26,189 research outputs found
Estimator selection: a new method with applications to kernel density estimation
Estimator selection has become a crucial issue in non parametric estimation.
Two widely used methods are penalized empirical risk minimization (such as
penalized log-likelihood estimation) or pairwise comparison (such as Lepski's
method). Our aim in this paper is twofold. First we explain some general ideas
about the calibration issue of estimator selection methods. We review some
known results, putting the emphasis on the concept of minimal penalty which is
helpful to design data-driven selection criteria. Secondly we present a new
method for bandwidth selection within the framework of kernel density density
estimation which is in some sense intermediate between these two main methods
mentioned above. We provide some theoretical results which lead to some fully
data-driven selection strategy
Minimal penalty for Goldenshluger-Lepski method
This paper is concerned with adaptive nonparametric estimation using the
Goldenshluger-Lepski selection method. This estimator selection method is based
on pairwise comparisons between estimators with respect to some loss function.
The method also involves a penalty term that typically needs to be large enough
in order that the method works (in the sense that one can prove some oracle
type inequality for the selected estimator). In the case of density estimation
with kernel estimators and a quadratic loss, we show that the procedure fails
if the penalty term is chosen smaller than some critical value for the penalty:
the minimal penalty. More precisely we show that the quadratic risk of the
selected estimator explodes when the penalty is below this critical value while
it stays under control when the penalty is above this critical value. This kind
of phase transition phenomenon for penalty calibration has already been
observed and proved for penalized model selection methods in various contexts
but appears here for the first time for the Goldenshluger-Lepski pairwise
comparison method. Some simulations illustrate the theoretical results and lead
to some hints on how to use the theory to calibrate the method in practice
Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality
We address the problem of density estimation with -loss by
selection of kernel estimators. We develop a selection procedure and derive
corresponding -risk oracle inequalities. It is shown that the
proposed selection rule leads to the estimator being minimax adaptive over a
scale of the anisotropic Nikol'skii classes. The main technical tools used in
our derivations are uniform bounds on the -norms of empirical
processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011),
to appear].Comment: Published in at http://dx.doi.org/10.1214/11-AOS883 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Spline-backfitted kernel smoothing of nonlinear additive autoregression model
Application of nonparametric and semiparametric regression techniques to
high-dimensional time series data has been hampered due to the lack of
effective tools to address the ``curse of dimensionality.'' Under rather weak
conditions, we propose spline-backfitted kernel estimators of the component
functions for the nonlinear additive time series data that are both
computationally expedient so they are usable for analyzing very
high-dimensional time series, and theoretically reliable so inference can be
made on the component functions with confidence. Simulation experiments have
provided strong evidence that corroborates the asymptotic theory.Comment: Published in at http://dx.doi.org/10.1214/009053607000000488 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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