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Estimation and model selection in generalized additive partial linear models for correlated data with diverging number of covariates
We propose generalized additive partial linear models for complex data which
allow one to capture nonlinear patterns of some covariates, in the presence of
linear components. The proposed method improves estimation efficiency and
increases statistical power for correlated data through incorporating the
correlation information. A unique feature of the proposed method is its
capability of handling model selection in cases where it is difficult to
specify the likelihood function. We derive the quadratic inference
function-based estimators for the linear coefficients and the nonparametric
functions when the dimension of covariates diverges, and establish asymptotic
normality for the linear coefficient estimators and the rates of convergence
for the nonparametric functions estimators for both finite and high-dimensional
cases. The proposed method and theoretical development are quite challenging
since the numbers of linear covariates and nonlinear components both increase
as the sample size increases. We also propose a doubly penalized procedure for
variable selection which can simultaneously identify nonzero linear and
nonparametric components, and which has an asymptotic oracle property.
Extensive Monte Carlo studies have been conducted and show that the proposed
procedure works effectively even with moderate sample sizes. A pharmacokinetics
study on renal cancer data is illustrated using the proposed method.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1194 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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