Nonparametric, Tuning-Free Estimation of S-Shaped Functions
Nonparametric, Tuning-Free Estimation of S-Shaped Functions
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S 形函数的非参数、免调整估计
DOI：
10.1111/rssb.12481
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Samworth, Richard J.
中科院分区：
文献类型：
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作者：
Feng, Oliver Y.;Chen, Yining;Han, Qiyang;Carroll, Raymond J.;Samworth, Richard J.
我们考虑S形回归函数的非参数估计。最小二乘估计器提供了一种非常自然的无调谐方法，但由于拐点未知，因此导致非凸优化问题。我们表明，估计仍然可以被视为一个投影到凸锥的有限联盟，这使我们能够提出一个混合的原始-对偶基算法，其高效，顺序计算。在开发了一个投影框架，证明了一致性和鲁棒性的误指定的估计，我们的主要理论结果提供了尖锐的甲骨文不等式，产生最坏情况下的自适应风险界的回归函数的估计，以及收敛速度的拐点估计。这些结果表明，不仅估计实现了极小极大最优收敛速度的回归函数的估计和它的拐点（在后一种情况下的对数因子），而且它是能够实现几乎参数率时，真正的回归函数是分段仿射没有太多的仿射片。模拟和空气污染建模的真实的数据应用也证实了所需的有限样本性质的估计，我们的算法是在R包Sshaped。
We consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimization problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst-case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm the desirable finite-sample properties of the estimator, and our algorithm is implemented in the R package Sshaped.