Improved convergence rates of nonparametric penalized regression under misspecified total variation

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.