Penalties that induce smoothness are common in nonparametric regression. In many settings, the amount of smoothness in the data generating function will not be known. Simon and Shojaie (2021) derived convergence rates for nonparametric estimators under misspecified smoothness. We show that their theoretical convergence rates can be improved by working with convenient approximating functions. Properties of convolutions and higher-order kernels allow these approximation functions to match the true functions more closely than those used in Simon and Shojaie (2021). As a result, we obtain tighter convergence rates.

翻译：诱导光滑性的惩罚项在非参数回归中十分常见。在许多场景下，数据生成函数的光滑程度是未知的。Simon与Shojaie（2021）推导了错误指定光滑条件下非参数估计量的收敛速率。我们证明，通过采用便捷的逼近函数，其理论收敛速率可得到改进。卷积与高阶核的性质使得这些逼近函数能够比Simon与Shojaie（2021）所用函数更精确地匹配真实函数。因此，我们获得了更紧凑的收敛速率。