Needs["StableM7`"];

Sums of Stable Random Variables

Below we show the histogram of 100,000 stable variables generated for parameters {α, β, γ, δ} = {1.5, 0.6, 1, 1}.  We then took the sample, partitioned it into groups of 10 random variables.  Each of the groups was summed and the second histogram shows the distribution of the sums of 10 random variables.  Note that the shape is the same, but that the scale and location have changed.

s=SRandom[100000,{1.5,0.6,1,1}];
s2=Plus@@Partition[s,10000];
w=Histogram[s,{-10,10, 20/1000},"ProbabilityDensity", PlotLabel -> "100,000 Stable Random Variables",PlotRange->All]
w2=Histogram[s2 ,{-10 10 ^(1/1.5),10 10^(1/1.5),20 10^(1/1.5)/1000},"ProbabilityDensity", PlotLabel -> "10,000 Stable Random Variables\nEvery 10 from above Sample Summed",PlotRange->All]

The parameter fits to each sample are shown below.

p=Quiet[CFFit[s]];
p2=Quiet[SMLFit[s2]];

TableForm[{p,p2},TableHeadings-> {{"100,000 RVs","10,000 RVs"},{"α","β","γ","δ"}},TableSpacing-> {1,5}]

For the 1-parameterization of stable distributions which we are using we would expect γ and δ to change by the following relationships.  The expected (Subscript[γ, s]) and (Subscript[δ, s]) are calculated from the parameters from the fit to 100,000 RVs.

(Subscript[γ, s])=γ (n^(1/α))

(Subscript[δ, s])=n δ

Clear[s,s2];(*this may be taking up too much memory to go on?????*)
w=p[[3]] 10^(1/p[[1]]);
CellPrint[Row[{TextCell["\t\t","Text"],ToExpression[SubscriptBox["γ","s"],TraditionalForm],TextCell[" = ","Text"],TextCell[ ToString[w],"Text"]}]];
w= 10 p[[4]];
CellPrint[Row[{TextCell["\t\t","Text"],ToExpression[SubscriptBox["δ","s"],TraditionalForm],TextCell[" = ","Text"],TextCell[ ToString[w],"Text"]}]];

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© Copyright 2008 mathestate    Fri 19 Dec 2008