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We study the problem of estimating an unknown vector $\theta$ from anobservation $X$ drawn according to the normal distribution with mean $\theta$and identity covariance matrix under the knowledge that $\theta$ belongs to aknown closed convex set $\Theta$. In this general setting, Chatterjee (2014)proved that the natural constrained least squares estimator is "approximatelyadmissible" for every $\Theta$. We extend this result by proving that the sameproperty holds for all convex penalized estimators as well. Moreover, wesimplify and shorten the original proof considerably. We also provide explicitupper and lower bounds for the universal constant underlying the notion ofapproximate admissibility.

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