Fisher diffusion
International audienceWe consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t1 < t2 < ..., the observations y(ti) are such that, given the process (x(t)), the random variables (y(ti)) are independent and the conditional distribution of y(ti) only depends on x(ti). When this conditional distribution has a specific form, we prove that the model ((x(ti),y(ti)), i1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations
Parametric inference for discrete observations of diffusion processes with mixed effects
Pré-publication, Document de travail HAL Id : hal-01332630, version 1Stochastic differential equations with mixed effects provide means to model intraindividual and in-terindividual variability in biomedical experiments based on longitudinal data. We consider N i.i.d. stochastic processes (Xi(t), t ∈ [0, T ]), i = 1,. .. , N , defined by a stochastic differential equation with linear mixed effects. We consider a parametric framework with distributions leading to explicit approximate likelihood functions and investigate the asymptotic behaviour of estimators under the double asymptotic framework: the number N of individuals (trajectories) and the number n of observations per individual tend to infinity within the fixed time interval [0, T ]. The estimation method is assessed on simulated data for various models comprised in our framework
Nonparametric Laguerre estimation in the multiplicative censoring model
International audienceWe study the model where the 's are {\em i.i.d.} with density, , the 's are {\em i.i.d.}, nonnegative with unknown density . The sequences are independent. We aim at estimating on from the observations . We propose projection estimators using a Laguerre basis. A data-driven procedure is described in order to select the dimension of the projection space, which performs automatically the bias variance compromise. Then, we give upper bounds on the -risk on specific Sobolev-Laguerre spaces. Lower bounds matching with the upper bounds within a logarithmic factor are proved. The method is illustrated on simulated data
Sobolev-Hermite versus Sobolev nonparametric density estimation on R
International audienceIn this paper, our aim is to revisit the nonparametric estimation of f assuming that f is square integrable on R, by using projection estimators on a Hermite basis. These estimators are defined and studied from the point of view of their mean integrated squared error on R. A model selection method is described and proved to perform an automatic bias variance compromise. Then, we present another collection of estimators, of deconvolution type, for which we define another model selection strategy. Considering Sobolev and Sobolev-Hermite spaces, the asymptotic rates of these estimators can be computed and compared: they are mainly proved to be equivalent. However, complexity evaluations prove that the Hermite estimators have a much lower computational cost than their deconvolution (or kernel) counterparts. These results are illustrated through a small simulation study
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