Monotone measures of statistical complexity

• The notion of monotonicity of the complexity measure of a probability distribution is introduced and discussed.
• The monotonicity properties of statistical measures of complexity are studied.
• The Cramer–Rao and Fisher–Shannon measures of complexity are shown to be monotone.

We introduce and discuss the notion of monotonicity for the complexity measures of general probability distributions, patterned after the resource theory of quantum entanglement. Then, we explore whether this property is satisfied by the three main intrinsic measures of complexity (Crámer–Rao, Fisher–Shannon, LMC) and some of their generalizations.