where θ is the unknown parameter of interest and ξ is the initial condition. With x θ (t) the solution vector corresponding to the parameter value θ, we observe
Given a sample from a discretely observed compound Poisson process, we
consider estimation of the density of the jump sizes. We propose a kernel type
nonparametric density estimator and study its asymptotic properties. An order
bound for the bias and an asymptotic expansion of the variance of the estimator
are given. Pointwise weak consistency and asymptotic normality are established.
The results show that, asymptotically, the estimator behaves very much like an
ordinary kernel estimator.Comment: Published at http://dx.doi.org/10.3150/07-BEJ6091 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Given a discrete time sample X 1 , . . . X n from a Lévy process X = (X t ) t≥0 of a finite jump activity, we study the problem of nonparametric estimation of the characteristic triplet (γ, σ 2 , ρ) corresponding to the process X. Based on Fourier inversion and kernel smoothing, we propose estimators of γ, σ 2 and ρ and study their asymptotic behaviour. The obtained results include derivation of upper bounds on the mean square error of the estimators of γ and σ 2 and an upper bound on the mean integrated square error of an estimator of ρ.
Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density r0 and intensity λ0. We take a nonparametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply the existence of Bayesian point estimates that converge to the true parameter pair (r0, λ0) at these rates. To the best of our knowledge, construction of nonparametric density estimators for inference in the class of discretely observed multidimensional Lévy processes, and the study of their rates of convergence is a new contribution to the literature.
Let $X_1,...,X_n$ be i.i.d. observations, where $X_i=Y_i+\sigma Z_i$ and
$Y_i$ and $Z_i$ are independent. Assume that unobservable $Y$'s are distributed
as a random variable $UV,$ where $U$ and $V$ are independent, $U$ has a
Bernoulli distribution with probability of zero equal to $p$ and $V$ has a
distribution function $F$ with density $f.$ Furthermore, let the random
variables $Z_i$ have the standard normal distribution and let $\sigma>0.$ Based
on a sample $X_1,..., X_n,$ we consider the problem of estimation of the
density $f$ and the probability $p.$ We propose a kernel type deconvolution
estimator for $f$ and derive its asymptotic normality at a fixed point. A
consistent estimator for $p$ 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
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