<- file stat 97basics.html -> Topics in this file: higher moments than S^2 other moments

higher moments than S^2

=======================Ercan Kuruoglu, 08 Mar 1997==========ssm From: eek@eng.cam.ac.uk (E.E. Kuruoglu) Subject: Re: What is special about variance? Message-ID: <5fsgi3$59b@lyra.csx.cam.ac.uk> > > >Rich Ulrich, biostatistician wpilib+@pitt.edu > >The only > >place where I can immediately think of the 4th power being used > >is in the criteria for Varimax rotation for factor analysis > > > >If 2nd power is EASY, and it works, what reason is there for > >people to use higher powers or fractional powers? The reason is that, some distributions such as Cauchy do not have finite second order statistics, that is its variance is infinite. Moreover, what do we know about the optimality of estimators based on second order statistics if the underlying pdf is non-Gaussian? Finally, second order moments loose the phase information in the signal, and therefore we need third order moments to uncover this information. Higher order statistics is a well developed theory with tens of applications in the field of signal processing. Nikias, Mendel, Giannakis and Tekalp published a lot in this field. Also, in 1989 and 1991 there had been two international workshops on Higher order statistics. Ercan Kuruoglu U. of Cambridge

Why not use higher/other moments?

=======================Rich Ulrich, 05 Mar 1997==========ssm Subject: Re: What is special about variance? Message-ID: <5fko9v$5r@usenet.srv.cis.pitt.edu> E.E. Kuruoglu (eek@eng.cam.ac.uk) wrote: : Hello, : I would like to take people's opinions about : why we use variance, covariance, autocorrelation : (all second order statistics) as an indispensible part : of our analyses of various problems. : That is, why another power statistics such as : E(|x|^p) (e.g. p = 2.5, 1.5, etc.?) : is not used. Using variance leads to simple (linear) <<snip ... >> -- Well, as you mention, there are strong practical reasons for using the power of 2: computationally easy. Additive sums of squares. And there are strong theoretical reasons that overlap with the practical: first and second power happen to match the parameterization (location, scale, for instance) of a whole lot of distributions that are interesting. But the absolute deviations do get used in some applications (for 'robust' estimation, in particular). And occasionally someone gets interested in kurtosis (3rd power). The only place where I can immediately think of the 4th power being used is in the criteria for Varimax rotation for factor analysis If 2nd power is EASY, and it works, what reason is there for people to use higher powers or fractional powers? * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

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