Lower order statistics: A new approach for probability density functions defined on IR+

Probability Density Functions defined on IR⁺ can be successfully modeled with the help of the Mellin Transform : this rather underrated transform is well suited for such functions so that we propose the new definitions of "second kind" characteristic functions based on this transform. By this way, second kind moments and second kind cumulants can also be defined, so that multiplicative noise, which can be seen as a Mellin convolution of Probability Density Function, can be easily analysed. The estimation of PDF parameters can be improved with this new approach. Indeed, it is possible to deal with lower order statistics so that negative moments can be defined in such a way that the variances of the estimators are reduced. The analytical formulation of this variance is proposed and validated on numerical simulations for the Gamma law. With this new approach, a same estimator variance is reached with a more reduced set of samples than traditional methods.