Kendal Operator

Author

Eric W. Weisstein
August 13, 2002

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

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

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

Requiements

http://www.wolfram.com/products/applications/mathstatica/

Initialization

<<mathStatica`;

Code

ToKendallForm[l_]:=ToKendallForm[l,r]
ToKendallForm[l_,s_]:=l/.{(Subscript[(d:(κ|μ|(Overscript[μ, ′]))), k_]):>d[s^k]}

FromKendallForm[l_]:=FromKendallForm[l,r,{t}]
FromKendallForm[l_,r_,t_List]:=l/.(d:(κ|μ|(Overscript[μ, ′])))[p_]:>Subscript[d,Sequence@@Exponent[p,Prepend[t,r]]]/.Thread[t->1]

KendallStep[eqn_]:=KendallStep[eqn,{r,t}]
KendallStep[eqn_,r_,t_]:=Module[{eqn2,s,tpoly},
eqn2=D[#,s]&/@(eqn/.r->s)/.{(m_)'[tpoly_. (s^(k_.))]:>m[tpoly t (r^(k-1))],m_[tpoly_. (s^(k_.))]:>m[tpoly (r^k)],s:>1};
Expand[#/FactorTermsList[eqn2[[1]]][[1]]]&/@eqn2
]

(Subscript[μ, 2,n])

ColumnForm[l=NestList[KendallStep[#,r,t]&,ToKendallForm[CentralToCumulant[2]],2]]

TableForm[Transpose[{
FromKendallForm/@l,
Table[CentralToCumulant[{2-r,r}],{r,0,2}]
}]]

(Subscript[μ, 3,n])

ColumnForm[l=NestList[KendallStep[#,r,t]&,ToKendallForm[CentralToCumulant[3]],3]]

TableForm[Transpose[{
FromKendallForm/@l,
Table[CentralToCumulant[{3-r,r}],{r,0,3}]
}]]

(Subscript[μ, 4,n])

ColumnForm[l=NestList[KendallStep[#,r,t]&,ToKendallForm[CentralToCumulant[4]],4]]

TableForm[Transpose[{
FromKendallForm/@l,
Table[CentralToCumulant[{4-r,r}],{r,0,4}]
}]]

(Subscript[μ, 2,1,1])

CentralToCumulant[4]

k0=ToKendallForm[CentralToCumulant[4]]

k1=KendallStep[k0,r,t[1]]

k2=KendallStep[k1,r,t[2]]

{{FromKendallForm[k2,r,Array[t,2]],CentralToCumulant[{2,1,1}]}}//TableForm

(Subscript[μ, 2,2,1])

CentralToCumulant[5]

k0=ToKendallForm[CentralToCumulant[5]]

k1=Nest[KendallStep[#,r,t1]&,k0,2]

k2=KendallStep[k1,r,t2]

FromKendallForm[k2,r,{t1,t2}]

CentralToCumulant[{2,2,1}]

(Subscript[μ, 3,2])

(l=NestList[KendallStep,ToKendallForm[CentralToCumulant[6]],6])//ColumnForm

ColumnForm[ken=FromKendallForm/@l]

(msta=Table[CentralToCumulant[{6-n,n}],{n,0,6}])//ColumnForm

ken-msta

Last/@ken-Last/@msta

(Subscript[(μ'), 3,1])

(l=NestList[KendallStep,ToKendallForm[RawToCumulant[6]],6])//ColumnForm

ColumnForm[ken=FromKendallForm/@l]

(msta=Table[RawToCumulant[{6-n,n}],{n,0,6}])//ColumnForm