148 research outputs found
Confidence bands for densities, logarithmic point of view
Let be a probability density and be an interval on which is
bounded away from zero. By establishing the limiting distribution of the
uniform error of the kernel estimates of , Bickel and Rosenblatt
(1973) provide confidence bands for on with asymptotic level
. Each of the confidence intervals whose union gives
has an asymptotic level equal to one; pointwise moderate deviations principles
allow to prove that all these intervals share the same logarithmic asymptotic
level. Now, as soon as both pointwise and uniform moderate deviations
principles for exist, they share the same asymptotics. Taking this
observation as a starting point, we present a new approach for the construction
of confidence bands for , based on the use of moderate deviations
principles. The advantages of this approach are the following: (i) it enables
to construct confidence bands, which have the same width (or even a smaller
width) as the confidence bands provided by Bickel and Rosenblatt (1973), but
which have a better aymptotic level; (ii) any confidence band constructed in
that way shares the same logarithmic asymptotic level as all the confidence
intervals, which make up this confidence band; (iii) it allows to deal with all
the dimensions in the same way; (iv) it enables to sort out the problem of
providing confidence bands for on compact sets on which vanishes (or on
all \bb R^d), by introducing a truncating operation
Convergence rate and averaging of nonlinear two-time-scale stochastic approximation algorithms
The first aim of this paper is to establish the weak convergence rate of
nonlinear two-time-scale stochastic approximation algorithms. Its second aim is
to introduce the averaging principle in the context of two-time-scale
stochastic approximation algorithms. We first define the notion of asymptotic
efficiency in this framework, then introduce the averaged two-time-scale
stochastic approximation algorithm, and finally establish its weak convergence
rate. We show, in particular, that both components of the averaged
two-time-scale stochastic approximation algorithm simultaneously converge at
the optimal rate .Comment: Published at http://dx.doi.org/10.1214/105051606000000448 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
A companion for the Kiefer--Wolfowitz--Blum stochastic approximation algorithm
A stochastic algorithm for the recursive approximation of the location
of a maximum of a regression function was introduced by Kiefer and
Wolfowitz [Ann. Math. Statist. 23 (1952) 462--466] in the univariate framework,
and by Blum [Ann. Math. Statist. 25 (1954) 737--744] in the multivariate case.
The aim of this paper is to provide a companion algorithm to the
Kiefer--Wolfowitz--Blum algorithm, which allows one to simultaneously
recursively approximate the size of the maximum of the regression
function. A precise study of the joint weak convergence rate of both algorithms
is given; it turns out that, unlike the location of the maximum, the size of
the maximum can be approximated by an algorithm which converges at the
parametric rate. Moreover, averaging leads to an asymptotically efficient
algorithm for the approximation of the couple .Comment: Published in at http://dx.doi.org/10.1214/009053606000001451 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Limit of normalized quadrangulations: The Brownian map
Consider a random pointed quadrangulation chosen equally likely among
the pointed quadrangulations with faces. In this paper we show that, when
goes to , suitably normalized converges weakly in a certain
sense to a random limit object, which is continuous and compact, and that we
name the Brownian map. The same result is shown for a model of rooted
quadrangulations and for some models of rooted quadrangulations with random
edge lengths. A metric space of rooted (resp. pointed) abstract maps that
contains the model of discrete rooted (resp. pointed) quadrangulations and the
model of the Brownian map is defined. The weak convergences hold in these
metric spaces.Comment: Published at http://dx.doi.org/10.1214/009117906000000557 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Large and moderate deviations principles for recursive kernel estimators of a multivariate density and its partial derivatives
In this paper we prove large and moderate deviations principles for the
recursive kernel estimator of a probability density function and its partial
derivatives. Unlike the density estimator, the derivatives estimators exhibit a
quadratic behavior not only for the moderate deviations scale but also for the
large deviations one. We provide results both for the pointwise and the uniform
deviations.Comment: 26 page
The stochastic approximation method for the estimation of a multivariate probability density
We apply the stochastic approximation method to construct a large class of
recursive kernel estimators of a probability density, including the one
introduced by Hall and Patil (1994). We study the properties of these
estimators and compare them with Rosenblatt's nonrecursive estimator. It turns
out that, for pointwise estimation, it is preferable to use the nonrecursive
Rosenblatt's kernel estimator rather than any recursive estimator. A contrario,
for estimation by confidence intervals, it is better to use a recursive
estimator rather than Rosenblatt's estimator.Comment: 28 page
Large and moderate deviations principles for kernel estimators of the multivariate regression
In this paper, we prove large deviations principle for the Nadaraya-Watson
estimator and for the semi-recursive kernel estimator of the regression in the
multidimensional case. Under suitable conditions, we show that the rate
function is a good rate function. We thus generalize the results already
obtained in the unidimensional case for the Nadaraya-Watson estimator.
Moreover, we give a moderate deviations principle for these two estimators. It
turns out that the rate function obtained in the moderate deviations principle
for the semi-recursive estimator is larger than the one obtained for the
Nadaraya-Watson estimator.Comment: 31 page
Large and Moderate Deviations Principles for Recursive Kernel Estimator of a Multivariate Density and its Partial Derivatives
2000 Mathematics Subject Classification: 62G07, 60F10.In this paper we prove large and moderate deviations principles for the recursive kernel estimator of a probability density function and its partial derivatives. Unlike the density estimator, the derivatives estimators exhibit a quadratic behaviour not only for the moderate deviations scale but also for the large deviations one. We provide results both for the pointwise and the uniform deviations
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