  Cram\'er type moderate deviation theorems quantify the accuracy of the
relative error of the normal approximation and provide theoretical
justifications for many commonly used methods in statistics. In this paper, we
develop a new randomized concentration inequality and establish a Cram\'er type
moderate deviation theorem for general self-normalized processes which include
many well-known Studentized nonlinear statistics. In particular, a sharp
moderate deviation theorem under optimal moment conditions is established for
Studentized $U$-statistics.
