Cumulant

Author

Eric W. Weisstein
August 6, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/Cumulant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Cumulant.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Requirements

mathStatica Application package
http://www.wolfram.com/products/applications/mathstatica/

Initialization

<<mathStatica`

Cumulants in terms of Raw Moments

One Coefficient at a Time

kappa[n_]:=Simplify[SeriesCoefficient[Series[Log[1+Sum[(I t)^k/k! (Subscript[(Overscript[μ, ′]), k]),{k,n}]],{t,0,n}],n]n!/I^n]

Table[kappa[n],{n,5}]//ColumnForm

All of them

cum=Simplify[Series[Log[1+Sum[(I t)^k/k! (Subscript[(Overscript[μ, ′]), k]),{k,1,30}]],{t,0,6}]]

l=CoefficientList[Normal[cum],t]

From the Series

(cum1=Table[(Subscript[κ, n])->l[[n+1]]n!/I^n,{n,Length[l]-1}])//ColumnForm

Using mathStatica

(cum2=Table[CumulantToRaw[n],{n,6}])//ColumnForm

Simplify[Last/@cum1-Last/@cum2]

Cumulants in Terms of Central Moments

Using mathStatica

(cum2=Table[CumulantToCentral[n],{n,6}])//ColumnForm

Simplify[Last/@cum1-Last/@cum2]

Multavariate Cumulants in Terms of Raw Moments

Using mathStatica

Table[CumulantToRaw[{n,1}],{n,5}]//ColumnForm

Multavariate Cumulants in Terms of Central Moments

Table[CumulantToCentral[{n,1}],{n,5}]//ColumnForm