Stable Densities

Needs["StableM7`"];
Needs["PlotLegends`"];

The goal of this section is to present some of the variety of shapes of the class of stable densities.  The computations are performed using a transform of inverse Fourier integrand.  (Zolotarev transform)  The general stable distribution is described by four parameters, α, β, γ, and δ.  α is limited to the range (0, 2]; it controls the general shape of the distribution.  In the graphic below, β and δ are set to 0 and γ is set to a value of 1.

Plot[{SPDF[x,{(2/3)},1],SPDF[x,{1},1],SPDF[x,{(4/3)},1],SPDF[x,{(5/3)},1],SPDF[x,{2},1]},{x,-10,10},PerformanceGoal->"Speed",PlotStyle->{{Thickness[0.005],Red},{Thickness[0.005],Blue},{Thickness[0.005],Green},{Thickness[0.005],Orange},{Thickness[0.005],Brown}},PlotLegend->{"α 0.67","α 1.00","α 1.33","α 1.67","α 2.00"},PlotRange->All,PlotLabel->"Stable Densities",ImageSize->400]

When α = 2, the distribution is Normal; as α becomes smaller the distribution becomes more peaked and narrower and the tails become fatter containing more of the weight of the distribution.  For the Normal distribution, the second moment or the variance exists, but it does not exist for any other stable distribution.  When α ≤ 1, the tails are so heavy that the mean or expectation no longer exists.

β, in the range [-1, 1], is the skewness parameter.  When it is negative the distribution is skewed to the left and skewed to the right when it is positive.  When β = 0, the distribution is symmetric.  As α approaches 2, the effect of β becomes negligible and the Normal distribution is symmetric.

Plot[{SPDF[x,{1.5`,-1},1],SPDF[x,{1.5`,-0.5`},1],SPDF[x,{1.5`,0},1],SPDF[x,{1.5`,0.5`},1],SPDF[x,{1.5`,1},1]},{x,-10,10},PerformanceGoal->"Speed",PlotStyle->{{Thickness[0.005],Red},{Thickness[0.005],Blue},{Thickness[0.005],Green},{Thickness[0.005],Orange},{Thickness[0.005],Brown}},PlotLegend->{"β -1.0","β -0.5","β 0","β 0.5","β 1.0"},PlotRange->All,PlotLabel->"Stable Densities α = 1.5",ImageSize->400]

γ is the scale parameter, it changes the size of the distribution, but not its shape.

Plot[{SPDF[x,{1.5`,-1,0.5`},1],SPDF[x,{1.5`,-1,1},1],SPDF[x,{1.5`,-1,1.5`},1]},{x,-10,10},PerformanceGoal->"Speed",PlotStyle->{{Thickness[0.005],Red},{Thickness[0.005],Blue},{Thickness[0.005],Green}},PlotLegend->{"γ 0.5","γ 1.0","γ 1.5"},LegendPosition-> {0.4,-1},PlotRange->All,PlotLabel->"Stable Densities α = 1.5, β = -1",ImageSize->400]

δ is the location parameter, when α is greater than one it is the location of the mean.

Plot[{SPDF[x,{1.5`,-1,1,-0.5`},1],SPDF[x,{1.5`,-1,1,1},1],SPDF[x,{1.5`,-1,1,3},1]},{x,-10,10},PerformanceGoal->"Speed",PlotStyle->{{Thickness[0.005],Red},{Thickness[0.005],Blue},{Thickness[0.005],Green}},PlotLegend->{"δ -0.5","δ 1.0","δ 3.0"},LegendPosition-> {0.5,-1},PlotRange->All,PlotLabel->"Stable Densities α = 1.5, β = -1, γ = 1",ImageSize->400]

Here is a link to a program on the Wolfram Demonstration Project that shows stable densities dynamically.

A link to an interactive webMathematica site where you may interactively experience stable density.

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t2 = Graphics[Text[Hyperlink[Framed[Style["Next", {txcolor, Bold, Larger}]], "../s02/StableProbability.html"], {0, -0.5}]];
t3 = Graphics[Text[Hyperlink[Framed[Style["Download Notebook", {txcolor, Bold, Larger}]], "../s01/StableDensity.nb"], {5, 0}]];
t4 = Graphics[Text[Hyperlink[Framed[Style["Previous", {txcolor, Bold, Larger}]], "../s00/StableDistributions.html"], {0, 0.5}]];
Show[ t1, t2, t3, t4, ImageSize -> 500]
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© Copyright 2010 mathestate    Wed 24 Mar 2010