  The R\'enyi entanglement entropy (REE) is an entanglement quantifier
considered as a natural generalisation of the entanglement entropy. When it
comes to stochastic local operations and classical communication (SLOCC),
however, only a limited class of the REEs satisfy the monotonicity condition,
while their statistical properties beyond mean values have not been fully
investigated. Here, we establish a general condition that the probability
distribution of the REE of any order obeys under SLOCC. The condition is
obtained by introducing a family of entanglement monotones that contain the
higher-order moments of the REEs. The contribution from the higher-order
moments imposes a strict limitation on entanglement distillation via SLOCC. We
find that the upper bound on success probabilities for entanglement
distillation exponentially decreases as the amount of raised entanglement
increases, which cannot be captured from the monotonicity of the REE. Based on
the strong restriction on entanglement transformation under SLOCC, we design a
new method to estimate entanglement in quantum many-body systems from
experimentally observable quantities.
