  We consider the problems of variable selection and estimation in
nonparametric additive regression models for high-dimensional data. In recent
years, several methods have been proposed to model nonlinear relationships when
the number of covariates exceeds the number of observations by using spline
basis functions and group penalties. Nonlinear {\it monotone} effects on the
response play a central role in many situations, in particular in medicine and
biology. We construct the monotone splines lasso (MS-lasso) to select variables
and estimate effects using monotone spline (I-splines). The additive components
in the model are represented by their I-spline basis function expansion and the
component selection becomes that of selecting the groups of coefficients in the
I-spline basis function expansion. We use a recent procedure, called
cooperative lasso, to select sign-coherent groups, that is selecting the groups
with either exclusively non-negative or non-positive coefficients. This leads
to the selection of important covariates that have nonlinear monotone
increasing or decreasing effect on the response. We also introduce an adaptive
version of the MS-lasso which reduces both the bias and the number of false
positive selections considerably. We compare the MS-lasso and the adaptive
MS-lasso with other existing methods for variable selection in high dimensions
by simulation and illustrate the method on two relevant genomic data sets.
Results indicate that the (adaptive) MS-lasso has excellent properties compared
to the other methods both by means of estimation and selection, and can be
recommended for high-dimensional monotone regression.
