Do there exist bounds similar to the Cramer-Rao bound on the variance, but for higher-order central moments such as the skewness or kurtosis?
I have looked at the derivation of the CR bound, and it involves an application of Cauchy-Schwarz theorem, which yields the square exponent necessary to obtain the variance. However, since Cauchy-Schwarz does not hold for general exponents, I don't think one can easily generalize it.