Title: Finite-sample risk bounds for maximum likelihood estimation with arbitrary penalties,
Abstract: The MDL two-part coding $ \textit{index of resolvability} $ provides a
finite-sample upper bound on the statistical risk of penalized likelihood
estimators over countable models. However, the bound does not apply to
unpenalized maximum likelihood estimation or procedures with exceedingly small
penalties. In this paper, we point out a more general inequality that holds for
arbitrary penalties. In addition, this approach makes it possible to derive
exact risk bounds of order $1/n$ for iid parametric models, which improves on
the order $(\log n)/n$ resolvability bounds. We conclude by discussing
implications for adaptive estimation.