2,800 research outputs found
Pivotal estimation via square-root Lasso in nonparametric regression
We propose a self-tuning method that simultaneously
resolves three important practical problems in high-dimensional regression
analysis, namely it handles the unknown scale, heteroscedasticity and (drastic)
non-Gaussianity of the noise. In addition, our analysis allows for badly
behaved designs, for example, perfectly collinear regressors, and generates
sharp bounds even in extreme cases, such as the infinite variance case and the
noiseless case, in contrast to Lasso. We establish various nonasymptotic bounds
for including prediction norm rate and sparsity. Our
analysis is based on new impact factors that are tailored for bounding
prediction norm. In order to cover heteroscedastic non-Gaussian noise, we rely
on moderate deviation theory for self-normalized sums to achieve Gaussian-like
results under weak conditions. Moreover, we derive bounds on the performance of
ordinary least square (ols) applied to the model selected by accounting for possible misspecification of the selected model. Under
mild conditions, the rate of convergence of ols post
is as good as 's rate. As an application, we consider
the use of and ols post as
estimators of nuisance parameters in a generic semiparametric problem
(nonlinear moment condition or -problem), resulting in a construction of
-consistent and asymptotically normal estimators of the main
parameters.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1204 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
There is a VaR beyond usual approximations
Basel II and Solvency 2 both use the Value-at-Risk (VaR) as the risk measure
to compute the Capital Requirements. In practice, to calibrate the VaR, a
normal approximation is often chosen for the unknown distribution of the yearly
log returns of financial assets. This is usually justified by the use of the
Central Limit Theorem (CLT), when assuming aggregation of independent and
identically distributed (iid) observations in the portfolio model. Such a
choice of modeling, in particular using light tail distributions, has proven
during the crisis of 2008/2009 to be an inadequate approximation when dealing
with the presence of extreme returns; as a consequence, it leads to a gross
underestimation of the risks. The main objective of our study is to obtain the
most accurate evaluations of the aggregated risks distribution and risk
measures when working on financial or insurance data under the presence of
heavy tail and to provide practical solutions for accurately estimating high
quantiles of aggregated risks. We explore a new method, called Normex, to
handle this problem numerically as well as theoretically, based on properties
of upper order statistics. Normex provides accurate results, only weakly
dependent upon the sample size and the tail index. We compare it with existing
methods.Comment: 33 pages, 5 figure
Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail
We prove precise deviations results in the sense of Cram\'er and Petrov for
the upper tail of the distribution of the maximal value for a special class of
determinantal point processes that play an important role in random matrix
theory. Here we cover all three regimes of moderate, large and superlarge
deviations for which we determine the leading order description of the tail
probabilities. As a corollary of our results we identify the region within the
regime of moderate deviations for which the limiting Tracy-Widom law still
predicts the correct leading order behavior. Our proofs use that the
determinantal point process is given by the Christoffel-Darboux kernel for an
associated family of orthogonal polynomials. The necessary asymptotic
information on this kernel has mostly been obtained in [Kriecherbauer T.,
Schubert K., Sch\"uler K., Venker M., Markov Process. Related Fields 21 (2015),
639-694]. In the superlarge regime these results of do not suffice and we put
stronger assumptions on the point processes. The results of the present paper
and the relevant parts of [Kriecherbauer T., Schubert K., Sch\"uler K., Venker
M., Markov Process. Related Fields 21 (2015), 639-694] have been proved in the
dissertation [Sch\"uler K., Ph.D. Thesis, Universit\"at Bayreuth, 2015].Comment: 18 page
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