5,287 research outputs found
Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains
We consider triangular arrays of Markov chains that converge weakly to a
diffusion process. Second order Edgeworth type expansions for transition
densities are proved. The paper differs from recent results in two respects. We
allow nonhomogeneous diffusion limits and we treat transition densities with
time lag converging to zero. Small time asymptotics are motivated by
statistical applications and by resulting approximations for the joint density
of diffusion values at an increasing grid of points.Comment: 58 page
Additive isotone regression
This paper is about optimal estimation of the additive components of a
nonparametric, additive isotone regression model. It is shown that
asymptotically up to first order, each additive component can be estimated as
well as it could be by a least squares estimator if the other components were
known. The algorithm for the calculation of the estimator uses backfitting.
Convergence of the algorithm is shown. Finite sample properties are also
compared through simulation experiments.Comment: Published at http://dx.doi.org/10.1214/074921707000000355 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Empirical risk minimization in inverse problems
We study estimation of a multivariate function
when the observations are available from the function , where is a
known linear operator. Both the Gaussian white noise model and density
estimation are studied. We define an -empirical risk functional which is
used to define a -net minimizer and a dense empirical risk minimizer.
Upper bounds for the mean integrated squared error of the estimators are given.
The upper bounds show how the difficulty of the estimation depends on the
operator through the norm of the adjoint of the inverse of the operator and on
the underlying function class through the entropy of the class. Corresponding
lower bounds are also derived. As examples, we consider convolution operators
and the Radon transform. In these examples, the estimators achieve the optimal
rates of convergence. Furthermore, a new type of oracle inequality is given for
inverse problems in additive models.Comment: Published in at http://dx.doi.org/10.1214/09-AOS726 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A New Look at the Easy-Hard-Easy Pattern of Combinatorial Search Difficulty
The easy-hard-easy pattern in the difficulty of combinatorial search problems
as constraints are added has been explained as due to a competition between the
decrease in number of solutions and increased pruning. We test the generality
of this explanation by examining one of its predictions: if the number of
solutions is held fixed by the choice of problems, then increased pruning
should lead to a monotonic decrease in search cost. Instead, we find the
easy-hard-easy pattern in median search cost even when the number of solutions
is held constant, for some search methods. This generalizes previous
observations of this pattern and shows that the existing theory does not
explain the full range of the peak in search cost. In these cases the pattern
appears to be due to changes in the size of the minimal unsolvable subproblems,
rather than changing numbers of solutions.Comment: See http://www.jair.org/ for any accompanying file
Nonparametric Estimation of an Additive Model With a Link Function
This paper describes an estimator of the additive components of a
nonparametric additive model with a known link function. When the additive
components are twice continuously differentiable, the estimator is
asymptotically normally distributed with a rate of convergence in probability
of n^{-2/5}. This is true regardless of the (finite) dimension of the
explanatory variable. Thus, in contrast to the existing asymptotically normal
estimator, the new estimator has no curse of dimensionality. Moreover, the
estimator has an oracle property. The asymptotic distribution of each additive
component is the same as it would be if the other components were known with
certainty.Comment: Published at http://dx.doi.org/10.1214/009053604000000814 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bandwidth selection for smooth backfitting in additive models
The smooth backfitting introduced by Mammen, Linton and Nielsen [Ann.
Statist. 27 (1999) 1443-1490] is a promising technique to fit additive
regression models and is known to achieve the oracle efficiency bound. In this
paper, we propose and discuss three fully automated bandwidth selection methods
for smooth backfitting in additive models. The first one is a penalized least
squares approach which is based on higher-order stochastic expansions for the
residual sums of squares of the smooth backfitting estimates. The other two are
plug-in bandwidth selectors which rely on approximations of the average squared
errors and whose utility is restricted to local linear fitting. The large
sample properties of these bandwidth selection methods are given. Their finite
sample properties are also compared through simulation experiments.Comment: Published at http://dx.doi.org/10.1214/009053605000000101 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A simple smooth backfitting method for additive models
In this paper a new smooth backfitting estimate is proposed for additive
regression models. The estimate has the simple structure of Nadaraya--Watson
smooth backfitting but at the same time achieves the oracle property of local
linear smooth backfitting. Each component is estimated with the same asymptotic
accuracy as if the other components were known.Comment: Published at http://dx.doi.org/10.1214/009053606000000696 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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