27,373 research outputs found
Sparse Additive Models
We present a new class of methods for high-dimensional nonparametric
regression and classification called sparse additive models (SpAM). Our methods
combine ideas from sparse linear modeling and additive nonparametric
regression. We derive an algorithm for fitting the models that is practical and
effective even when the number of covariates is larger than the sample size.
SpAM is closely related to the COSSO model of Lin and Zhang (2006), but
decouples smoothing and sparsity, enabling the use of arbitrary nonparametric
smoothers. An analysis of the theoretical properties of SpAM is given. We also
study a greedy estimator that is a nonparametric version of forward stepwise
regression. Empirical results on synthetic and real data are presented, showing
that SpAM can be effective in fitting sparse nonparametric models in high
dimensional data
Markov-switching generalized additive models
We consider Markov-switching regression models, i.e. models for time series
regression analyses where the functional relationship between covariates and
response is subject to regime switching controlled by an unobservable Markov
chain. Building on the powerful hidden Markov model machinery and the methods
for penalized B-splines routinely used in regression analyses, we develop a
framework for nonparametrically estimating the functional form of the effect of
the covariates in such a regression model, assuming an additive structure of
the predictor. The resulting class of Markov-switching generalized additive
models is immensely flexible, and contains as special cases the common
parametric Markov-switching regression models and also generalized additive and
generalized linear models. The feasibility of the suggested maximum penalized
likelihood approach is demonstrated by simulation and further illustrated by
modelling how energy price in Spain depends on the Euro/Dollar exchange rate
Variable selection in nonparametric additive models
We consider a nonparametric additive model of a conditional mean function in
which the number of variables and additive components may be larger than the
sample size but the number of nonzero additive components is "small" relative
to the sample size. The statistical problem is to determine which additive
components are nonzero. The additive components are approximated by truncated
series expansions with B-spline bases. With this approximation, the problem of
component selection becomes that of selecting the groups of coefficients in the
expansion. We apply the adaptive group Lasso to select nonzero components,
using the group Lasso to obtain an initial estimator and reduce the dimension
of the problem. We give conditions under which the group Lasso selects a model
whose number of components is comparable with the underlying model, and the
adaptive group Lasso selects the nonzero components correctly with probability
approaching one as the sample size increases and achieves the optimal rate of
convergence. The results of Monte Carlo experiments show that the adaptive
group Lasso procedure works well with samples of moderate size. A data example
is used to illustrate the application of the proposed method.Comment: Published in at http://dx.doi.org/10.1214/09-AOS781 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Smooth backfitting in generalized additive models
Generalized additive models have been popular among statisticians and data
analysts in multivariate nonparametric regression with non-Gaussian responses
including binary and count data. In this paper, a new likelihood approach for
fitting generalized additive models is proposed. It aims to maximize a smoothed
likelihood. The additive functions are estimated by solving a system of
nonlinear integral equations. An iterative algorithm based on smooth
backfitting is developed from the Newton--Kantorovich theorem. Asymptotic
properties of the estimator and convergence of the algorithm are discussed. It
is shown that our proposal based on local linear fit achieves the same bias and
variance as the oracle estimator that uses knowledge of the other components.
Numerical comparison with the recently proposed two-stage estimator [Ann.
Statist. 32 (2004) 2412--2443] is also made.Comment: Published in at http://dx.doi.org/10.1214/009053607000000596 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Comparisons of Concordance in Additive Models
In this note we compare bivariate additive models with respect to their Pearson correlation coecients, Kendall's concordance coecients, and Blomqvist medial correlation coefcients. The conditions that enable the comparisons involve variability stochastic orders such as the dispersive and the peakedness orders. Specically we show that we can compare the Kendall's concordance coecients of Cheriyan and Ramabhadran's bivariate gamma distributions, in spite of the fact that it is hard (and not necessary) to compute the
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