  We study the cost sharing problem for cooperative games in situations where
the cost function $C$ is not available via oracle queries, but must instead be
derived from data, represented as tuples $(S, C(S))$, for different subsets $S$
of players. We formalize this approach, which we call statistical cost sharing,
and consider the computation of the core and the Shapley value, when the tuples
are drawn from some distribution $\mathcal{D}$.
  Previous work by Balcan et al. in this setting showed how to compute cost
shares that satisfy the core property with high probability for limited classes
of functions. We expand on their work and give an algorithm that computes such
cost shares for any function with a non-empty core. We complement these results
by proving an inapproximability lower bound for a weaker relaxation.
  We then turn our attention to the Shapley value. We first show that when cost
functions come from the family of submodular functions with bounded curvature,
$\kappa$, the Shapley value can be approximated from samples up to a $\sqrt{1 -
\kappa}$ factor, and that the bound is tight. We then define statistical
analogues of the Shapley axioms, and derive a notion of statistical Shapley
value. We show that these can always be approximated arbitrarily well for
general functions over any distribution $\mathcal{D}$.
