  Establishing a Large Deviation Principle (LDP) proves to be a powerful result
for a vast number of stochastic models in many application areas of probability
theory. The key object of an LDP is the large deviations rate function, from
which probabilistic estimates of rare events can be determined. In order make
these results empirically applicable, it would be necessary to estimate the
rate function from observations. This is the question we address in this
article for the best known and most widely used LDP: Cram\'er's theorem for
random walks.
  We establish that even when only a narrow LDP holds for Cram\'er's Theorem,
as occurs for heavy-tailed increments, one gets a LDP for estimating the random
walk's rate function in the space of convex lower-semicontinuous functions
equipped with the Attouch-Wets topology via empirical estimates of the moment
generating function. This result may seem surprising as it is saying that for
Cram\'er's theorem, one can quickly form non-parametric estimates of the
function that governs the likelihood of rare events.
