An improved complex fractional moment-based approach for the probabilistic characterization of random variables

Recently, the concept of complex fractional moments has been introduced as an extension to ordinary moments. It has been shown that complex fractional moments never diverge and they are able to represent both probability density function (PDF) and characteristic function (CF). In this paper, we develop an improved complex fractional moments-based approach to recover the PDF and the CF, in which the PDF is re-expressed in two parts such that the complex fractional moment coincide with Mellin transform in each part. The proposed PDF decomposition scheme is a most natural way to connect complex fractional moments and Mellin transform while it represent the PDF (or CF) in a more straightforward and more distinct way. By virtue of inverse Mellin transform theorem, the PDF (or CF) can be exactly recovered in terms of complex fractional moments. In addition, the developed approach is further extended to estimate the PDF (or CF) from a finite number of samples of variables. Three illustrative distributions are used to demonstrate the proposed complex fractional moment-based approach.