21,836 research outputs found
On examples of difference operators for -valued functions over finite sets
Recently V.I.Arnold have formulated a geometrical concept of monads and apply
it to the study of difference operators on the sets of -valued
sequences of length . In the present note we show particular examples of
these monads and indicate one question arising here
Sharp adaptive estimation of the drift function for ergodic diffusions
The global estimation problem of the drift function is considered for a large
class of ergodic diffusion processes. The unknown drift is supposed
to belong to a nonparametric class of smooth functions of order , but
the value of is not known to the statistician. A fully data-driven
procedure of estimating the drift function is proposed, using the estimated
risk minimization method. The sharp adaptivity of this procedure is proven up
to an optimal constant, when the quality of the estimation is measured by the
integrated squared error weighted by the square of the invariant density.Comment: Published at http://dx.doi.org/10.1214/009053605000000615 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Optimization of self-similar factor approximants
The problem is analyzed of extrapolating power series, derived for an
asymptotically small variable, to the region of finite values of this variable.
The consideration is based on the self-similar approximation theory. A new
method is suggested for defining the odd self-similar factor approximants by
employing an optimization procedure. The method is illustrated by several
examples having the mathematical structure typical of the problems in
statistical and chemical physics. It is shown that the suggested method
provides a good accuracy even when the number of terms in the perturbative
power series is small.Comment: Latex file, 16 page
Trees under attack: a Ray-Knight representation of Feller's branching diffusion with logistic growth
We obtain a representation of Feller's branching diffusion with logistic
growth in terms of the local times of a reflected Brownian motion with a
drift that is affine linear in the local time accumulated by at its current
level. As in the classical Ray-Knight representation, the excursions of are
the exploration paths of the trees of descendants of the ancestors at time
, and the local time of at height measures the population size at
time (see e.g. \cite{LG4}). We cope with the dependence in the reproduction
by introducing a pecking order of individuals: an individual explored at time
and living at time is prone to be killed by any of its
contemporaneans that have been explored so far. The proof of our main result
relies on approximating with a sequence of Harris paths which figure
in a Ray-Knight representation of the total mass of a branching particle
system. We obtain a suitable joint convergence of together with its local
times {\em and} with the Girsanov densities that introduce the dependence in
the reproduction
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