375 research outputs found
Group Lasso for high dimensional sparse quantile regression models
This paper studies the statistical properties of the group Lasso estimator
for high dimensional sparse quantile regression models where the number of
explanatory variables (or the number of groups of explanatory variables) is
possibly much larger than the sample size while the number of variables in
"active" groups is sufficiently small. We establish a non-asymptotic bound on
the -estimation error of the estimator. This bound explains
situations under which the group Lasso estimator is potentially
superior/inferior to the -penalized quantile regression estimator in
terms of the estimation error. We also propose a data-dependent choice of the
tuning parameter to make the method more practical, by extending the original
proposal of Belloni and Chernozhukov (2011) for the -penalized
quantile regression estimator. As an application, we analyze high dimensional
additive quantile regression models. We show that under a set of suitable
regularity conditions, the group Lasso estimator can attain the convergence
rate arbitrarily close to the oracle rate. Finally, we conduct simulations
experiments to examine our theoretical results.Comment: 37 pages. Some errors are correcte
Estimation in functional linear quantile regression
This paper studies estimation in functional linear quantile regression in
which the dependent variable is scalar while the covariate is a function, and
the conditional quantile for each fixed quantile index is modeled as a linear
functional of the covariate. Here we suppose that covariates are discretely
observed and sampling points may differ across subjects, where the number of
measurements per subject increases as the sample size. Also, we allow the
quantile index to vary over a given subset of the open unit interval, so the
slope function is a function of two variables: (typically) time and quantile
index. Likewise, the conditional quantile function is a function of the
quantile index and the covariate. We consider an estimator for the slope
function based on the principal component basis. An estimator for the
conditional quantile function is obtained by a plug-in method. Since the
so-constructed plug-in estimator not necessarily satisfies the monotonicity
constraint with respect to the quantile index, we also consider a class of
monotonized estimators for the conditional quantile function. We establish
rates of convergence for these estimators under suitable norms, showing that
these rates are optimal in a minimax sense under some smoothness assumptions on
the covariance kernel of the covariate and the slope function. Empirical choice
of the cutoff level is studied by using simulations.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1066 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Comparison and anti-concentration bounds for maxima of Gaussian random vectors
Slepian and Sudakov-Fernique type inequalities, which compare expectations of
maxima of Gaussian random vectors under certain restrictions on the covariance
matrices, play an important role in probability theory, especially in empirical
process and extreme value theories. Here we give explicit comparisons of
expectations of smooth functions and distribution functions of maxima of
Gaussian random vectors without any restriction on the covariance matrices. We
also establish an anti-concentration inequality for the maximum of a Gaussian
random vector, which derives a useful upper bound on the L\'{e}vy concentration
function for the Gaussian maximum. The bound is dimension-free and applies to
vectors with arbitrary covariance matrices. This anti-concentration inequality
plays a crucial role in establishing bounds on the Kolmogorov distance between
maxima of Gaussian random vectors. These results have immediate applications in
mathematical statistics. As an example of application, we establish a
conditional multiplier central limit theorem for maxima of sums of independent
random vectors where the dimension of the vectors is possibly much larger than
the sample size.Comment: 22 pages; discussions and references update
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