Introduction to Second Kind Statistics: Application of Log-Moments and Log-Cumulants to the Analysis of Radar Image Distributions

Statistical methods classically used to analyse a probability density function (pdf) are founded on the Fourier transform, on which useful tools such the first and second characteristic function are based, yielding the definitions of moments and cumulants. Yet this transformation is badly adapted to the analysis of probability density functions defined on R+, for which the analytic expressions of the characteristic functions may become hard, or even impossible to formulate. In this article we propose to substitute the Fourier transform with the Mellin transform. It is then possible, inspired by the precedent definitions, to introduce second kind statistics: second kind characteristic functions, second kind moments (or log-moments), and second kind cumulants (or log-cumulants). Applied to traditional distributions like the gamma distribution or the Nakagami distribution, this approach gives results that are easier to apply than the classical approach. Moreover, for more complicated distributions, like the K distributions or the positive αstable distributions, the second kind statistics give expressions that are truly simple and easy to exploit. The new approach leads to innovative methods for estimating the parameters of distributions defined on R+. It is possible to compare the estimators obtained with estimators based on maximum likelihood theory and the method of moments. One can thus show that the new methods have variances that are considerably lower than those mentioned, and slightly higher than the Cramér-Rao bound.