The estimation of regression parameters in one dimensional broken stick models is a research area of statistics with an extensive literature. We are interested in extending such models by aiming to recover two or more intersecting (hyper)planes in multiple dimensions. In contrast to approaches aiming to recover a given number of piecewise linear components using either a grid search or local smoothing around the change points, we show how to use Nesterov smoothing to obtain a smooth and everywhere differentiable approximation to a piecewise linear regression model with a uniform error bound. The parameters of the smoothed approximation are then efficiently found by minimizing a least squares objective function using a quasi-Newton algorithm. Our main contribution is threefold: We show that the estimates of the Nesterov smoothed approximation of the broken plane model are also $\sqrt{n}$ consistent and asymptotically normal, where $n$ is the number of data points on the two planes. Moreover, we show that as the degree of smoothing goes to zero, the smoothed estimates converge to the unsmoothed estimates and present an algorithm to perform parameter estimation. We conclude by presenting simulation results on simulated data together with some guidance on suitable parameter choices for practical applications.

翻译：一维分段线性回归模型中的参数估计是统计学中一个文献广泛的研究领域。我们旨在通过恢复多维空间中两个或多个相交（超）平面来扩展此类模型。与通过网格搜索或变化点局部平滑来恢复给定数量分段线性分量的方法不同，我们展示了如何利用Nesterov平滑技术获得具有均匀误差界的分段线性回归模型的光滑且处处可微近似。随后通过拟牛顿算法最小化最小二乘目标函数，可高效求解平滑近似模型的参数。我们的主要贡献有三方面：首先证明分段平面模型的Nesterov平滑近似估计量同样具有√n相合性与渐近正态性，其中n表示两个平面上的数据点数量；其次证明当平滑度趋近于零时，平滑估计量收敛于非平滑估计量；最后提出执行参数估计的算法。我们通过模拟数据的仿真实验结果进行总结，并为实际应用提供合适的参数选择指导。