  Although distributional reinforcement learning (DRL) has been widely examined
in the past few years, very few studies investigate the validity of the
obtained Q-function estimator in the distributional setting. To fully
understand how the approximation errors of the Q-function affect the whole
training process, we do some error analysis and theoretically show how to
reduce both the bias and the variance of the error terms. With this new
understanding, we construct a new estimator \emph{Quantiled Expansion Mean}
(QEM) and introduce a new DRL algorithm (QEMRL) from the statistical
perspective. We extensively evaluate our QEMRL algorithm on a variety of Atari
and Mujoco benchmark tasks and demonstrate that QEMRL achieves significant
improvement over baseline algorithms in terms of sample efficiency and
convergence performance.
