Regression splines are smooth, flexible, and parsimonious nonparametric
function estimators. They are known to be sensitive to knot number and
placement, but if assumptions such as monotonicity or convexity may be imposed
on the regression function, the shape-restricted regression splines are robust
to knot choices. Monotone regression splines were introduced by Ramsay
[Statist. Sci. 3 (1998) 425--461], but were limited to quadratic and lower
order. In this paper an algorithm for the cubic monotone case is proposed, and
the method is extended to convex constraints and variants such as
increasing-concave. The restricted versions have smaller squared error loss
than the unrestricted splines, although they have the same convergence rates.
The relatively small degrees of freedom of the model and the insensitivity of
the fits to the knot choices allow for practical inference methods; the
computational efficiency allows for back-fitting of additive models. Tests of
constant versus increasing and linear versus convex regression function, when
implemented with shape-restricted regression splines, have higher power than
the standard version using ordinary shape-restricted regression.

本文提出了一种适用于三次单调、凸约束等变体的算法，该形状受限回归小波具有比无限制小波更小的平方误差损失，并且在实际推断方法中具有计算效率和数据自由度较小的优点。

利用形状受限制的回归样条进行推断

Inference using shape-restricted regression splines

We consider the generic regularized optimization problem
$\hat{\mathsf{\beta}}(\lambda)=\arg
\min_{\beta}L({\sf{y}},X{\sf{\beta}})+\lambda J({\sf{\beta}})$. Efron, Hastie,
Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407--499] have shown that for
the LASSO--that is, if $L$ is squared error loss and $J(\beta)=\|\beta\|_1$ is
the $\ell_1$ norm of $\beta$--the optimal coefficient path is piecewise linear,
that is, $\partial \hat{\beta}(\lambda)/\partial \lambda$ is piecewise
constant. We derive a general characterization of the properties of (loss $L$,
penalty $J$) pairs which give piecewise linear coefficient paths. Such pairs
allow for efficient generation of the full regularized coefficient paths. We
investigate the nature of efficient path following algorithms which arise. We
use our results to suggest robust versions of the LASSO for regression and
classification, and to develop new, efficient algorithms for existing problems
in the literature, including Mammen and van de Geer's locally adaptive
regression splines.

对一种正则化优化问题，引入一种求系数路径的方法及其性质，为回归和分类的 LASSO 提供了鲁棒性解决方案，同时提出了现有问题的新型算法，包括 Mammen 与 van de Geer 的局部自适应回归样条。