We consider a one-dimensional recurrent random walk in random environment
(RWRE) when the environment is i.i.d. with a parametric, finitely supported
distribution. Based on a single observation of the path, we provide a maximum
likelihood estimation procedure of the parameters of the environment. Unlike
most of the classical maximum likelihood approach, the limit of the criterion
function is in general a nondegenerate random variable and convergence does not
hold in probability. Not only the leading term but also the second order
asymptotics is needed to fully identify the unknown parameter. We present
different frameworks to illustrate these facts. We also explore the numerical
performance of our estimation procedure
