8,861 research outputs found
Partially linear additive quantile regression in ultra-high dimension
We consider a flexible semiparametric quantile regression model for analyzing
high dimensional heterogeneous data. This model has several appealing features:
(1) By considering different conditional quantiles, we may obtain a more
complete picture of the conditional distribution of a response variable given
high dimensional covariates. (2) The sparsity level is allowed to be different
at different quantile levels. (3) The partially linear additive structure
accommodates nonlinearity and circumvents the curse of dimensionality. (4) It
is naturally robust to heavy-tailed distributions. In this paper, we
approximate the nonlinear components using B-spline basis functions. We first
study estimation under this model when the nonzero components are known in
advance and the number of covariates in the linear part diverges. We then
investigate a nonconvex penalized estimator for simultaneous variable selection
and estimation. We derive its oracle property for a general class of nonconvex
penalty functions in the presence of ultra-high dimensional covariates under
relaxed conditions. To tackle the challenges of nonsmooth loss function,
nonconvex penalty function and the presence of nonlinear components, we combine
a recently developed convex-differencing method with modern empirical process
techniques. Monte Carlo simulations and an application to a microarray study
demonstrate the effectiveness of the proposed method. We also discuss how the
method for a single quantile of interest can be extended to simultaneous
variable selection and estimation at multiple quantiles.Comment: Published at http://dx.doi.org/10.1214/15-AOS1367 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Local Quantile Regression
Quantile regression is a technique to estimate conditional quantile curves.
It provides a comprehensive picture of a response contingent on explanatory
variables. In a flexible modeling framework, a specific form of the conditional
quantile curve is not a priori fixed. % Indeed, the majority of applications do
not per se require specific functional forms. This motivates a local parametric
rather than a global fixed model fitting approach. A nonparametric smoothing
estimator of the conditional quantile curve requires to balance between local
curvature and stochastic variability. In this paper, we suggest a local model
selection technique that provides an adaptive estimator of the conditional
quantile regression curve at each design point. Theoretical results claim that
the proposed adaptive procedure performs as good as an oracle which would
minimize the local estimation risk for the problem at hand. We illustrate the
performance of the procedure by an extensive simulation study and consider a
couple of applications: to tail dependence analysis for the Hong Kong stock
market and to analysis of the distributions of the risk factors of temperature
dynamics
Penalized single-index quantile regression
This article is made available through the Brunel Open Access Publishing Fund. Copyright for this article is retained by the author(s), with first publication rights granted to the journal.
This is an open-access article distributed under the terms and conditions of the Creative Commons Attribution
license (http://creativecommons.org/licenses/by/3.0/).The single-index (SI) regression and single-index quantile (SIQ) estimation methods product linear combinations of all the original predictors. However, it is possible that there are many unimportant predictors within the original predictors. Thus, the precision of parameter estimation as well as the accuracy of prediction will be effected by the existence of those unimportant predictors when the previous methods are used. In this article, an extension of the SIQ method of Wu et al. (2010) has been proposed, which considers Lasso and Adaptive Lasso for estimation and variable selection. Computational algorithms have been developed in order to calculate the penalized SIQ estimates. A simulation study and a real data application have been used to assess the performance of the methods under consideration
Penalized Composite Quasi-Likelihood for Ultrahigh-Dimensional Variable Selection
In high-dimensional model selection problems, penalized simple least-square
approaches have been extensively used. This paper addresses the question of
both robustness and efficiency of penalized model selection methods, and
proposes a data-driven weighted linear combination of convex loss functions,
together with weighted -penalty. It is completely data-adaptive and does
not require prior knowledge of the error distribution. The weighted
-penalty is used both to ensure the convexity of the penalty term and to
ameliorate the bias caused by the -penalty. In the setting with
dimensionality much larger than the sample size, we establish a strong oracle
property of the proposed method that possesses both the model selection
consistency and estimation efficiency for the true non-zero coefficients. As
specific examples, we introduce a robust method of composite L1-L2, and optimal
composite quantile method and evaluate their performance in both simulated and
real data examples
Intersection Bounds: Estimation and Inference
We develop a practical and novel method for inference on intersection bounds,
namely bounds defined by either the infimum or supremum of a parametric or
nonparametric function, or equivalently, the value of a linear programming
problem with a potentially infinite constraint set. We show that many bounds
characterizations in econometrics, for instance bounds on parameters under
conditional moment inequalities, can be formulated as intersection bounds. Our
approach is especially convenient for models comprised of a continuum of
inequalities that are separable in parameters, and also applies to models with
inequalities that are non-separable in parameters. Since analog estimators for
intersection bounds can be severely biased in finite samples, routinely
underestimating the size of the identified set, we also offer a
median-bias-corrected estimator of such bounds as a by-product of our
inferential procedures. We develop theory for large sample inference based on
the strong approximation of a sequence of series or kernel-based empirical
processes by a sequence of "penultimate" Gaussian processes. These penultimate
processes are generally not weakly convergent, and thus non-Donsker. Our
theoretical results establish that we can nonetheless perform asymptotically
valid inference based on these processes. Our construction also provides new
adaptive inequality/moment selection methods. We provide conditions for the use
of nonparametric kernel and series estimators, including a novel result that
establishes strong approximation for any general series estimator admitting
linearization, which may be of independent interest
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