(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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Mathematics of Expected Value with Stable, non-Normal Distributions"]], "Subtitle", PageWidth->PaperWidth, FontSize->16, FontSlant->"Plain", FontColor->GrayLevel[0]], Cell[TextData[{ "This monograph is intended to demonstrate that one may expect values of \ \[Alpha] between 1.2 and 1.8 and \[Beta] between 0 and .2 for distributions \ of investment returns on privately held real estate. It also extends the \ illustration contained in Appendix C to the case involving stable \ distributions. Finally, it attempts to provide more insight into the special \ form of discrete Fourier transforms used in this dissertation. This notebook \ was used to produce many of the illustrations in the text.\n\nIn general, the \ probability distribution function (pdf) and the characteristic function \ (ch.f.) of a random variable form a Fourier transform pair defined by the \ equations:\n\n", Cell[BoxData[ FormBox[ StyleBox[\(F \((u)\) = \ \[Integral]\_\(-\[Infinity]\)\%\[Infinity] f \((x)\) e\^\((iux)\)\ dx, \n\n f \((x)\) = \(1\/\(2 \[Pi]\)\) \(\[Integral]\_\(-\[Infinity]\)\%\ \[Infinity] F \((u)\) e\^\(-iux\)\ du\)\), FontColor->GrayLevel[0]], TextForm]], "Text", FontSize->14], "\n\nwhere x and u are respectively the probability distribution variable \ and the angular variable of the characteristic function.\n\nFor many \ distributions, including almost all of the stable distributions of interest \ here, the above integrations cannot be performed analytically and a closed \ analytical pdf is not defined. Therefore, stable distributions are usually \ defined by stating their ch.f.\n\nIt is sometimes desirable, as in our case, \ to study the effects on the pdf of varying the parameters of the cf. It is \ possible, subject the restrictions of sampling theory, to construct a sampled \ version of the ch.f. by evaluating the ch.f. at equally spaced values of the \ angle variable u. The usual procedure is to assign values to all parameters \ before sampling, which results in a numeric series. This series is then \ transformed into a numeric series in x using discrete fourier transform \ techniques. This can be accomplished to a high degree of accuracy provided \ the samples are properly spaced and cover all the significant range of the \ ch.f. \n\nOne of the drawbacks of using the numeric approach is that all \ parameters must be evaluated prior to the application of the transform \ procedure. We have elected to use a variation of the usual discrete transform \ which allows evaluation after transformation. This procedure is outlined \ below.\n\nWe will use the the S(\[Alpha],\[Beta],\[Gamma],\[Delta];0) \ parameterization as suggested by Nolan (1998 page 189 Eq. #3) substituting u \ for t, \[Gamma] for \[Sigma], and \[Delta] for \[Mu]. " }], "Text", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontSize->12, FontSlant->"Plain", FontColor->GrayLevel[0]], Cell[BoxData[ \(fu := Exp[\((\(-\[Gamma]^\[Alpha]\)\ Abs[u]^\[Alpha]\ \((1 + I\ \[Beta]\ \ Sign[ u]\ Tan[\[Pi]\ \[Alpha]/ 2]\ \((\[Gamma]\ Abs[u]^\((1 - \[Alpha])\) - 1)\))\) + \[Delta]\ u\ I)\)]\)], "Input", PageWidth->PaperWidth], Cell[TextData[StyleBox["Although fu is not directly integratable, we can \ apply the transform at the selected sampling points by use of an impulse \ function, DiracDelta[u-u1], which has value only when u= u1. The integration \ is performed over a band of +/- \[Epsilon] about each point where u= u1, then \ the limit is calculated as \[Epsilon]\[Rule]0. The result is a list of terms \ which are the Fourier transforms of each sample of u. x is now the \ independent variable. The pdf is the sum of these terms, ffx1 below. \ Mathematica performs a change of variable (u1\[Rule]2 \[Pi] f, to correct the \ scaling), evaluates the terms at selected points, -5<=f <=5 in steps of .01, \ and sums the terms in the ffx1 operation below.", FontSize->12, FontSlant->"Plain"]], "Text", PageWidth->PaperWidth, FontColor->GrayLevel[0]], Cell[CellGroupData[{ Cell[BoxData[ \(idf = Limit[1\/\(2\ \[Pi]\)\ \(\[Integral]\_\(u1 - \[Epsilon]\)\%\(u1 + \ \[Epsilon]\)DiracDelta[ u - u1] \((fu /. u -> u1)\)\ \(E\^\(\(-I\)\ u1\ x\)\) \[DifferentialD]u\), \ \[Epsilon] -> 0]\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ u1\ x + \[ImaginaryI]\ u1\ \ \[Delta] - \[Gamma]\^\[Alpha]\ Abs[u1]\^\[Alpha]\ \((1 + \[ImaginaryI]\ \ \[Beta]\ \((\(-1\) + \[Gamma]\ Abs[u1]\^\(1 - \[Alpha]\))\)\ Sign[u1]\ Tan[\(\ \[Pi]\ \[Alpha]\)\/2])\)\)\/\(2\ \[Pi]\)\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ StyleBox["Note, below, that a table of values, ", FontSlant->"Plain"], "ffx1,", StyleBox[" is produced with a specific range, {-5,5}, and step value (.01). \ ", FontSlant->"Plain"] }], "Text", PageWidth->PaperWidth, FontSize->12, FontColor->GrayLevel[0]], Cell[BoxData[ \(\(ffx1 = Apply[Plus, Table[\((idf /. u1 -> 2\ \[Pi]\ f)\), {f, \(-5\), 5, .01}]];\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["To correct amplitude scaling, we make use of the fact that ", FontSize->12, FontColor->GrayLevel[0]], Cell[BoxData[ \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity] x \[DifferentialD]x = 1. \)], PageWidth->PaperWidth, CellMargins->{{Inherited, 0}, {Inherited, Inherited}}, CellSize->{483.188, Inherited}, FontSize->12, FontSlant->"Plain", FontColor->GrayLevel[0]], "\n", StyleBox["We can perform the integration numerically after assigning \ values to the parameters. The scaling factor is independent of the assigned \ parameters so long as the integration is performed over a range which covers \ all significant values of the pdf.", FontSize->12, FontColor->GrayLevel[0]] }], "Text", PageWidth->PaperWidth, FontSlant->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ \(m1 = NIntegrate[ Evaluate[ Abs[ffx1 /. {\[Alpha] \[Rule] 2, \[Beta] \[Rule] 0, \[Gamma] \[Rule] 1, \[Delta] \[Rule] 0}]], {x, \(-30\), 30}]\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(15.915494309191605`\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ StyleBox["The value of the constant term \"m1\" above is ONLY useful for \ that certain range and step value (in ", FontSize->12, FontSlant->"Plain", FontColor->GrayLevel[0]], StyleBox["ffx1", FontSize->12, FontColor->GrayLevel[0]], StyleBox[") specified above. This constant, a divisor of the function that \ creates the \"simulated\" pdf, must be reset if range or step value changes.\n\ \nAfter application of the scaling factor, we are able to plot the pdf for a \ wide range of parameters. The \"simulated\" probability distribution \ function, above, is now operational within the bounds of the range and step \ value leading to the m1 constant above. Below, a stable distribution is \ plotted using input values for \[Alpha], \[Beta], \[Gamma] and \[Delta].", FontSize->12, FontSlant->"Plain", FontColor->GrayLevel[0]] }], "Text", PageWidth->PaperWidth, FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(pf1 = Plot[Evaluate[{Abs[ ffx1\/m1] /. {\[Alpha] \[Rule] 2, \[Beta] \[Rule] 0, \[Gamma] \[Rule] 1, \[Delta] \[Rule] 0}, Abs[ffx1\/m1] /. {\[Alpha] \[Rule] 1.4, \[Beta] \[Rule] 1, \[Gamma] \[Rule] 1, \[Delta] \[Rule] 0}}], {x, \(-5\), 25}, PlotStyle -> {Hue[ .6], Hue[ .9]}, PlotRange -> All, GridLines -> Automatic]\)], "Input", PageWidth->PaperWidth], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.18254 0.031746 0.0147151 2.08655 [ 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