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.

中文翻译：

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S 形函数的非参数、无调整估计

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、无需调整的方法，但会导致非凸优化问题，因为拐点是未知的。我们表明，估计器仍然可以被视为对凸锥的有限联合的投影，这使我们能够提出一种混合原始双基算法，以实现其高效的顺序计算。在开发了一个投影框架来证明估计量的错误指定的一致性和稳健性之后，我们的主要理论结果提供了尖锐的预言不等式，这些不等式产生了回归函数估计的最坏情况和自适应风险界限，以及收敛速度拐点的估计。这些结果不仅表明估计器在回归函数的估计及其拐点（在后一种情况下达到对数因子）的估计中都实现了极小极大最优收敛速度，而且它能够实现几乎-当真正的回归函数是分段仿射且没有太多仿射块时的参数率。模拟和空气污染建模的真实数据应用也证实了估计器的理想有限样本属性，我们的算法在 而且当真正的回归函数是分段仿射且没有太多仿射块时，它能够实现几乎参数化的速率。模拟和空气污染建模的真实数据应用也证实了估计器的理想有限样本属性，我们的算法在 而且当真正的回归函数是分段仿射且没有太多仿射块时，它能够实现几乎参数化的速率。模拟和空气污染建模的真实数据应用也证实了估计器的理想有限样本属性，我们的算法在R包S形。