96 research outputs found
Simulation of conditioned diffusions
In this paper, we propose some algorithms for the simulation of the
distribution of certain diffusions conditioned on terminal point. We prove that
the conditional distribution is absolutely continuous with respect to the
distribution of another diffusion which is easy for simulation, and the formula
for the density is given explicitly
On the acceleration of some empirical means with application to nonparametric regression
Let be an i.i.d. sequence of random variables in ,
, for some function , under regularity conditions,
we show that \begin{align*} n^{1/2} \left(n^{-1} \sum_{i=1}^n
\frac{\varphi(X_i)}{\w f^{(i)}(X_i)}-\int_{} \varphi(x)dx \right)
\overset{\P}{\lr} 0, \end{align*} where \w f^{(i)} is the classical
leave-one-out kernel estimator of the density of . This result is striking
because it speeds up traditional rates, in root , derived from the central
limit theorem when \w f^{(i)}=f. As a consequence, it improves the classical
Monte Carlo procedure for integral approximation. The paper mainly addressed
with theoretical issues related to the later result (rates of convergence,
bandwidth choice, regularity of ) but also interests some statistical
applications dealing with random design regression. In particular, we provide
the asymptotic normality of the estimation of the linear functionals of a
regression function on which the only requirement is the H\"older regularity.
This leads us to a new version of the \textit{average derivative estimator}
introduced by H\"ardle and Stoker in \cite{hardle1989} which allows for
\textit{dimension reduction} by estimating the \textit{index space} of a
regression
Integral approximation by kernel smoothing
Let be an i.i.d. sequence of random variables in
, . We show that, for any function , under regularity conditions, where
is the classical kernel estimator of the density of . This
result is striking because it speeds up traditional rates, in root , derived
from the central limit theorem when . Although this paper
highlights some applications, we mainly address theoretical issues related to
the later result. We derive upper bounds for the rate of convergence in
probability. These bounds depend on the regularity of the functions
and , the dimension and the bandwidth of the kernel estimator
. Moreover, they are shown to be accurate since they are used as
renormalizing sequences in two central limit theorems each reflecting different
degrees of smoothness of . As an application to regression modelling
with random design, we provide the asymptotic normality of the estimation of
the linear functionals of a regression function. As a consequence of the above
result, the asymptotic variance does not depend on the regression function.
Finally, we debate the choice of the bandwidth for integral approximation and
we highlight the good behavior of our procedure through simulations.Comment: Published at http://dx.doi.org/10.3150/15-BEJ725 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin
note: text overlap with arXiv:1312.449
Bootstrap Testing of the Rank of a Matrix via Least Squared Constrained Estimation
In order to test if an unknown matrix has a given rank (null hypothesis), we
consider the family of statistics that are minimum squared distances between an
estimator and the manifold of fixed-rank matrix. Under the null hypothesis,
every statistic of this family converges to a weighted chi-squared
distribution. In this paper, we introduce the constrained bootstrap to build
bootstrap estimate of the law under the null hypothesis of such statistics. As
a result, the constrained bootstrap is employed to estimate the quantile for
testing the rank. We provide the consistency of the procedure and the
simulations shed light one the accuracy of the constrained bootstrap with
respect to the traditional asymptotic comparison. More generally, the results
are extended to test if an unknown parameter belongs to a sub-manifold locally
smooth. Finally, the constrained bootstrap is easy to compute, it handles a
large family of tests and it works under mild assumptions
Exponential inequalities for dependent processes
We give exponential inequalities and Gaussian approximation results for sums of weakly dependent variables. These results lead to generalizations of Bernstein and Hoeffding inequalities, where an extra control term is added; this term contains conditional moments of the variables
Convergence rate of the powers of an operator. Applications to stochastic systems
International audienceWe extend the traditional operator theoretic approach for the study of dynamical systems in order to handle the problem of non-geometric convergence. We show that the probabilistic treatment developed and popularized under Richard Tweedie's impulsion, can be placed into an operator framework in the spirit of Yosida-Kakutani's approach. General Theorems as well as specific results for Markov chains are given. Application examples to general classes of Markov chains and dynamical systems are presented
Convex domains and K-spectral sets
Let be an open convex domain of the complex plane. We study
constants K such that is K-spectral or complete K-spectral for each
continuous linear Hilbert space operator with numerical range included in
. Several approaches are discussed.Comment: the introduction was changed and some remarks have been added. 26
pages ; to appear in Math.
Integral estimation based on Markovian design
Suppose that a mobile sensor describes a Markovian trajectory in the ambient
space. At each time the sensor measures an attribute of interest, e.g., the
temperature. Using only the location history of the sensor and the associated
measurements, the aim is to estimate the average value of the attribute over
the space. In contrast to classical probabilistic integration methods, e.g.,
Monte Carlo, the proposed approach does not require any knowledge on the
distribution of the sensor trajectory. Probabilistic bounds on the convergence
rates of the estimator are established. These rates are better than the
traditional "root n"-rate, where n is the sample size, attached to other
probabilistic integration methods. For finite sample sizes, the good behaviour
of the procedure is demonstrated through simulations and an application to the
evaluation of the average temperature of oceans is considered.Comment: 45 page
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