132 research outputs found
Functional approach for excess mass estimation in the density model
We consider a multivariate density model where we estimate the excess mass of
the unknown probability density at a given level from i.i.d.
observed random variables. This problem has several applications such as
multimodality testing, density contour clustering, anomaly detection,
classification and so on. For the first time in the literature we estimate the
excess mass as an integrated functional of the unknown density . We suggest
an estimator and evaluate its rate of convergence, when belongs to general
Besov smoothness classes, for several risk measures. A particular care is
devoted to implementation and numerical study of the studied procedure. It
appears that our procedure improves the plug-in estimator of the excess mass.Comment: Published in at http://dx.doi.org/10.1214/07-EJS079 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Thresholding methods to estimate the copula density
This paper deals with the problem of the multivariate copula density
estimation. Using wavelet methods we provide two shrinkage procedures based on
thresholding rules for which the knowledge of the regularity of the copula
density to be estimated is not necessary. These methods, said to be adaptive,
are proved to perform very well when adopting the minimax and the maxiset
approaches. Moreover we show that these procedures can be discriminated in the
maxiset sense. We produce an estimation algorithm whose qualities are evaluated
thanks some simulation. Last, we propose a real life application for financial
data
Thresholding methods to estimate the copula density
This paper deals with the problem of the multivariate copula density
estimation. Using wavelet methods we provide two shrinkage procedures based on
thresholding rules for which the knowledge of the regularity of the copula
density to be estimated is not necessary. These methods, said to be adaptive,
are proved to perform very well when adopting the minimax and the maxiset
approaches. Moreover we show that these procedures can be discriminated in the
maxiset sense. We produce an estimation algorithm whose qualities are evaluated
thanks some simulation. Last, we propose a real life application for financial
data
A test of goodness-of-fit for the copula densities
We consider the problem of testing hypotheses on the copula density from
bi-dimensional observations. We wish to test the null hypothesis characterized
by a parametric class against a composite nonparametric alternative. Each
density under the alternative is separated in the -norm from any density
lying in the null hypothesis. The copula densities under consideration are
supposed to belong to a range of Besov balls. According to the minimax
approach, the testing problem is solved in an adaptive framework: it leads to a
term loss in the minimax rate of testing in comparison with the
non-adaptive case. A smoothness-free test statistic that achieves the minimax
rate is proposed. The lower bound is also proved. Besides, the empirical
performance of the test procedure is demonstrated with both simulated and real
data
Grouping Strategies and Thresholding for High Dimensional Linear Models
The estimation problem in a high regression model with structured sparsity is
investigated. An algorithm using a two steps block thresholding procedure
called GR-LOL is provided. Convergence rates are produced: they depend on
simple coherence-type indices of the Gram matrix -easily checkable on the data-
as well as sparsity assumptions of the model parameters measured by a
combination of within-blocks with between-blocks norms. The
simplicity of the coherence indicator suggests ways to optimize the rates of
convergence when the group structure is not naturally given by the problem and
is unknown. In such a case, an auto-driven procedure is provided to determine
the regressors groups (number and contents). An intensive practical study
compares our grouping methods with the standard LOL algorithm. We prove that
the grouping rarely deteriorates the results but can improve them very
significantly. GR-LOL is also compared with group-Lasso procedures and exhibits
a very encouraging behavior. The results are quite impressive, especially when
GR-LOL algorithm is combined with a grouping pre-processing
Adaptive estimation of the transition density of a particular hidden Markov chain
We study the following model of hidden Markov chain: , with a real-valued positive recurrent and stationary
Markov chain and a noise independent of the
sequence having a known distribution. We present an adaptive estimator
of the transition density based on the quotient of a deconvolution estimator of
the density of and an estimator of the density of . These
estimators are obtained by contrast minimization and model selection. We
evaluate the risk and its rate of convergence for ordinary smooth and
supersmooth noise with regard to ordinary smooth and supersmooth chains. Some
examples are also detailed
Nonparametric estimation of the stationary density and the transition density of a Markov chain
In this paper, we study first the problem of nonparametric estimation of the
stationary density of a discrete-time Markov chain . We consider a
collection of projection estimators on finite dimensional linear spaces. We
select an estimator among the collection by minimizing a penalized contrast.
The same technique enables to estimate the density of and
so to provide an adaptive estimator of the transition density . We
give bounds in norm for these estimators and we show that they are
adaptive in the minimax sense over a large class of Besov spaces. Some examples
and simulations are also provided
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