We study maximum penalized likelihood density estimation using the first roughness penalty functional of Good. We prove a simple pointwise comparison result with a kernel estimator based on the two-sided exponential kernel. This leads to $L^1$ convergence results similar to those for kernel estimators. We also prove Hellinger distance bounds for the roughness penalized estimator.
"Optimal convergence rates for Good's nonparametric maximum likelihood density estimator." Ann. Statist. 27 (5) 1600 - 1615, October 1999. https://doi.org/10.1214/aos/1017939143