594 research outputs found
Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression
We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the
elastic-net penalized Huber loss regression and quantile regression in high
dimensional settings. Unlike existing coordinate descent type algorithms, the
SNCD updates each regression coefficient and its corresponding subgradient
simultaneously in each iteration. It combines the strengths of the coordinate
descent and the semismooth Newton algorithm, and effectively solves the
computational challenges posed by dimensionality and nonsmoothness. We
establish the convergence properties of the algorithm. In addition, we present
an adaptive version of the "strong rule" for screening predictors to gain extra
efficiency. Through numerical experiments, we demonstrate that the proposed
algorithm is very efficient and scalable to ultra-high dimensions. We
illustrate the application via a real data example
Sparse least trimmed squares regression.
Sparse model estimation is a topic of high importance in modern data analysis due to the increasing availability of data sets with a large number of variables. Another common problem in applied statistics is the presence of outliers in the data. This paper combines robust regression and sparse model estimation. A robust and sparse estimator is introduced by adding an L1 penalty on the coefficient estimates to the well known least trimmed squares (LTS) estimator. The breakdown point of this sparse LTS estimator is derived, and a fast algorithm for its computation is proposed. Both the simulation study and the real data example show that the LTS has better prediction performance than its competitors in the presence of leverage points.Breakdown point; Outliers; Penalized regression; Robust regression; Trimming;
Stability
Reproducibility is imperative for any scientific discovery. More often than
not, modern scientific findings rely on statistical analysis of
high-dimensional data. At a minimum, reproducibility manifests itself in
stability of statistical results relative to "reasonable" perturbations to data
and to the model used. Jacknife, bootstrap, and cross-validation are based on
perturbations to data, while robust statistics methods deal with perturbations
to models. In this article, a case is made for the importance of stability in
statistics. Firstly, we motivate the necessity of stability for interpretable
and reliable encoding models from brain fMRI signals. Secondly, we find strong
evidence in the literature to demonstrate the central role of stability in
statistical inference, such as sensitivity analysis and effect detection.
Thirdly, a smoothing parameter selector based on estimation stability (ES),
ES-CV, is proposed for Lasso, in order to bring stability to bear on
cross-validation (CV). ES-CV is then utilized in the encoding models to reduce
the number of predictors by 60% with almost no loss (1.3%) of prediction
performance across over 2,000 voxels. Last, a novel "stability" argument is
seen to drive new results that shed light on the intriguing interactions
between sample to sample variability and heavier tail error distribution (e.g.,
double-exponential) in high-dimensional regression models with predictors
and independent samples. In particular, when
and the error distribution is
double-exponential, the Ordinary Least Squares (OLS) is a better estimator than
the Least Absolute Deviation (LAD) estimator.Comment: Published in at http://dx.doi.org/10.3150/13-BEJSP14 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Adaptive estimation with partially overlapping models
In many problems, one has several models of interest that capture key parameters describing the distribution of the data. Partially overlapping models are taken as models in which at least one covariate effect is common to the models. A priori knowledge of such structure enables efficient estimation of all model parameters. However, in practice, this structure may be unknown. We propose adaptive composite M-estimation (ACME) for partially overlapping models using a composite loss function, which is a linear combination of loss functions defining the individual models. Penalization is applied to pairwise differences of parameters across models, resulting in data driven identification of the overlap structure. Further penalization is imposed on the individual parameters, enabling sparse estimation in the regression setting. The recovery of the overlap structure enables more efficient parameter estimation. An oracle result is established. Simulation studies illustrate the advantages of ACME over existing methods that fit individual models separately or make strong a priori assumption about the overlap structure
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