Probability Distributions Approximation via Fractional Moments and Maximum Entropy: Theoretical and Computational Aspects

In the literature, the use of fractional moments to express the available information in the framework of maximum entropy (MaxEnt) approximation of a distribution F having finite or unbounded positive support, has been essentially considered as a computational tool to improve the performance of the analogous procedure based on integer moments. No attention has been paid to two formal aspects concerning fractional moments, such as conditions for the existence of the maximum entropy approximation based on them or convergence in entropy of this approximation to F. This paper aims to fill this gap by providing proofs of these two fundamental results. In fact, convergence in entropy can be involved in the optimal selection of the order of fractional moments for accelerating the convergence of the MaxEnt approximation to F, to clarify the entailment relationships of this type of convergence with other types of convergence useful in statistical applications, and to preserve some important prior features of the underlying F distribution.