A Framework for Mellin Kind Series Expansion Methods
Mellin kind statistics (MKS) is the framework which arises if the Fourier transform is replaced with the Mellin transform when computing the characteristic function from the probability density function. We may then proceed to retrieve logarithmic moments and cumulants, that have important applications in the analysis of heavy-tailed distribution models for nonnegative random variables. In this paper we present a framework for series expansion methods based on MKS. The series expansions recently proposed in  are derived independently and in a different way, showing that the methods truly are Mellin kind analogies to the classical Gram-Charlier and Edgeworth series expansion. From this new approach, a novel series expansion is also derived. In achieving this we demonstrate the role of two differential operators, which are called Mellin derivatives in , but have not been used in the context of Mellin kind statistics before. Also, the Bell polynomials  are used in new ways to simplify the derivation and representation of the the Mellin kind Edgeworth series expansion. The underlying assumption of the nature of the observations which validates that series is also investigated. Finally, a thorough review of the performance of several probability density function estimation methods is conducted. This includes classical ,  and recent methods , ,  in addition to the novel series expansion presented in this paper. The comparison is based on synthesized data and sheds new light on the strengths and weaknesses of methods based on classical and Mellin kind statistics.
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