A note on the approximate admissibility of regularized estimators in the Gaussian sequence model

We study the problem of estimating an unknown vector theta from an observation X drawn according to the normal distribution with mean theta and identity covariance matrix under the knowledge that theta belongs to a known closed convex set circle dot. In this general setting, Chatterjee (2014) proved that the natural constrained least squares estimator is " approximately admissible" for every circle dot We extend this result by proving that the same property holds for all convex penalized estimators as well. Moreover, we simplify and shorten the original proof considerably. We also provide explicit upper and lower bounds for the universal constant underlying the notion of approximate admissibility.