Introduction to second kind statistics: application of Log-moments and Log-cumulants to SAR image law analysis

摘要

Statistics methods classicaly used to analyse a probability density function (p.d.f.) are based on Fourier Transform,udon which usefull tools as first and second characteristic functions are based, yielding the definitions of momentsudand cumulants. Yet this transform does not well match with p.d.f. defined on R+ as analytical expressions can beudrather heavy in this case. In this article, we propose to start with a rather misknown transform: the Mellin transform,udin order to define second kind statistics. By this way, second kind characteristic functions, second kindudmoments (log-moments) and second kind cumulants (log-cumulants) can be defined by mimicing the traditionaluddefinitions. For classical p.d.f. defined on R+, as Gamma and Nakagami laws, this approach seems to be simplerudthan previous one. More, for complicated p.d.f., as the famous K law or positive α-stable distributions, second kindudstatistics yield oversimple results.udThis new approach provides new methods for estimating the parameters of p.d.f. defined on R+. Comparisons canudbe done with traditional methods as Maximum Likehood Method and Moment Method: the variance of the newudmethods estimators are lower than Moment Method ones, and slightly upper than Cramer Rao bounds.