12,879 research outputs found
The Bayesian Regularized Quantile Varying Coefficient Model
The quantile varying coefficient (VC) model can flexibly capture dynamical
patterns of regression coefficients. In addition, due to the quantile check
loss function, it is robust against outliers and heavy-tailed distributions of
the response variable, and can provide a more comprehensive picture of modeling
via exploring the conditional quantiles of the response variable. Although
extensive studies have been conducted to examine variable selection for the
high-dimensional quantile varying coefficient models, the Bayesian analysis has
been rarely developed. The Bayesian regularized quantile varying coefficient
model has been proposed to incorporate robustness against data heterogeneity
while accommodating the non-linear interactions between the effect modifier and
predictors. Selecting important varying coefficients can be achieved through
Bayesian variable selection. Incorporating the multivariate spike-and-slab
priors further improves performance by inducing exact sparsity. The Gibbs
sampler has been derived to conduct efficient posterior inference of the sparse
Bayesian quantile VC model through Markov chain Monte Carlo (MCMC). The merit
of the proposed model in selection and estimation accuracy over the
alternatives has been systematically investigated in simulation under specific
quantile levels and multiple heavy-tailed model errors. In the case study, the
proposed model leads to identification of biologically sensible markers in a
non-linear gene-environment interaction study using the NHS data
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
Conditional Transformation Models
The ultimate goal of regression analysis is to obtain information about the
conditional distribution of a response given a set of explanatory variables.
This goal is, however, seldom achieved because most established regression
models only estimate the conditional mean as a function of the explanatory
variables and assume that higher moments are not affected by the regressors.
The underlying reason for such a restriction is the assumption of additivity of
signal and noise. We propose to relax this common assumption in the framework
of transformation models. The novel class of semiparametric regression models
proposed herein allows transformation functions to depend on explanatory
variables. These transformation functions are estimated by regularised
optimisation of scoring rules for probabilistic forecasts, e.g. the continuous
ranked probability score. The corresponding estimated conditional distribution
functions are consistent. Conditional transformation models are potentially
useful for describing possible heteroscedasticity, comparing spatially varying
distributions, identifying extreme events, deriving prediction intervals and
selecting variables beyond mean regression effects. An empirical investigation
based on a heteroscedastic varying coefficient simulation model demonstrates
that semiparametric estimation of conditional distribution functions can be
more beneficial than kernel-based non-parametric approaches or parametric
generalised additive models for location, scale and shape
Model-based Boosting in R: A Hands-on Tutorial Using the R Package mboost
We provide a detailed hands-on tutorial for the R add-on package mboost. The package implements boosting for optimizing general risk functions utilizing component-wise (penalized) least squares estimates as base-learners for fitting various kinds of generalized linear and generalized additive models to potentially high-dimensional data. We give a theoretical background and demonstrate how mboost can be used to fit interpretable models of different complexity. As an example we use mboost to predict the body fat based on anthropometric measurements throughout the tutorial
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
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