1,497 research outputs found
Quantile regression in partially linear varying coefficient models
Semiparametric models are often considered for analyzing longitudinal data
for a good balance between flexibility and parsimony. In this paper, we study a
class of marginal partially linear quantile models with possibly varying
coefficients. The functional coefficients are estimated by basis function
approximations. The estimation procedure is easy to implement, and it requires
no specification of the error distributions. The asymptotic properties of the
proposed estimators are established for the varying coefficients as well as for
the constant coefficients. We develop rank score tests for hypotheses on the
coefficients, including the hypotheses on the constancy of a subset of the
varying coefficients. Hypothesis testing of this type is theoretically
challenging, as the dimensions of the parameter spaces under both the null and
the alternative hypotheses are growing with the sample size. We assess the
finite sample performance of the proposed method by Monte Carlo simulation
studies, and demonstrate its value by the analysis of an AIDS data set, where
the modeling of quantiles provides more comprehensive information than the
usual least squares approach.Comment: Published in at http://dx.doi.org/10.1214/09-AOS695 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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
Disease Mapping via Negative Binomial Regression M-quantiles
We introduce a semi-parametric approach to ecological regression for disease
mapping, based on modelling the regression M-quantiles of a Negative Binomial
variable. The proposed method is robust to outliers in the model covariates,
including those due to measurement error, and can account for both spatial
heterogeneity and spatial clustering. A simulation experiment based on the
well-known Scottish lip cancer data set is used to compare the M-quantile
modelling approach and a random effects modelling approach for disease mapping.
This suggests that the M-quantile approach leads to predicted relative risks
with smaller root mean square error than standard disease mapping methods. The
paper concludes with an illustrative application of the M-quantile approach,
mapping low birth weight incidence data for English Local Authority Districts
for the years 2005-2010.Comment: 23 pages, 7 figure
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