47 research outputs found
Nonparametric estimation of the characteristic triplet of a discretely observed L\'evy process
Given a discrete time sample from a L\'evy process
of a finite jump activity, we study the problem of
nonparametric estimation of the characteristic triplet
corresponding to the process Based on Fourier inversion and kernel
smoothing, we propose estimators of and and study
their asymptotic behaviour. The obtained results include derivation of upper
bounds on the mean square error of the estimators of and
and an upper bound on the mean integrated square error of an estimator of
Comment: 29 page
Nonparametric inference for discretely sampled L\'evy processes
Given a sample from a discretely observed L\'evy process
of the finite jump activity, the problem of nonparametric estimation of the
L\'evy density corresponding to the process is studied. An estimator
of is proposed that is based on a suitable inversion of the
L\'evy-Khintchine formula and a plug-in device. The main results of the paper
deal with upper risk bounds for estimation of over suitable classes of
L\'evy triplets. The corresponding lower bounds are also discussed.Comment: 38 page
Bayesian linear inverse problems in regularity scales
We obtain rates of contraction of posterior distributions in inverse problems
defined by scales of smoothness classes. We derive abstract results for general
priors, with contraction rates determined by Galerkin approximation. The rate
depends on the amount of prior concentration near the true function and the
prior mass of functions with inferior Galerkin approximation. We apply the
general result to non-conjugate series priors, showing that these priors give
near optimal and adaptive recovery in some generality, Gaussian priors, and
mixtures of Gaussian priors, where the latter are also shown to be near optimal
and adaptive. The proofs are based on general testing and approximation
arguments, without explicit calculations on the posterior distribution. We are
thus not restricted to priors based on the singular value decomposition of the
operator. We illustrate the results with examples of inverse problems resulting
from differential equations.Comment: 34 page
Parametric inference for stochastic differential equations: a smooth and match approach
We study the problem of parameter estimation for a univariate discretely
observed ergodic diffusion process given as a solution to a stochastic
differential equation. The estimation procedure we propose consists of two
steps. In the first step, which is referred to as a smoothing step, we smooth
the data and construct a nonparametric estimator of the invariant density of
the process. In the second step, which is referred to as a matching step, we
exploit a characterisation of the invariant density as a solution of a certain
ordinary differential equation, replace the invariant density in this equation
by its nonparametric estimator from the smoothing step in order to arrive at an
intuitively appealing criterion function, and next define our estimator of the
parameter of interest as a minimiser of this criterion function. Our main
results show that under suitable conditions our estimator is
-consistent, and even asymptotically normal. We also discuss a way of
improving its asymptotic performance through a one-step Newton-Raphson type
procedure and present results of a small scale simulation study.Comment: 26 page
Deconvolution for an atomic distribution
Let be i.i.d. observations, where and
and are independent. Assume that unobservable 's are distributed
as a random variable where and are independent, has a
Bernoulli distribution with probability of zero equal to and has a
distribution function with density Furthermore, let the random
variables have the standard normal distribution and let Based
on a sample we consider the problem of estimation of the
density and the probability We propose a kernel type deconvolution
estimator for and derive its asymptotic normality at a fixed point. A
consistent estimator for is given as well. Our results demonstrate that our
estimator behaves very much like the kernel type deconvolution estimator in the
classical deconvolution problem.Comment: Published in at http://dx.doi.org/10.1214/07-EJS121 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A non-parametric Bayesian approach to decompounding from high frequency data
Given a sample from a discretely observed compound Poisson process, we
consider non-parametric estimation of the density of its jump sizes, as
well as of its intensity We take a Bayesian approach to the
problem and specify the prior on as the Dirichlet location mixture of
normal densities. An independent prior for is assumed to be
compactly supported and to possess a positive density with respect to the
Lebesgue measure. We show that under suitable assumptions the posterior
contracts around the pair at essentially (up to a logarithmic
factor) the -rate, where is the number of observations and
is the mesh size at which the process is sampled. The emphasis is on
high frequency data, , but the obtained results are also valid for
fixed . In either case we assume that . Our
main result implies existence of Bayesian point estimates converging (in the
frequentist sense, in probability) to at the same rate.
We also discuss a practical implementation of our approach. The computational
problem is dealt with by inclusion of auxiliary variables and we develop a
Markov Chain Monte Carlo algorithm that samples from the joint distribution of
the unknown parameters in the mixture density and the introduced auxiliary
variables. Numerical examples illustrate the feasibility of this approach
Consistent non-parametric Bayesian estimation for a time-inhomogeneous Brownian motion
We establish posterior consistency for non-parametric Bayesian estimation of
the dispersion coefficient of a time-inhomogeneous Brownian motion.Comment: 12 page