529 research outputs found
Spatial aggregation of local likelihood estimates with applications to classification
This paper presents a new method for spatially adaptive local (constant)
likelihood estimation which applies to a broad class of nonparametric models,
including the Gaussian, Poisson and binary response models. The main idea of
the method is, given a sequence of local likelihood estimates (``weak''
estimates), to construct a new aggregated estimate whose pointwise risk is of
order of the smallest risk among all ``weak'' estimates. We also propose a new
approach toward selecting the parameters of the procedure by providing the
prescribed behavior of the resulting estimate in the simple parametric
situation. We establish a number of important theoretical results concerning
the optimality of the aggregated estimate. In particular, our ``oracle'' result
claims that its risk is, up to some logarithmic multiplier, equal to the
smallest risk for the given family of estimates. The performance of the
procedure is illustrated by application to the classification problem. A
numerical study demonstrates its reasonable performance in simulated and
real-life examples.Comment: Published in at http://dx.doi.org/10.1214/009053607000000271 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Statistical inference for time-inhomogeneous volatility models
This paper offers a new approach for estimating and forecasting the
volatility of financial time series. No assumption is made about the parametric
form of the processes. On the contrary, we only suppose that the volatility can
be approximated by a constant over some interval. In such a framework, the main
problem consists of filtering this interval of time homogeneity; then the
estimate of the volatility can be simply obtained by local averaging.
We construct a locally adaptive volatility estimate (LAVE) which can perform
this task and investigate it both from the theoretical point of view and
through Monte Carlo simulations. Finally, the LAVE procedure is applied to a
data set of nine exchange rates and a comparison with a standard GARCH model is
also provided. Both models appear to be capable of explaining many of the
features of the data; nevertheless, the new approach seems to be superior to
the GARCH method as far as the out-of-sample results are concerned
Parameter tuning in pointwise adaptation using a propagation approach
This paper discusses the problem of adaptive estimation of a univariate
object like the value of a regression function at a given point or a linear
functional in a linear inverse problem. We consider an adaptive procedure
originated from Lepski [Theory Probab. Appl. 35 (1990) 454--466.] that selects
in a data-driven way one estimate out of a given class of estimates ordered by
their variability. A serious problem with using this and similar procedures is
the choice of some tuning parameters like thresholds. Numerical results show
that the theoretically recommended proposals appear to be too conservative and
lead to a strong oversmoothing effect. A careful choice of the parameters of
the procedure is extremely important for getting the reasonable quality of
estimation. The main contribution of this paper is the new approach for
choosing the parameters of the procedure by providing the prescribed behavior
of the resulting estimate in the simple parametric situation. We establish a
non-asymptotical "oracle" bound, which shows that the estimation risk is, up to
a logarithmic multiplier, equal to the risk of the "oracle" estimate that is
optimally selected from the given family. A numerical study demonstrates a good
performance of the resulting procedure in a number of simulated examples.Comment: Published in at http://dx.doi.org/10.1214/08-AOS607 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
An adaptive multiclass nearest neighbor classifier
We consider a problem of multiclass classification, where the training sample
is generated from the model , , and are
unknown -Holder continuous functions.Given a test point , our goal
is to predict its label. A widely used -nearest-neighbors classifier
constructs estimates of and uses a plug-in rule
for the prediction. However, it requires a proper choice of the smoothing
parameter , which may become tricky in some situations. In our
solution, we fix several integers , compute corresponding
-nearest-neighbor estimates for each and each and apply an
aggregation procedure. We study an algorithm, which constructs a convex
combination of these estimates such that the aggregated estimate behaves
approximately as well as an oracle choice. We also provide a non-asymptotic
analysis of the procedure, prove its adaptation to the unknown smoothness
parameter and to the margin and establish rates of convergence under
mild assumptions.Comment: Accepted in ESAIM: Probability & Statistics. The original publication
is available at www.esaim-ps.or
Two convergence results for an alternation maximization procedure
Andresen and Spokoiny's (2013) ``critical dimension in semiparametric
estimation`` provide a technique for the finite sample analysis of profile
M-estimators. This paper uses very similar ideas to derive two convergence
results for the alternating procedure to approximate the maximizer of random
functionals such as the realized log likelihood in MLE estimation. We manage to
show that the sequence attains the same deviation properties as shown for the
profile M-estimator in Andresen and Spokoiny (2013), i.e. a finite sample Wilks
and Fisher theorem. Further under slightly stronger smoothness constraints on
the random functional we can show nearly linear convergence to the global
maximizer if the starting point for the procedure is well chosen
- …