S23 Distribution

Version Date: 12/26/13

Eventually this will be converted to a package by pasting tested algorithms into a Package template.  Evaluate the whole notebook.

Modification of StableDistribution to take a parameter set: {α, β, γ, δ} for use in the type 1 parameterization.  γ is scale parameter and δ location parameter.  The modification allows the parameter sets from fitting to be used more easily.

Unprotect[Evaluate[StableDistribution]];
StableDistribution::usage="StableDistribution[type,α,β,μ,σ] represents the stable distribution with index of stability α, skewness parameter β, location parameter μ, and scale parameter σ.\n
StableDistribution[type,{α,β,γ,δ}] represents a stable distribution with the Nolan parameter order, type can be omitted if type = 1.";
StableDistribution[{a_, b_: 0, c_: 1, d_: 0}] := 
  StableDistribution[a, b, d, c];
StableDistribution[0, {a_, b_: 0, c_: 1, d_: 0}] := 
  StableDistribution[0, a, b, d, c];
Protect[StableDistribution];

S23OrderDistribution using Mathematica function OrderDistribution[].

S23OrderDistribution/:PDF[S23OrderDistribution[a_,b_:0,c_:1,d_:0,type_:1],x_]:=PDF[OrderDistribution[{StableDistribution[type,a/2,b,d,c],3},2],x];
S23OrderDistribution/:PDF[S23OrderDistribution[{a_,b_:0,c_:1,d_:0},type_:1],x_]:=PDF[OrderDistribution[{StableDistribution[type,a/2,b,d,c],3},2],x]/;VectorQ[{a,b,c,d},NumericQ]

S23OrderDistribution/:CDF[S23OrderDistribution[a_,b_:0,c_:1,d_:0,type_:1],x_]:=CDF[OrderDistribution[{StableDistribution[type,a/2,b,d,c],3},2],x];
S23OrderDistribution/:CDF[S23OrderDistribution[{a_,b_:0,c_:1,d_:0},type_:1],x_]:=CDF[OrderDistribution[{StableDistribution[type,a/2,b,d,c],3},2],x]/;VectorQ[{a,b,c,d},NumericQ]

S23OrderDistribution/:Quantile[S23OrderDistribution[a_,b_:0,c_:1,d_:0,type_:1],p_]:=Quantile[OrderDistribution[{StableDistribution[type,a/2,b,d,c],3},2],p];
S23OrderDistribution/:Quantile[S23OrderDistribution[{a_,b_:0,c_:1,d_:0},type_:1],p_]:=Quantile[OrderDistribution[{StableDistribution[type,a/2,b,d,c],3},2],p]/;VectorQ[{a,b,c,d},NumericQ]

(*S23OrderDistribution[a_,b_:0,c_:1,d_:0,type_:1]:=OrderDistribution[{StableDistribution[type,a/2,b,d,c],3},2];
S23OrderDistribution[{a_,b_:0,c_:1,d_:0},type_:1]:=OrderDistribution[{StableDistribution[type,a/2,b,d,c],3},2]/;VectorQ[{a,b,c,d},NumericQ]*)

Parameter assumptions and domain

S23OrderDistribution/:DistributionDomain[S23OrderDistribution[a_,b_:0,c_:1,d_:0,type_:1]]:=Interval[{-∞,∞}];

S23OrderDistribution/:DistributionParameterQ[S23OrderDistribution[a_,b_:0,c_:1,d_:0,type_:1]]:=TrueQ[a>0&&a≤2&&b≥-1&&b≤1&&d∈Reals&&c>0];

S23OrderDistribution/:DistributionParameterAssumptions[S23OrderDistribution[a_,b_:0,c_:1,d_:0,type_:1]]:=a>0&&a≤2&&b≥-1&&b≤1&&d∈Reals&&c>0;

pdf30 Interpolation

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pdf15 Interpolation

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S23Convert

S23Convert[{a_,b_,c_,d_},input_Integer,output_Integer]:= Which[input==output,{a,b,c,d},
input==0&&output==1,{a,b,c,If[a==2,d-b 2/Pi c Log[c],d-b c Tan[Pi (a/2)/2]]},
input==1&&output==0,{a,b,c,If[a==2,d+b 2/Pi c Log[c],d+b c Tan[Pi (a/2)/2]]}];

S23PDF

Clear[S23PDF];

S23PDF[x1_,{a1_,b1_:0,c1_:1,d1_:0},iparam_Integer:1]:=Module[{a,b,c,d},
{a,b,c,d}=S23Convert[{a1,b1,c1,d1},iparam,0];
If[a<2,
pdf15[If[b<0,-1,1] (x1-d)/c,a,Abs[b]]/c,pdf30[If[b<0,-1,1] (x1-d)/c,a,Abs[b]]/c]
];

S23PDF[x1_List,{a1_,b1_:0,c1_:1,d1_:0},iparam_Integer:1]:=Module[{a,b,c,d},
{a,b,c,d}=S23Convert[{a1,b1,c1,d1},iparam,0];
If[a<2,
pdf15[If[b<0,-1,1] (x1-d)/c,a,Abs[b]]/c,pdf30[If[b<0,-1,1] (x1-d)/c,a,Abs[b]]/c]
]/;VectorQ[x1,NumericQ]&&1<a1≤4&&-1≤b1≤1&&c1>0
(*eventually need to remove error checking here to improve speed*)

S23Random   generate Stable  2:3 order statistic random variables

S23Random[n_,{a_,b_,c_,d_},type_:1]:= Median/@Partition[RandomVariate[StableDistribution[type,a/2,b,d,c],3n],3];

RandomVariate

S23OrderDistribution /: 
  Random`DistributionVector[
   S23OrderDistribution[a_,b_,c_,d_,type_:1], n_, prec_] :=
	Median/@Partition[RandomVariate[StableDistribution[type,a/2,b,d,c],3n],3];

S23OrderDistribution /: 
  Random`DistributionVector[
   S23OrderDistribution[{a_,b_,c_,d_},type_:1], n_, prec_] :=
	Median/@Partition[RandomVariate[StableDistribution[type,a/2,b,d,c],3n],3];

ML Fitting

Clear[s23maxlike];
s23maxlike[s_,a_?NumericQ,b_?NumericQ,c_?NumericQ,d_?NumericQ]:=Total[Log[Select[S23PDF[s,{a,b,c,d},0],#>0&]]];

(*Need to build wrapper which estimates initial parameters*)
(*Need to add AccuracyGoal, PrecisionGoal*)
Clear[S23MLFit];
S23MLFit[s_,{a0_,b0_,c0_,d0_},type_:1,findMaxOpts___]:=Module[{za,za0,b,c,d,w,aa0,bb0,cc0,dd0,optsFM},
optsFM=FilterRules[{findMaxOpts},Options[FindMaximum]];
{aa0,bb0,cc0,dd0}=S23Convert[{a0,b0,c0,d0},type,0];
za0=N[Sqrt[-Log[aa0/4]]];
(*7/2(Exp[-(za)^2]+1/7), SPDF,SCDF routines have minimum alpha 0.5 *)
w=FindMaximum[s23maxlike[s,3(Exp[-(za)^2]+1/3),If[Abs[b]>1,Sign[b],b],Abs[c],d],{{za,za0,-20,20},{b,bb0,-1.,1.},{c,cc0,-Infinity,Infinity},{d,dd0,-Infinity,Infinity}},Evaluate[Sequence@@optsFM]];
S23Convert[{3(Exp[-(za)^2]+1/3),If[Abs[b]>1,Sign[b],b],Abs[c],d}/.w[[2]],0, type]

];