80 research outputs found
Discussion on the paper: Hypotheses testing by convex optimization by Goldenshluger, Juditsky and Nemirovski
We briefly discuss some interesting questions related to the paper
"Hypotheses testing by convex optimization" by Goldenshluger, Juditsky and
Nemirovski.Comment: To appear in the EJ
Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case
Asymptotic local equivalence in the sense of Le Cam is established for
inference on the drift in multidimensional ergodic diffusions and an
accompanying sequence of Gaussian shift experiments. The nonparametric local
neighbourhoods can be attained for any dimension, provided the regularity of
the drift is sufficiently large. In addition, a heteroskedastic Gaussian
regression experiment is given, which is also locally asymptotically equivalent
and which does not depend on the centre of localisation. For one direction of
the equivalence an explicit Markov kernel is constructed.Comment: 03 May 2005, 23 page
Second-order asymptotic expansion for a non-synchronous covariation estimator
In this paper, we consider the problem of estimating the covariation of two
diffusion processes when observations are subject to non-synchronicity.
Building on recent papers \cite{Hay-Yos03, Hay-Yos04}, we derive second-order
asymptotic expansions for the distribution of the Hayashi-Yoshida estimator in
a fairly general setup including random sampling schemes and non-anticipative
random drifts. The key steps leading to our results are a second-order
decomposition of the estimator's distribution in the Gaussian set-up, a
stochastic decomposition of the estimator itself and an accurate evaluation of
the Malliavin covariance. To give a concrete example, we compute the constants
involved in the resulting expansions for the particular case of sampling scheme
generated by two independent Poisson processes
Simple proof of the risk bound for denoising by exponential weights for asymmetric noise distributions
In this note, we consider the problem of aggregation of estimators in order
to denoise a signal. The main contribution is a short proof of the fact that
the exponentially weighted aggregate satisfies a sharp oracle inequality. While
this result was already known for a wide class of symmetric noise
distributions, the extension to asymmetric distributions presented in this note
is new
Sharp Oracle Inequalities for Aggregation of Affine Estimators
We consider the problem of combining a (possibly uncountably infinite) set of
affine estimators in non-parametric regression model with heteroscedastic
Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a
PAC-Bayesian type inequality that leads to sharp oracle inequalities in
discrete but also in continuous settings. The framework is general enough to
cover the combinations of various procedures such as least square regression,
kernel ridge regression, shrinking estimators and many other estimators used in
the literature on statistical inverse problems. As a consequence, we show that
the proposed aggregate provides an adaptive estimator in the exact minimax
sense without neither discretizing the range of tuning parameters nor splitting
the set of observations. We also illustrate numerically the good performance
achieved by the exponentially weighted aggregate
Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression
We consider the problem of testing a particular type of composite null
hypothesis under a nonparametric multivariate regression model. For a given
quadratic functional , the null hypothesis states that the regression
function satisfies the constraint , while the alternative
corresponds to the functions for which is bounded away from zero. On the
one hand, we provide minimax rates of testing and the exact separation
constants, along with a sharp-optimal testing procedure, for diagonal and
nonnegative quadratic functionals. We consider smoothness classes of
ellipsoidal form and check that our conditions are fulfilled in the particular
case of ellipsoids corresponding to anisotropic Sobolev classes. In this case,
we present a closed form of the minimax rate and the separation constant. On
the other hand, minimax rates for quadratic functionals which are neither
positive nor negative makes appear two different regimes: "regular" and
"irregular". In the "regular" case, the minimax rate is equal to
while in the "irregular" case, the rate depends on the smoothness class and is
slower than in the "regular" case. We apply this to the issue of testing the
equality of norms of two functions observed in noisy environments
Statistical inference in compound functional models
We consider a general nonparametric regression model called the compound
model. It includes, as special cases, sparse additive regression and
nonparametric (or linear) regression with many covariates but possibly a small
number of relevant covariates. The compound model is characterized by three
main parameters: the structure parameter describing the "macroscopic" form of
the compound function, the "microscopic" sparsity parameter indicating the
maximal number of relevant covariates in each component and the usual
smoothness parameter corresponding to the complexity of the members of the
compound. We find non-asymptotic minimax rate of convergence of estimators in
such a model as a function of these three parameters. We also show that this
rate can be attained in an adaptive way
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