  We derive a family of entanglement criteria for continuous variable systems
based on the R\'enyi entropy of complementary distributions. We show that these
entanglement witnesses can be more sensitive than those based on second-order
moments, as well as previous tests involving the Shannon entropy [Phys. Rev.
Lett. \textbf{103}, 160505 (2009)]. We extend our results to include the case
of discrete sampling, and develop another set of entanglement tests using the
discrete Tsallis entropy. We provide several numerical results which show that
our criteria can be used to identify entanglement in a number of experimentally
relevant quantum states.
