k-Statistic

Author

Eric W. Weisstein
November 21, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/k-Statistic.nb.

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

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

Requirements

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

Initialization

<<mathStatica`

In terms of power sums

Table[KStatistic[i],{i,5}]//TableForm//TraditionalForm

In terms of sample central moments

Table[KStatisticToSampleCentral[i], {i,4}] // TableForm//TraditionalForm

Variance of (Subscript[k, n])

Table[CentralMomentToCumulant[2,KStatistic[k][[2]]],{k,4}]//ColumnForm//TraditionalForm

Unbiased Estimators

Of var((Subscript[k, 2]))

TraditionalForm[vk2=CentralMomentToCumulant[2,KStatistic[2][[2]]]]

Works if there are no cross-terms

TraditionalForm[k21=Simplify[vk2/.(Subscript[κ, r_]^(p_.)):>PolyK[Table[r,{p}]][[2]]]]

Works in general

TraditionalForm[k21=ListForm[vk2,κ]/.κ[x_]:>PolyK[x][[2]]//Simplify]

TraditionalForm[k22=Simplify[((2((Subscript[k, 2])^2)n+(n-1)(Subscript[k, 4]))/(n(n+1)))/.(Subscript[k, r_]^(p_.)):>KStatistic[r][[2]]^p]]

k21-k22