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", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["M", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["ACLACHLAN", FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["AND ", FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["J", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["OHN ", FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["E. R", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["EITH", FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "Title"], Cell[CellGroupData[{ Cell["1. INTRODUCTION", "Section"], Cell[TextData[{ "Since Pareto many attempts have been made", StyleBox[" ", FontSize->10], " to discover a mathematical relationship that describes precisely the \ skewed distribution of income. Pareto's distribution with its two parameters \ is parsimonious, but the fit to most data sets is poor. On the other \ extreme, the generalized beta with five adjustable parameters (McDonald & Xu \ 1995) can be made to fit income data almost perfectly. In selecting the \ optimal point of tradeoff between parsimony and fit, a judgment has to be \ made about the relative importance of the two qualities. We shall argue \ below that there are slight but systematic imperfections in the income data \ used to test the models, meaning that a better fit does not necessarily imply \ a better model. In considering the trade-off between parsimony and fit, \ therefore, the value of parsimony ought to be given greater weight than if \ the data were perfect. \n\nMcDonald (1984) identifies four two-parameter \ models in the modern literature. He ranks the four models by goodness of fit \ to family income data from 1970, 1975 and 1980 as follows: the gamma, \ Weibull, Fisk, and lognormal. \n\nThe gamma distribution, originally fitted \ to income data by Amorosa (1925), was introduced to English speaking readers \ by A. B. Z. Salem and T. D. Mount (1974). " }], "Text"], Cell[BoxData[ \(<< Statistics`ContinuousDistributions`\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?GammaDistribution\)\)], "Input"], Cell[BoxData[ RowBox[{"\<\"GammaDistribution[alpha, beta] represents the gamma \ distribution with shape parameter alpha and scale parameter beta.\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"Statistics`ContinuousDistributions`", Active->True, ButtonStyle->"AddOnsLink"]}]], "Print", CellTags->"Info3318953887-5940970"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CDF[GammaDistribution[\[Alpha], \[Beta]], x]\)], "Input"], Cell[BoxData[ \(GammaRegularized[\[Alpha], 0, x\/\[Beta]]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PDF[GammaDistribution[\[Alpha], \[Beta]], x]\)], "Input"], Cell[BoxData[ \(\(\[ExponentialE]\^\(-\(x\/\[Beta]\)\)\ x\^\(\(-1\) + \[Alpha]\)\ \ \[Beta]\^\(-\[Alpha]\)\)\/Gamma[\[Alpha]]\)], "Output"] }, Open ]], Cell["\<\ The Weibull distribution,used by Swedish engineer Wallobi Weibull to model \ metallurgical failure rates, was later found to have a wide variety of \ applications outside of engineering (Weibull 1951).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?WeibullDistribution\)\)], "Input"], Cell[BoxData[ RowBox[{"\<\"WeibullDistribution[alpha, beta] represents the Weibull \ distribution with shape parameter alpha and scale parameter beta.\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"Statistics`ContinuousDistributions`", Active->True, ButtonStyle->"AddOnsLink"]}]], "Print", CellTags->"Info3318953887-1236196"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CDF[WeibullDistribution[\[Alpha], \[Beta]], x] t\)], "Input"], Cell[BoxData[ \(\((1 - \[ExponentialE]\^\(-\((x\/\[Beta])\)\^\[Alpha]\))\)\ t\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PDF[WeibullDistribution[\[Alpha], \[Beta]], x]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-\((x\/\[Beta])\)\^\[Alpha]\)\ x\^\(\(-1\) + \ \[Alpha]\)\ \[Alpha]\ \[Beta]\^\(-\[Alpha]\)\)], "Output"] }, Open ]], Cell["\<\ Peter R.Fisk (1961) proposed what he called the sech squared distributionfor \ incomes within specific occupations.\ \>", "Text"], Cell[BoxData[ \(FiskCDF[ x_] := \((x/\[Beta])\)^\[Alpha]/\((\((x/\[Beta])\)^\[Alpha]\ + 1)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(FiskCDF[x] // TraditionalForm\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\((x\/\[Beta])\)\^\[Alpha]\/\(\((x\/\[Beta])\)\^\ \[Alpha] + 1\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FiskPDF = D[FiskCDF[x], x] // TraditionalForm\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(\[Alpha]\ \((x\/\[Beta])\)\^\(\[Alpha] - \ 1\)\)\/\(\((\((x\/\[Beta])\)\^\[Alpha] + 1)\)\ \[Beta]\) - \(\[Alpha]\ \((x\/\ \[Beta])\)\^\(2\ \[Alpha] - 1\)\)\/\(\((\((x\/\[Beta])\)\^\[Alpha] + 1)\)\^2\ \ \[Beta]\)\)], "Output"] }, Open ]], Cell["\<\ Finally,the lognormal distribution was introduced as a model of the size \ distribution of incomes by French economist, Robert Gibrat (1931).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?LogNormalDistribution\)\)], "Input"], Cell[BoxData[ RowBox[{"\<\"LogNormalDistribution[mu, sigma] represents the log-normal \ distribution based on a normal distribution having mean mu and standard \ deviation sigma.\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"Statistics`ContinuousDistributions`", Active->True, ButtonStyle->"AddOnsLink"]}]], "Print", CellTags->"Info3318953887-7025858"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CDF[LogNormalDistribution[\[Mu], \[Sigma]], x]\)], "Input"], Cell[BoxData[ \(1\/2\ \((1 + Erf[\(\(-\[Mu]\) + Log[x]\)\/\(\@2\ \[Sigma]\)])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PDF[LogNormalDistribution[\[Mu], \[Sigma]], x]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-\(\((\(-\[Mu]\) + Log[x])\)\^2\/\(2\ \ \[Sigma]\^2\)\)\)\/\(\@\(2\ \[Pi]\)\ x\ \[Sigma]\)\)], "Output"] }, Open ]], Cell["\<\ In this paper we introduce an additional two-parameter model. Using 23 years \ of data on male and female incomes we show that it consistently provides a \ better fit to the data than the lognormal or Fisk, and is competitive with \ the gamma and Weibull. Furthermore,unlike any of the existing two-parameter \ distributions, the distribution presented here allows for the possibility of \ negative incomes. If capital losses are to be included in income \ measures,the possibility of negative incomes becomes an important \ consideration.\ \>", "Text"], Cell[BoxData[ StyleBox[\(Off[General::spell1]\), FontSize->9]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["2. THE DISTRIBUTION", "Section"], Cell["\<\ The distribution has its origin in the textile market research group at \ duPont in the 1970s. With access to consumer spending data from a sample of \ roughly 10,000 households, the group was able to examine sales volume at \ different prices for various products. Products studied included various \ textiles, panty hose, carpet fibers, and men's and boys\[CloseCurlyQuote] \ dress shirts. Plots of the data revealed a distribution that was stable \ through time and across products. Apparently following a lognormal \ distribution, a better fit was found with a distribution whose CDF is given \ by: \ \>", "Text"], Cell[BoxData[ \(\(BeamanCDF[ x_] := \((1 + \((b - a)\)^3/\((x - a)\)^3)\)^\(-1\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(BeamanCDF[x] // TraditionalForm\)], "Input"], Cell[BoxData[ \(TraditionalForm\`1\/\(\((b - a)\)\^3\/\((x - a)\)\^3 + 1\)\)], "Output"] }, Open ]], Cell["The PDF is given by:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[BeamanCDF[x], x] // TraditionalForm\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(3\ \((b - a)\)\^3\)\/\(\((x - a)\)\^4\ \((\((b - \ a)\)\^3\/\((x - a)\)\^3 + 1)\)\^2\)\)], "Output"] }, Open ]], Cell["\<\ The two parameters have direct economic interpretation. One represents the \ minimum price (or income), while the other represents the median. The figure \ below illustrates the fitted distribution using data points for the income \ for males in 2002, and shows the position of the two parameters along the \ horizontal axis. \t\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(<< Statistics`NonlinearFit`\), "\[IndentingNewLine]", \(<< Graphics`Graphics`\), "\[IndentingNewLine]", \(\(maledata02 = {{4999.`, 0.07776252367447967`}, {9999.`, 0.16094637792987343`}, {14999.`, 0.25587477828661903`}, {24999.`, 0.433936928179896`}, {34999.`, 0.5821266446873967`}, {49999.`, 0.7397059855097153`}, {74999.`, 0.8773737110561073`}};\)\), "\[IndentingNewLine]", \(parm02 = \(NonlinearRegress[ maledata02, \ \((1 + \((b - a)\)^3/\((x - a)\)^3)\)^\(-1\), \ x, \ \ {{b, 25000, 45000}, {a, \ \(-25000\), 0}}, \ RegressionReport \[Rule] BestFitParameters]\)[\([1, 2]\)]; DisplayTogether[ Plot[\((1 + \((b - a)\)^3/\((x - a)\)^3)\)^\(-1\) /. parm02, \ {x, \ \(-17000\), \ 120000}], \[IndentingNewLine]\ Graphics[Text["\", \ {\(-17076\), .06}]], 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ool00?l0ooooJ@3oool00?l0ooooJ@3oool00?l0ooooJ@3oool00?l0ooooJ@3oool00?l0ooooJ@3o ool002L0oooo0`0000060?ooo`H000001@3oool300000?l0oooo:`3oool002P0oooo00<000000?oo o`3oool01P3oool00`000000oooo0?ooo`020?ooo`030000003oool0oooo00<0oooo00<000000?oo o`3oool0o`3ooolZ0?ooo`00:03oool00`000000oooo0?ooo`060?ooo`030000003oool0oooo00<0 oooo00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`3o0?ooobX0oooo000X0?ooo`030000 003oool0oooo00H0oooo00<000000?ooo`3oool0103oool01@000000oooo0?ooo`3oool000000?l0 oooo;03oool002P0oooo1@0000040?ooo`030000003oool0oooo00@0oooo00D000000?ooo`3oool0 oooo000000020?ooo`030000003oool0oooo0?l0oooo9`3oool002P0oooo00<000000?ooo`3oool0 0P3oool01@000000oooo0?ooo`3oool0000000H0oooo00@000000?ooo`3oool0oooo1000003o0?oo obT0oooo000X0?ooo`030000003oool0oooo0080oooo00D000000?ooo`3oool0oooo000000060?oo o`050000003oool0oooo0?ooo`0000000P3oool00`000000oooo0?ooo`3o0?ooobL0oooo000X0?oo o`030000003oool0oooo0080oooo00D000000?ooo`3oool0oooo000000050?ooo`030000003oool0 oooo0080oooo00<000000?ooo`3oool0o`3ooolZ0?ooo`00:03oool00`000000oooo0?ooo`020?oo o`050000003oool0oooo0?ooo`000000103oool00`000000oooo0?ooo`030?ooo`030000003oool0 oooo0080oooo00<000000?ooo`3oool0o`3ooolU0?ooo`009`3oool6000000<0oooo1P0000050?oo o`L00000o`3ooolW0?ooo`00o`3ooomY0?ooo`00o`3ooomY0?ooo`00o`3ooomY0?ooo`00o`3ooomY 0?ooo`00\ \>"], ImageRangeCache->{{{0, 359}, {221.375, 0}} -> {-26931.5, -2.49389*^-06, \ 640.883, 1.34589*^-07}}] }, Open ]], Cell["\<\ Reith (1986) first introduced the equation in an unpublished paper in which \ he presents a number of results based on his market research at duPont. He \ labeled the distribution after the late Ralph G. Beaman, the colleague in the \ market research group at duPont who contributed the most in its formulation. \ While it was not possible to recover the underlying data from the product \ market studies, we discovered that the distribution provides a remarkably \ good fit to publicly available income data. \ \>", "Text"], Cell[TextData[{ "A notable difference between the results of the product market studies and \ our work on income distribution presented here is the large negative value \ that we find in estimating the minimum value parameter. Over the 23 years, \ the magnitude of the negative intercept is usually over one half the \ estimated median. For example, in 2002 the male median income is estimated \ at $29,549, while the minimum is estimated at negative $17,076. For product \ markets, estimates of the minimum price were rarely negative, and only when \ the market prices were very low. ", StyleBox["\n", FontWeight->"Bold"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["3. THE DATA", "Section"], Cell[TextData[{ "The data used to test the fit are drawn from the Statistical Abstract of \ the United States (1995-2004) and cover the 23 years, 1970, 1980-1993, \ 1995-2002. We consider individual incomes, labeled \ \[OpenCurlyDoubleQuote]income of people\[CloseCurlyDoubleQuote] by the \ Census, for males and females. Since many studies of income distribution \ are addressing the social justice issues of poverty and inequality, it is \ common to examine family income data. Our focus, however, is on the purely \ empirical question of the market determined distribution of claims to output \ across individuals.\n\nAnother result of the traditional focus on inequality \ and poverty is the way in which the Census measures income. Of particular \ importance to our study is the fact that government transfer payments are \ included, and capital gains are not. The Census Bureau (2003, p. 2) provides \ the following definition:\n\n", StyleBox["Money income (MI) is collected for all people in the sample 15 \ years old and over. Money income includes earnings, unemployment \ compensation, workers\[CloseCurlyQuote] compensation, social security, \ supplemental security income, public assistance, veterans\[CloseCurlyQuote] \ payments, survivor benefits, income from estates, trusts, educational \ assistance, alimony, child support, assistance from outside the household, \ and other miscellaneous sources. It is income before deduction for taxes or \ other expenses and does not include lump-sum payments or capital gains. \n\n", FontWeight->"Bold"], "If lower income individuals are more likely to receive a significant \ portion of their income in the form of government transfers, then one should \ expect the official data points on the left hand tail of the distribution to \ lie below the pure market determined values. For example, if the U.S. Census \ says that 8% of the population earns less than $5,000, then the percentage \ when transfers are not included would be larger than 8%. Interestingly, the \ fitted Beaman distribution is more likely than not to cut above the minimum \ data point", StyleBox[".", FontVariations->{"CompatibilityType"->0}] }], "Text"], Cell[CellGroupData[{ Cell["Raw Data: 1980 to 1993.", "Subsubsection"], Cell["\<\ The following are the incomes levels used for the data for 1980 to 1993.\ \>", "Text"], Cell[BoxData[ \(\(incomes = {2499, 4999, 9999, 14999, 24999, 49999, 74999};\)\)], "Input"], Cell["\<\ The following are the percentage shares in each of the income categories from \ 1995 SAUS. For example, the first entry is the percentage of males earning \ $2499 or less.\ \>", "Text"], Cell[BoxData[{ \(\(male1970\ = {8.3, \ 5.9, \ 10.8, \ 9.8, \ 19.5, \ 35.6, \ 7.1};\)\), "\[IndentingNewLine]", \(\(male1980\ = {7.2, \ 5.1, \ 11.9, \ 11.1, \ 20.0, \ 32.9, \ 8.1};\)\), "\[IndentingNewLine]", \(\(male1981 = \ {7.6, \ 5.8, \ 12.1, \ 10.9, \ 20.1, \ 32.1, \ 7.8};\)\), "\[IndentingNewLine]", \(\(male1982 = {8.2, 5.6, 12.2, 10.6, 21.1, 30.6, 7.6};\)\), "\[IndentingNewLine]", \(\(male1983\ = {8.3, 5.6, 12.1, 11.7, 20.1, 30.1, 8.4};\)\), "\[IndentingNewLine]", \(\(male1984 = {7.6, 5.5, 12.6, 11.1, 19.0, 31.3, 8.7};\)\), "\[IndentingNewLine]", \(\(male1985\ = {7.3, \ 5.5, \ 12.0, 11.6, 19.8, 30.6, 8.9};\)\), "\[IndentingNewLine]", \(\(male1986 = {7.0, \ 5.1, \ 11.8, \ 11.1, \ 19.3, \ 31.3, 9.6};\)\), "\[IndentingNewLine]", \(\(male1987 = {6.7, 5.1, 11.6, 11.1, 19.7, 31.0, 9.8};\)\), "\[IndentingNewLine]", \(\(male1988 = {6.7, \ 4.9, 11.7, 11.0, 20.1, 31.1, 9.5};\)\), "\[IndentingNewLine]", \(\(male1989 = {6.3, 4.7, 11.6, 11.2, 20.5, 30.7, 9.4};\)\), "\[IndentingNewLine]", \(\(male1990 = {6.4, 4.7, 12.3, 11.5, 20.9, 30.4, 9.0};\)\), "\[IndentingNewLine]", \(\(male1991 = {6.4, 5.1, 12.5, 12.4, 20.3, 30.2, 8.4};\)\), "\[IndentingNewLine]", \(\(male1992 = {6.9, 5.1, 12.9, 12.3, 20.6, 29.0, 8.4};\)\), "\[IndentingNewLine]", \(\(male1993 = {7.0, 4.9, 12.1, 12.4, 20.8, 28.8, 8.7};\)\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[BoxData[ \(\(m70 = Transpose[{incomes, \ .01\ male1970}];\)\)], "Input"], Cell[BoxData[ \(\(m80 = Transpose[{incomes, \ .01\ male1980}];\)\)], "Input"], Cell[BoxData[ \(\(m81 = Transpose[{incomes, \ .01\ male1981}];\)\)], "Input"], Cell[BoxData[ \(\(m82 = Transpose[{incomes, \ .01\ male1982}];\)\)], "Input"], Cell[BoxData[ \(\(m83 = Transpose[{incomes, \ .01\ male1983}];\)\)], "Input"], Cell[BoxData[ \(\(m84 = Transpose[{incomes, \ .01\ male1984}];\)\)], "Input"], Cell[BoxData[ \(\(m85 = Transpose[{incomes, \ .01\ male1985}];\)\)], "Input"], Cell[BoxData[ \(\(m86 = Transpose[{incomes, \ .01\ male1986}];\)\)], "Input"], Cell[BoxData[ \(\(m87 = Transpose[{incomes, \ .01\ male1987}];\)\)], "Input"], Cell[BoxData[ \(\(m88 = Transpose[{incomes, \ .01\ male1988}];\)\)], "Input"], Cell[BoxData[{ \(\(m89 = Transpose[{incomes, \ .01\ male1989}];\)\), "\[IndentingNewLine]", \(\(m90 = Transpose[{incomes, \ .01\ male1990}];\)\)}], "Input"], Cell[BoxData[ \(\(m91 = Transpose[{incomes, \ .01\ male1991}];\)\)], "Input"], Cell[BoxData[ \(\(m92 = Transpose[{incomes, \ .01\ male1992}];\)\)], "Input"], Cell[BoxData[ \(\(m93 = Transpose[{incomes, \ .01\ male1993}];\)\)], "Input"], Cell[BoxData[{ \(\(female1970 = {20.8, 16.3, 20.4, 15.0, 18.0, 8.9, 0.5};\)\), "\[IndentingNewLine]", \(\(female1980 = {20.3, 12.8, 22.3, 14.4, 18.3, 10.8, 0.8};\)\), "\[IndentingNewLine]", \(\(female1981 = {19.9, 12.6, 22.8, 14.1, 18.6, 11.0, 0.8};\)\), "\[IndentingNewLine]", \(\(female1982 = {19.8, 12.2, 22.4, 13.6, 18.9, 11.7, 1.0};\)\), "\[IndentingNewLine]", \(\(female1983 = {18.9, \ 12.1, 21.7, \ 14.6, 18.4, 12.7, 1.2};\)\), "\[IndentingNewLine]", \(\(female1984 = {18.1, \ 11.9, \ 22.0, 14.1, 18.2, 13.9, 1.4};\)\), "\[IndentingNewLine]", \(\(female1985 = {17.7, 12.0, 21.7, 14.0, 18.1, 14.4, 1.5};\)\), "\[IndentingNewLine]", \(\(female1986 = {17.1, 11.6, 21.6, 13.7, 18.4, 15.2, 1.7};\)\), "\[IndentingNewLine]", \(\(female1987 = {16.0, 11.4, 21.2, 13.8, 19.1, 15.8, 1.9};\)\), "\[IndentingNewLine]", \(\(female1988 = {15.9, 11.1, 21.3, 14.0, 18.9, 16.1, 2.1};\)\), "\[IndentingNewLine]", \(\(female1989 = {15.1, 11.1, 20.6, 13.9, 19.6, 16.6, 2.2};\)\), "\[IndentingNewLine]", \(\(female1990 = {14.9, 11.2, \ 20.9, \ 14.4, 19.1, 16.4, 2.3};\)\), "\[IndentingNewLine]", \(\(female1991 = {14.3, 10.8, 21.7, 14.9, 19.0, 16.3, 2.2};\)\), "\[IndentingNewLine]", \(\(female1992 = {14.6, 10.9, 21.3, 14.7, 18.7, 16.7, 2.2};\)\), "\[IndentingNewLine]", \(\(\(female1993 = {14.1, 10.9, 21.4, 14.4, 19.1, 16.5, 2.6};\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[BoxData[ \(\(f70 = Transpose[{incomes, \ .01\ female1970}];\)\)], "Input"], Cell[BoxData[ \(\(f80 = Transpose[{incomes, \ .01\ female1980}];\)\)], "Input"], Cell[BoxData[ \(\(f81 = Transpose[{incomes, \ .01\ female1981}];\)\)], "Input"], Cell[BoxData[ \(\(f82 = Transpose[{incomes, \ .01\ female1982}];\)\)], "Input"], Cell[BoxData[ \(\(f83 = Transpose[{incomes, \ .01\ female1983}];\)\)], "Input"], Cell[BoxData[ \(\(f84 = Transpose[{incomes, \ .01\ female1984}];\)\)], "Input"], Cell[BoxData[ \(\(f85 = Transpose[{incomes, \ .01\ female1985}];\)\)], "Input"], Cell[BoxData[ \(\(f86 = Transpose[{incomes, \ .01\ female1986}];\)\)], "Input"], Cell[BoxData[ \(\(f87 = Transpose[{incomes, \ .01\ female1987}];\)\)], "Input"], Cell[BoxData[ \(\(f88 = Transpose[{incomes, \ .01\ female1988}];\)\)], "Input"], Cell[BoxData[ \(\(f89 = Transpose[{incomes, \ .01\ female1989}];\)\)], "Input"], Cell[BoxData[ \(\(f90 = Transpose[{incomes, \ .01\ female1990}];\)\)], "Input"], Cell[BoxData[ \(\(f91 = Transpose[{incomes, \ .01\ female1991}];\)\)], "Input"], Cell[BoxData[ \(\(f92 = Transpose[{incomes, \ .01\ female1992}];\)\)], "Input"], Cell[BoxData[ \(\(f93 = Transpose[{incomes, \ .01\ female1993}];\)\)], "Input"], Cell["", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Raw Data: 1995-2002", "Subsubsection"], Cell["\<\ **** No data from 1994 appear in the SAUS. ****\ \>", "Text"], Cell["\<\ The following are the incomes levels used for the data for 1995 to 2002.\ \>", "Text"], Cell[BoxData[ \(\(incomes2 = {4999, 9999, 14999, 24999, 34999, 49999, 74999};\)\)], "Input"], Cell[TextData[StyleBox["1995 (from 1997 SAUS)", "Text"]], "Text"], Cell[BoxData[ \(\(males1995 = {9548, 10224, 10793, 19018, 14135, 13153, 9383, 5811};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(m95 = Transpose[{incomes2, Delete[males1995/Total[males1995], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.10370933579536197`}, {9999.`, 0.11105197414869929`}, {14999.`, 0.1172323901591267`}, {24999.`, 0.2065714440884158`}, {34999.`, 0.15353283006571444`}, {49999.`, 0.1428664530494759`}, {74999.`, 0.10191712377124858`}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(females1995 = {21135, \ 19966, 13954, 18775, 10658, 7131, 3136, 1252};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f95 = Transpose[{incomes2, Delete[females1995/Total[females1995], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.22014019811055444`}, {9999.`, 0.2079640026248086`}, {14999.`, 0.14534356869811577`}, {24999.`, 0.1955586571812472`}, {34999.`, 0.11101273865447311`}, {49999.`, 0.0742758340537669`}, {74999.`, 0.03266428489589301`}}\)], "Output"] }, Open ]], Cell["\<\ 1996?(labelled 1995 but references 1996 Population Report) from 1998 SAUS\ \>", "Text"], Cell[BoxData[ \(\(males1996 = {9014, 10402, 10931, 18145, 14516, 13996, 9912, 6253};\)\)], "Input"], Cell["\<\ *** Total for males in 1998 SAUS is 93439, slightly different from the sum of \ above. ***\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(m96 = Transpose[{incomes2, Delete[males1996/Total[males1996], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.09674891863173374`}, {9999.`, 0.11164657772435037`}, {14999.`, 0.11732443194624821`}, {24999.`, 0.19475361976623126`}, {34999.`, 0.155802895812985`}, {49999.`, 0.15022164024514592`}, {74999.`, 0.10638731767004046`}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(females1996 = {19495, 19854, 13970, 19237, 11278, 7505, 3659, 1560};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f96 = Transpose[{incomes2, Delete[females1996/Total[females1996], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.20189937654052487`}, {9999.`, 0.20561734915801902`}, {14999.`, 0.1446798815219868`}, {24999.`, 0.19922740736137864`}, {34999.`, 0.11680026512562398`}, {49999.`, 0.07772530499803228`}, {74999.`, 0.0378943225833178`}}\)], "Output"] }, Open ]], Cell["1997 from 1999 SAUS", "Text"], Cell[BoxData[ \(\(males1997 = {8578, 9598, 10695, 17800, 14740, 14367, 10938, 7451};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(m97 = Transpose[{incomes2, Delete[males1997/Total[males1997], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.09109348285492795`}, {9999.`, 0.1019253029192817`}, {14999.`, 0.1135748192041798`}, {24999.`, 0.18902587955440867`}, {34999.`, 0.1565304193613474`}, {49999.`, 0.15256937143585333`}, {74999.`, 0.11615534104304055`}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(females1997 = {18456, 19077, 14123, 19472, 11813, 8207, 4351, 1947};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f97 = Transpose[{incomes2, Delete[females1997/Total[females1997], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.18939720460562773`}, {9999.`, 0.19576996490363893`}, {14999.`, 0.1449315518338362`}, {24999.`, 0.1998234919853047`}, {34999.`, 0.12122611497649981`}, {49999.`, 0.08422100445374875`}, {74999.`, 0.044650370461588984`}}\)], "Output"] }, Open ]], Cell["1998 from 2000 SAUS", "Text"], Cell[BoxData[ \(\(males1998 = {8360, 9142, 9548, 17620, 14718, 15234, 11763, 8562};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(m98 = Transpose[{incomes2, Delete[males1998/Total[males1998], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.08804912214182649`}, {9999.`, 0.09628529600724615`}, {14999.`, 0.10056136581461236`}, {24999.`, 0.1855772167630362`}, {34999.`, 0.15501279661284717`}, {49999.`, 0.1604474075010269`}, {74999.`, 0.12389017030553888`}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(females1998 = {18456, 19077, 14123, 19472, 11813, 8207, 4351, 1947};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f98 = Transpose[{incomes2, Delete[females1998/Total[females1998], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.18939720460562773`}, {9999.`, 0.19576996490363893`}, {14999.`, 0.1449315518338362`}, {24999.`, 0.1998234919853047`}, {34999.`, 0.12122611497649981`}, {49999.`, 0.08422100445374875`}, {74999.`, 0.044650370461588984`}}\)], "Output"] }, Open ]], Cell["1999 from 2001 SAUS", "Text"], Cell[BoxData[ \(\(males1999 = {8341, 8695, 9380, 17386, 14566, 15235, 12557, 9863};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(m99 = Transpose[{incomes2, Delete[males1999/Total[males1999], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.08686460535496704`}, {9999.`, 0.09055122210303782`}, {14999.`, 0.09768492965227081`}, {24999.`, 0.18106078751965674`}, {34999.`, 0.1516928235943472`}, {49999.`, 0.15865990439790467`}, {74999.`, 0.1307707528404653`}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(females1999 = {17797, 17559, 13667, 19849, 12939, 9663, 5466, 2674};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f99 = Transpose[{incomes2, Delete[females1999/Total[females1999], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.1786596261569659`}, {9999.`, 0.17627040375850783`}, {14999.`, 0.13719959041901741`}, {24999.`, 0.19925914028148653`}, {34999.`, 0.1298913807296163`}, {49999.`, 0.09700443712731142`}, {74999.`, 0.05487180516794828`}}\)], "Output"] }, Open ]], Cell["2000 from 2002 SAUS", "Text"], Cell[BoxData[ \(\(males2000 = {7716, 8278, 9352, 17556, 14573, 15346, 13347, 10815};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(m00 = Transpose[{incomes2, Delete[males2000/Total[males2000], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.07956033531649877`}, {9999.`, 0.08535516533825516`}, {14999.`, 0.09642927110936968`}, {24999.`, 0.18102141612447542`}, {34999.`, 0.1502634482331955`}, {49999.`, 0.15823391728447253`}, {74999.`, 0.13762205747399028`}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(females2000 = {16767, 16879, 13481, 19411, 13629, 10443, 6495, 2869};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f00 = Transpose[{incomes2, Delete[females2000/Total[females2000], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.16771360553743975`}, {9999.`, 0.16883389681317143`}, {14999.`, 0.13484505971552604`}, {24999.`, 0.19416048172524855`}, {34999.`, 0.13632544461560006`}, {49999.`, 0.10445715886130394`}, {74999.`, 0.06496689139176186`}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["2001 from 2003 SAUS", "Subsubsection"]], "Text"], Cell[BoxData[ \(\(males2001 = {7923, 8453, 9290, 17147, 15093, 15802, 13478, 11690};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(m01 = Transpose[{incomes2, Delete[males2001/Total[males2001], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.08013066871637202`}, {9999.`, 0.08549091791739148`}, {14999.`, 0.09395606618390712`}, {24999.`, 0.17341923216958616`}, {34999.`, 0.1526457380962013`}, {49999.`, 0.15981633561228206`}, {74999.`, 0.13631214854969861`}}\)], "Output"] }, Open ]], Cell["\<\ *** total in SAUS is 98873, slightly different from the sum of above. ***\ \>", "Text"], Cell[BoxData[ \(\(females2001 = {16452, 16904, 13656, 20076, 13517, 10986, 6819, 3529};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f01 = Transpose[{incomes2, Delete[females2001/Total[females2001], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.16139063557617792`}, {9999.`, 0.16582465984559394`}, {14999.`, 0.13396246775032128`}, {24999.`, 0.19694130803715948`}, {34999.`, 0.13259890718959377`}, {49999.`, 0.10777033323850538`}, {74999.`, 0.06689294578130058`}}\)], "Output"] }, Open ]], Cell["2002 from 2004 SAUS", "Text"], Cell[BoxData[ \(\(males2002 = \ {7760, 8301, 9473, 17769, 14788, 15725, 13738, 12237};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(m02 = Transpose[{incomes2, Delete[males2002/Total[males2002], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.07776252367447967`}, {9999.`, 0.08318385425539378`}, {14999.`, 0.0949284003567456`}, {24999.`, 0.17806214989327696`}, {34999.`, 0.14818971650750068`}, {49999.`, 0.15757934082231864`}, {74999.`, 0.13766772554639195`}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(females2002 = \ {16621, 16405, 13658, 19770, 13917, 11534, 6964, 3617};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f02 = Transpose[{incomes2, Delete[females2002/Total[females2002], 8]}] // N\)], "Input"], Cell[BoxData[ \({{4999.`, 0.1621782487364128`}, {9999.`, 0.16007064379525007`}, {14999.`, 0.13326698280740784`}, {24999.`, 0.1929043966980856`}, {34999.`, 0.13579415725074645`}, {49999.`, 0.11254220088597466`}, {74999.`, 0.0679507444919306`}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Non-cumulative Data Points", FontSize->10]], "Subsection"], Cell[BoxData[ \(\(maledata = {m70, \ m80, \ m81, \ m82, \ m83, \ m84, \ m85, \ m86, \ m87, \ m88, \ m89, \ m90, \ m91, \ m92, \ m93, \ m95, \ m96, \ m97, \ m98, \ m99, \ m00, \ m01, \ m02} // TableForm;\)\)], "Input"], Cell[BoxData[ \(\(femaledata = {f70, \ f80, \ f81, \ f82, \ f83, \ f84, \ f85, \ f86, \ f87, \ f88, \ f89, \ f90, \ f91, \ f92, \ f93, \ f95, \ f96, \ f97, \ f98, \ f99, \ f00, \ f01, \ f02} // TableForm;\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Cumulative (regression-ready) Data Points", 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COMPARING THE FIT", "Section"], Cell[TextData[{ "In order to compare the fit of the five two-parameters models, we find \ parameter estimates using the Levenberg-Marquardt algorithm in the ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["5.0 ", FontSlant->"Italic"], " nonlinear regression package. An added advantage of the Beaman \ distribution is that the parameters can also be estimated with a simple \ linear regression, since the quantile function is linear in the parameters. \ The quantile function represents the level of income, ", StyleBox["x", FontSlant->"Italic"], ", as a function of the cumulative proportion, ", StyleBox["p", FontSlant->"Italic"], ", of the population earning ", StyleBox["x", FontSlant->"Italic"], " or less. " }], "Text"], Cell[BoxData[ \(q[p_] := a + \ \((b - a)\) \((\((p/\((1 - p)\))\))\)^\((1/3)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(q[p] // TraditionalForm\)], "Input"], Cell[BoxData[ \(TraditionalForm\`a + \((b - a)\)\ \@\(p\/\(1 - p\)\)\%3\)], "Output"] }, Open ]], Cell["\<\ Being able to readily estimate the parameters of the model with a simple \ linear regression contributes to the usefulness of the Beaman distribution \ for practical and pedagogical purposes. \ \>", "Text"], Cell[CellGroupData[{ Cell["Lognormal Distribution", "Subsection"], Cell["", "Text"], Cell[BoxData[ \(\(\(Table[ NonlinearRegress[mdata[\([i]\)], \ CDF[LogNormalDistribution[\[Mu], \[Sigma]], \ x], \ x, \ {\ {\[Mu], 5, 15}, {\[Sigma], \ .5, 1.5}}], \ {i, 23}];\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(\(Table[ NonlinearRegress[fdata[\([i]\)], \ CDF[LogNormalDistribution[\[Mu], \[Sigma]], \ x], \ x, \ {\ {\[Mu], 5, 15}, {\[Sigma], \ .5, 1.5}}], \ {i, 23}]\ ;\)\)], "Input"], Cell[CellGroupData[{ 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Open ]], Cell[CellGroupData[{ Cell["Beaman Distribution", "Subsection"], Cell[BoxData[ \(\(Table[ NonlinearRegress[ mdata[\([i]\)], \((1 + \((b - a)\)^3/\((x - a)\)^3)\)^\(-1\), \ x, \ \ {{b, 25000, 45000}, {a, \ \(-25000\), 0}}], \ {i, 23}];\)\)], "Input"], Cell[BoxData[ \(\(Table[ NonlinearRegress[ fdata[\([i]\)], \((1 + \((b - a)\)^3/\((x - a)\)^3)\)^\(-1\), \ x, \ \ {{b, 25000, 45000}, {a, \ \(-25000\), 0}}], \ {i, 23}];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(mBsse = Table[\(\(NonlinearRegress[ mdata[\([i]\)], \((1 + \((b - a)\)^3/\((x - a)\)^3)\)^\(-1\), \ x, \ \ {{b, 25000, 45000}, {a, \ \(-25000\), 0}}\ , \ RegressionReport \[Rule] ANOVATable]\)[\([1, 2]\)]\)[\([1, \ 2, \ 2]\)], \ {i, 23}]\)], "Input"], Cell[BoxData[ \({0.003798645072985225`, 0.00152124405720972`, 0.001562627473931631`, 0.0009106916139395209`, 0.0008948499959340442`, 0.0013935317174262713`, 0.0009071324038021629`, 0.000970852629140878`, 0.0008040841161165918`, 0.0007099959759953995`, 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-0.90428 1.24464 Mabsadd L .64493 .23188 -3.80423 1.23607 Mabsadd L .64493 .23188 -1.46316 -0.47541 Mabsadd L .64493 .23188 -2.35114 -3.23607 Mabsadd L .64493 .23188 0 -1.53846 Mabsadd L .64493 .23188 2.35114 -3.23607 Mabsadd L .64493 .23188 1.46316 -0.47541 Mabsadd L .64493 .23188 3.80423 1.23607 Mabsadd L .64493 .23188 .90428 1.24464 Mabsadd L .64493 .23188 0 4 Mabsadd L closepath F .68634 .13471 0 4 Mabsadd m .68634 .13471 -0.90428 1.24464 Mabsadd L .68634 .13471 -3.80423 1.23607 Mabsadd L .68634 .13471 -1.46316 -0.47541 Mabsadd L .68634 .13471 -2.35114 -3.23607 Mabsadd L .68634 .13471 0 -1.53846 Mabsadd L .68634 .13471 2.35114 -3.23607 Mabsadd L .68634 .13471 1.46316 -0.47541 Mabsadd L .68634 .13471 3.80423 1.23607 Mabsadd L .68634 .13471 .90428 1.24464 Mabsadd L .68634 .13471 0 4 Mabsadd L closepath F .72774 .1366 0 4 Mabsadd m .72774 .1366 -0.90428 1.24464 Mabsadd L .72774 .1366 -3.80423 1.23607 Mabsadd L .72774 .1366 -1.46316 -0.47541 Mabsadd L .72774 .1366 -2.35114 -3.23607 Mabsadd L .72774 .1366 0 -1.53846 Mabsadd L .72774 .1366 2.35114 -3.23607 Mabsadd L .72774 .1366 1.46316 -0.47541 Mabsadd L .72774 .1366 3.80423 1.23607 Mabsadd L .72774 .1366 .90428 1.24464 Mabsadd L .72774 .1366 0 4 Mabsadd L closepath F .76915 .13339 0 4 Mabsadd m .76915 .13339 -0.90428 1.24464 Mabsadd L .76915 .13339 -3.80423 1.23607 Mabsadd L .76915 .13339 -1.46316 -0.47541 Mabsadd L .76915 .13339 -2.35114 -3.23607 Mabsadd L .76915 .13339 0 -1.53846 Mabsadd L .76915 .13339 2.35114 -3.23607 Mabsadd L .76915 .13339 1.46316 -0.47541 Mabsadd L .76915 .13339 3.80423 1.23607 Mabsadd L .76915 .13339 .90428 1.24464 Mabsadd L .76915 .13339 0 4 Mabsadd L closepath F .81056 .15208 0 4 Mabsadd m .81056 .15208 -0.90428 1.24464 Mabsadd L .81056 .15208 -3.80423 1.23607 Mabsadd L .81056 .15208 -1.46316 -0.47541 Mabsadd L .81056 .15208 -2.35114 -3.23607 Mabsadd L .81056 .15208 0 -1.53846 Mabsadd L .81056 .15208 2.35114 -3.23607 Mabsadd L .81056 .15208 1.46316 -0.47541 Mabsadd L .81056 .15208 3.80423 1.23607 Mabsadd L .81056 .15208 .90428 1.24464 Mabsadd L .81056 .15208 0 4 Mabsadd L closepath F .85197 .14474 0 4 Mabsadd m .85197 .14474 -0.90428 1.24464 Mabsadd L .85197 .14474 -3.80423 1.23607 Mabsadd L .85197 .14474 -1.46316 -0.47541 Mabsadd L .85197 .14474 -2.35114 -3.23607 Mabsadd L .85197 .14474 0 -1.53846 Mabsadd L .85197 .14474 2.35114 -3.23607 Mabsadd L .85197 .14474 1.46316 -0.47541 Mabsadd L .85197 .14474 3.80423 1.23607 Mabsadd L .85197 .14474 .90428 1.24464 Mabsadd L .85197 .14474 0 4 Mabsadd L closepath F .89337 .12726 0 4 Mabsadd m .89337 .12726 -0.90428 1.24464 Mabsadd L .89337 .12726 -3.80423 1.23607 Mabsadd L .89337 .12726 -1.46316 -0.47541 Mabsadd L .89337 .12726 -2.35114 -3.23607 Mabsadd L .89337 .12726 0 -1.53846 Mabsadd L .89337 .12726 2.35114 -3.23607 Mabsadd L .89337 .12726 1.46316 -0.47541 Mabsadd L .89337 .12726 3.80423 1.23607 Mabsadd L .89337 .12726 .90428 1.24464 Mabsadd L .89337 .12726 0 4 Mabsadd L closepath F .93478 .13489 0 4 Mabsadd m .93478 .13489 -0.90428 1.24464 Mabsadd L .93478 .13489 -3.80423 1.23607 Mabsadd L .93478 .13489 -1.46316 -0.47541 Mabsadd L .93478 .13489 -2.35114 -3.23607 Mabsadd L .93478 .13489 0 -1.53846 Mabsadd L .93478 .13489 2.35114 -3.23607 Mabsadd L .93478 .13489 1.46316 -0.47541 Mabsadd L .93478 .13489 3.80423 1.23607 Mabsadd L .93478 .13489 .90428 1.24464 Mabsadd L .93478 .13489 0 4 Mabsadd L closepath F .97619 .11635 0 4 Mabsadd m .97619 .11635 -0.90428 1.24464 Mabsadd L .97619 .11635 -3.80423 1.23607 Mabsadd L .97619 .11635 -1.46316 -0.47541 Mabsadd L .97619 .11635 -2.35114 -3.23607 Mabsadd L .97619 .11635 0 -1.53846 Mabsadd L .97619 .11635 2.35114 -3.23607 Mabsadd L .97619 .11635 1.46316 -0.47541 Mabsadd L .97619 .11635 3.80423 1.23607 Mabsadd L .97619 .11635 .90428 1.24464 Mabsadd L .97619 .11635 0 4 Mabsadd L closepath F .001 w [ .5 3 ] 0 Mabsdash .06522 .56308 m .10663 .35058 L .14803 .36845 L .18944 .34715 L .23085 .3335 L .27226 .32283 L .31366 .28702 L .35507 .28305 L .39648 .26009 L .43789 .25705 L .4793 .21366 L .5207 .21897 L .56211 .21169 L .60352 .20457 L .64493 .19735 L .68634 .12606 L .72774 .13067 L .76915 .12172 L .81056 .1332 L .85197 .12216 L .89337 .10357 L .93478 .10864 L .97619 .09127 L s .0001 w [ ] 0 setdash .06522 .56308 -2.25 0 Mabsadd m .06522 .56308 0 3 Mabsadd L .06522 .56308 2.25 0 Mabsadd L .06522 .56308 0 -3 Mabsadd L .06522 .56308 -2.25 0 Mabsadd L closepath F .10663 .35058 -2.25 0 Mabsadd m .10663 .35058 0 3 Mabsadd L .10663 .35058 2.25 0 Mabsadd L .10663 .35058 0 -3 Mabsadd L .10663 .35058 -2.25 0 Mabsadd L closepath F .14803 .36845 -2.25 0 Mabsadd m .14803 .36845 0 3 Mabsadd L .14803 .36845 2.25 0 Mabsadd L .14803 .36845 0 -3 Mabsadd L .14803 .36845 -2.25 0 Mabsadd L closepath F .18944 .34715 -2.25 0 Mabsadd m .18944 .34715 0 3 Mabsadd L .18944 .34715 2.25 0 Mabsadd L .18944 .34715 0 -3 Mabsadd L .18944 .34715 -2.25 0 Mabsadd L closepath F .23085 .3335 -2.25 0 Mabsadd m .23085 .3335 0 3 Mabsadd L .23085 .3335 2.25 0 Mabsadd L .23085 .3335 0 -3 Mabsadd L .23085 .3335 -2.25 0 Mabsadd L closepath F .27226 .32283 -2.25 0 Mabsadd m .27226 .32283 0 3 Mabsadd L .27226 .32283 2.25 0 Mabsadd L .27226 .32283 0 -3 Mabsadd L .27226 .32283 -2.25 0 Mabsadd L closepath F .31366 .28702 -2.25 0 Mabsadd m .31366 .28702 0 3 Mabsadd L .31366 .28702 2.25 0 Mabsadd L .31366 .28702 0 -3 Mabsadd L .31366 .28702 -2.25 0 Mabsadd L closepath F .35507 .28305 -2.25 0 Mabsadd m .35507 .28305 0 3 Mabsadd L .35507 .28305 2.25 0 Mabsadd L .35507 .28305 0 -3 Mabsadd L .35507 .28305 -2.25 0 Mabsadd L closepath F .39648 .26009 -2.25 0 Mabsadd m .39648 .26009 0 3 Mabsadd L .39648 .26009 2.25 0 Mabsadd L .39648 .26009 0 -3 Mabsadd L .39648 .26009 -2.25 0 Mabsadd L closepath F .43789 .25705 -2.25 0 Mabsadd m .43789 .25705 0 3 Mabsadd L .43789 .25705 2.25 0 Mabsadd L .43789 .25705 0 -3 Mabsadd L .43789 .25705 -2.25 0 Mabsadd L closepath F .4793 .21366 -2.25 0 Mabsadd m .4793 .21366 0 3 Mabsadd L .4793 .21366 2.25 0 Mabsadd L .4793 .21366 0 -3 Mabsadd L .4793 .21366 -2.25 0 Mabsadd L closepath F .5207 .21897 -2.25 0 Mabsadd m .5207 .21897 0 3 Mabsadd L .5207 .21897 2.25 0 Mabsadd L .5207 .21897 0 -3 Mabsadd L .5207 .21897 -2.25 0 Mabsadd L closepath F .56211 .21169 -2.25 0 Mabsadd m .56211 .21169 0 3 Mabsadd L .56211 .21169 2.25 0 Mabsadd L .56211 .21169 0 -3 Mabsadd L .56211 .21169 -2.25 0 Mabsadd L closepath F .60352 .20457 -2.25 0 Mabsadd m .60352 .20457 0 3 Mabsadd L .60352 .20457 2.25 0 Mabsadd L .60352 .20457 0 -3 Mabsadd L .60352 .20457 -2.25 0 Mabsadd L closepath F .64493 .19735 -2.25 0 Mabsadd m .64493 .19735 0 3 Mabsadd L .64493 .19735 2.25 0 Mabsadd L .64493 .19735 0 -3 Mabsadd L .64493 .19735 -2.25 0 Mabsadd L closepath F .68634 .12606 -2.25 0 Mabsadd m .68634 .12606 0 3 Mabsadd L .68634 .12606 2.25 0 Mabsadd L .68634 .12606 0 -3 Mabsadd L .68634 .12606 -2.25 0 Mabsadd L closepath F .72774 .13067 -2.25 0 Mabsadd m .72774 .13067 0 3 Mabsadd L .72774 .13067 2.25 0 Mabsadd L .72774 .13067 0 -3 Mabsadd L .72774 .13067 -2.25 0 Mabsadd L closepath F .76915 .12172 -2.25 0 Mabsadd m .76915 .12172 0 3 Mabsadd L .76915 .12172 2.25 0 Mabsadd L .76915 .12172 0 -3 Mabsadd L .76915 .12172 -2.25 0 Mabsadd L closepath F .81056 .1332 -2.25 0 Mabsadd m .81056 .1332 0 3 Mabsadd L .81056 .1332 2.25 0 Mabsadd L .81056 .1332 0 -3 Mabsadd L .81056 .1332 -2.25 0 Mabsadd L closepath F .85197 .12216 -2.25 0 Mabsadd m .85197 .12216 0 3 Mabsadd L .85197 .12216 2.25 0 Mabsadd L .85197 .12216 0 -3 Mabsadd L .85197 .12216 -2.25 0 Mabsadd L closepath F .89337 .10357 -2.25 0 Mabsadd m .89337 .10357 0 3 Mabsadd L .89337 .10357 2.25 0 Mabsadd L .89337 .10357 0 -3 Mabsadd L .89337 .10357 -2.25 0 Mabsadd L closepath F .93478 .10864 -2.25 0 Mabsadd m .93478 .10864 0 3 Mabsadd L .93478 .10864 2.25 0 Mabsadd L .93478 .10864 0 -3 Mabsadd L .93478 .10864 -2.25 0 Mabsadd L closepath F .97619 .09127 -2.25 0 Mabsadd m .97619 .09127 0 3 Mabsadd L .97619 .09127 2.25 0 Mabsadd L .97619 .09127 0 -3 Mabsadd L .97619 .09127 -2.25 0 Mabsadd L closepath F .001 w [ .5 4 5 4 ] 0 Mabsdash .06522 .12456 m .10663 .0587 L .14803 .0599 L .18944 .04105 L .23085 .04059 L .27226 .05501 L .31366 .04095 L .35507 .04279 L .39648 .03797 L .43789 .03525 L .4793 .02995 L .5207 .03254 L .56211 .03769 L .60352 .03464 L .64493 .02796 L .68634 .02447 L .72774 .03205 L .76915 .03082 L .81056 .02627 L .85197 .02702 L .89337 .0304 L .93478 .02944 L .97619 .03464 L s .0001 w [ ] 0 setdash .06522 .12456 -2.5 2.5 Mabsadd m .06522 .12456 2.5 2.5 Mabsadd L .06522 .12456 2.5 -2.5 Mabsadd L .06522 .12456 -2.5 -2.5 Mabsadd L .06522 .12456 -2.5 2.5 Mabsadd L s .10663 .0587 -2.5 2.5 Mabsadd m .10663 .0587 2.5 2.5 Mabsadd L .10663 .0587 2.5 -2.5 Mabsadd L .10663 .0587 -2.5 -2.5 Mabsadd L .10663 .0587 -2.5 2.5 Mabsadd L s .14803 .0599 -2.5 2.5 Mabsadd m .14803 .0599 2.5 2.5 Mabsadd L .14803 .0599 2.5 -2.5 Mabsadd L .14803 .0599 -2.5 -2.5 Mabsadd L .14803 .0599 -2.5 2.5 Mabsadd L s .18944 .04105 -2.5 2.5 Mabsadd m .18944 .04105 2.5 2.5 Mabsadd L .18944 .04105 2.5 -2.5 Mabsadd L .18944 .04105 -2.5 -2.5 Mabsadd L .18944 .04105 -2.5 2.5 Mabsadd L s .23085 .04059 -2.5 2.5 Mabsadd m .23085 .04059 2.5 2.5 Mabsadd L .23085 .04059 2.5 -2.5 Mabsadd L .23085 .04059 -2.5 -2.5 Mabsadd L .23085 .04059 -2.5 2.5 Mabsadd L s .27226 .05501 -2.5 2.5 Mabsadd m .27226 .05501 2.5 2.5 Mabsadd L .27226 .05501 2.5 -2.5 Mabsadd L .27226 .05501 -2.5 -2.5 Mabsadd L .27226 .05501 -2.5 2.5 Mabsadd L s .31366 .04095 -2.5 2.5 Mabsadd m .31366 .04095 2.5 2.5 Mabsadd L .31366 .04095 2.5 -2.5 Mabsadd L .31366 .04095 -2.5 -2.5 Mabsadd L .31366 .04095 -2.5 2.5 Mabsadd L s .35507 .04279 -2.5 2.5 Mabsadd m .35507 .04279 2.5 2.5 Mabsadd L .35507 .04279 2.5 -2.5 Mabsadd L .35507 .04279 -2.5 -2.5 Mabsadd L .35507 .04279 -2.5 2.5 Mabsadd L s .39648 .03797 -2.5 2.5 Mabsadd m .39648 .03797 2.5 2.5 Mabsadd L .39648 .03797 2.5 -2.5 Mabsadd L .39648 .03797 -2.5 -2.5 Mabsadd L .39648 .03797 -2.5 2.5 Mabsadd L s .43789 .03525 -2.5 2.5 Mabsadd m .43789 .03525 2.5 2.5 Mabsadd L .43789 .03525 2.5 -2.5 Mabsadd L .43789 .03525 -2.5 -2.5 Mabsadd L .43789 .03525 -2.5 2.5 Mabsadd L s .4793 .02995 -2.5 2.5 Mabsadd m .4793 .02995 2.5 2.5 Mabsadd L .4793 .02995 2.5 -2.5 Mabsadd L .4793 .02995 -2.5 -2.5 Mabsadd L .4793 .02995 -2.5 2.5 Mabsadd L s .5207 .03254 -2.5 2.5 Mabsadd m .5207 .03254 2.5 2.5 Mabsadd L .5207 .03254 2.5 -2.5 Mabsadd L .5207 .03254 -2.5 -2.5 Mabsadd L .5207 .03254 -2.5 2.5 Mabsadd L s .56211 .03769 -2.5 2.5 Mabsadd m .56211 .03769 2.5 2.5 Mabsadd L .56211 .03769 2.5 -2.5 Mabsadd L .56211 .03769 -2.5 -2.5 Mabsadd L .56211 .03769 -2.5 2.5 Mabsadd L s .60352 .03464 -2.5 2.5 Mabsadd m .60352 .03464 2.5 2.5 Mabsadd L .60352 .03464 2.5 -2.5 Mabsadd L .60352 .03464 -2.5 -2.5 Mabsadd L .60352 .03464 -2.5 2.5 Mabsadd L s .64493 .02796 -2.5 2.5 Mabsadd m .64493 .02796 2.5 2.5 Mabsadd L .64493 .02796 2.5 -2.5 Mabsadd L .64493 .02796 -2.5 -2.5 Mabsadd L .64493 .02796 -2.5 2.5 Mabsadd L s .68634 .02447 -2.5 2.5 Mabsadd m .68634 .02447 2.5 2.5 Mabsadd L .68634 .02447 2.5 -2.5 Mabsadd L .68634 .02447 -2.5 -2.5 Mabsadd L .68634 .02447 -2.5 2.5 Mabsadd L s .72774 .03205 -2.5 2.5 Mabsadd m .72774 .03205 2.5 2.5 Mabsadd L .72774 .03205 2.5 -2.5 Mabsadd L .72774 .03205 -2.5 -2.5 Mabsadd L .72774 .03205 -2.5 2.5 Mabsadd L s .76915 .03082 -2.5 2.5 Mabsadd m .76915 .03082 2.5 2.5 Mabsadd L .76915 .03082 2.5 -2.5 Mabsadd L .76915 .03082 -2.5 -2.5 Mabsadd L .76915 .03082 -2.5 2.5 Mabsadd L s .81056 .02627 -2.5 2.5 Mabsadd m .81056 .02627 2.5 2.5 Mabsadd L .81056 .02627 2.5 -2.5 Mabsadd L .81056 .02627 -2.5 -2.5 Mabsadd L .81056 .02627 -2.5 2.5 Mabsadd L s .85197 .02702 -2.5 2.5 Mabsadd m .85197 .02702 2.5 2.5 Mabsadd L .85197 .02702 2.5 -2.5 Mabsadd L .85197 .02702 -2.5 -2.5 Mabsadd L .85197 .02702 -2.5 2.5 Mabsadd L s .89337 .0304 -2.5 2.5 Mabsadd m .89337 .0304 2.5 2.5 Mabsadd L .89337 .0304 2.5 -2.5 Mabsadd L .89337 .0304 -2.5 -2.5 Mabsadd L .89337 .0304 -2.5 2.5 Mabsadd L s .93478 .02944 -2.5 2.5 Mabsadd m .93478 .02944 2.5 2.5 Mabsadd L .93478 .02944 2.5 -2.5 Mabsadd L .93478 .02944 -2.5 -2.5 Mabsadd L .93478 .02944 -2.5 2.5 Mabsadd L s .97619 .03464 -2.5 2.5 Mabsadd m .97619 .03464 2.5 2.5 Mabsadd L .97619 .03464 2.5 -2.5 Mabsadd L .97619 .03464 -2.5 -2.5 Mabsadd L .97619 .03464 -2.5 2.5 Mabsadd L s .001 w [ 5 2 ] 0 Mabsdash .06522 .14958 m .10663 .05856 L .14803 .06186 L .18944 .05997 L .23085 .04939 L .27226 .04711 L .31366 .03728 L .35507 .0383 L .39648 .03303 L .43789 .03403 L .4793 .02884 L .5207 .02695 L .56211 .02483 L .60352 .02452 L .64493 .0284 L .68634 .02114 L .72774 .01856 L .76915 .0196 L .81056 .01937 L .85197 .01947 L .89337 .02008 L .93478 .02072 L .97619 .02198 L s .0001 w [ ] 0 setdash .06522 .14958 0 4 Mabsadd m .06522 .14958 -3.464 -2 Mabsadd L .06522 .14958 3.464 -2 Mabsadd L .06522 .14958 0 4 Mabsadd L s .10663 .05856 0 4 Mabsadd m .10663 .05856 -3.464 -2 Mabsadd L .10663 .05856 3.464 -2 Mabsadd L .10663 .05856 0 4 Mabsadd L s .14803 .06186 0 4 Mabsadd m .14803 .06186 -3.464 -2 Mabsadd L .14803 .06186 3.464 -2 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-0.47541 Mabsadd L .39648 .44329 -2.35114 -3.23607 Mabsadd L .39648 .44329 0 -1.53846 Mabsadd L .39648 .44329 2.35114 -3.23607 Mabsadd L .39648 .44329 1.46316 -0.47541 Mabsadd L .39648 .44329 3.80423 1.23607 Mabsadd L .39648 .44329 .90428 1.24464 Mabsadd L .39648 .44329 0 4 Mabsadd L closepath F .43789 .45334 0 4 Mabsadd m .43789 .45334 -0.90428 1.24464 Mabsadd L .43789 .45334 -3.80423 1.23607 Mabsadd L .43789 .45334 -1.46316 -0.47541 Mabsadd L .43789 .45334 -2.35114 -3.23607 Mabsadd L .43789 .45334 0 -1.53846 Mabsadd L .43789 .45334 2.35114 -3.23607 Mabsadd L .43789 .45334 1.46316 -0.47541 Mabsadd L .43789 .45334 3.80423 1.23607 Mabsadd L .43789 .45334 .90428 1.24464 Mabsadd L .43789 .45334 0 4 Mabsadd L closepath F .4793 .46228 0 4 Mabsadd m .4793 .46228 -0.90428 1.24464 Mabsadd L .4793 .46228 -3.80423 1.23607 Mabsadd L .4793 .46228 -1.46316 -0.47541 Mabsadd L .4793 .46228 -2.35114 -3.23607 Mabsadd L .4793 .46228 0 -1.53846 Mabsadd L .4793 .46228 2.35114 -3.23607 Mabsadd L .4793 .46228 1.46316 -0.47541 Mabsadd L .4793 .46228 3.80423 1.23607 Mabsadd L .4793 .46228 .90428 1.24464 Mabsadd L .4793 .46228 0 4 Mabsadd L closepath F .5207 .423 0 4 Mabsadd m .5207 .423 -0.90428 1.24464 Mabsadd L .5207 .423 -3.80423 1.23607 Mabsadd L .5207 .423 -1.46316 -0.47541 Mabsadd L .5207 .423 -2.35114 -3.23607 Mabsadd L .5207 .423 0 -1.53846 Mabsadd L .5207 .423 2.35114 -3.23607 Mabsadd L .5207 .423 1.46316 -0.47541 Mabsadd L .5207 .423 3.80423 1.23607 Mabsadd L .5207 .423 .90428 1.24464 Mabsadd L .5207 .423 0 4 Mabsadd L closepath F .56211 .38924 0 4 Mabsadd m .56211 .38924 -0.90428 1.24464 Mabsadd L .56211 .38924 -3.80423 1.23607 Mabsadd L .56211 .38924 -1.46316 -0.47541 Mabsadd L .56211 .38924 -2.35114 -3.23607 Mabsadd L .56211 .38924 0 -1.53846 Mabsadd L .56211 .38924 2.35114 -3.23607 Mabsadd L .56211 .38924 1.46316 -0.47541 Mabsadd L .56211 .38924 3.80423 1.23607 Mabsadd L .56211 .38924 .90428 1.24464 Mabsadd L .56211 .38924 0 4 Mabsadd L closepath F .60352 .39973 0 4 Mabsadd m .60352 .39973 -0.90428 1.24464 Mabsadd L .60352 .39973 -3.80423 1.23607 Mabsadd L .60352 .39973 -1.46316 -0.47541 Mabsadd L .60352 .39973 -2.35114 -3.23607 Mabsadd L .60352 .39973 0 -1.53846 Mabsadd L .60352 .39973 2.35114 -3.23607 Mabsadd L .60352 .39973 1.46316 -0.47541 Mabsadd L .60352 .39973 3.80423 1.23607 Mabsadd L .60352 .39973 .90428 1.24464 Mabsadd L .60352 .39973 0 4 Mabsadd L closepath F .64493 .36896 0 4 Mabsadd m .64493 .36896 -0.90428 1.24464 Mabsadd L .64493 .36896 -3.80423 1.23607 Mabsadd L .64493 .36896 -1.46316 -0.47541 Mabsadd L .64493 .36896 -2.35114 -3.23607 Mabsadd L .64493 .36896 0 -1.53846 Mabsadd L .64493 .36896 2.35114 -3.23607 Mabsadd L .64493 .36896 1.46316 -0.47541 Mabsadd L .64493 .36896 3.80423 1.23607 Mabsadd L .64493 .36896 .90428 1.24464 Mabsadd L .64493 .36896 0 4 Mabsadd L closepath F .68634 .18679 0 4 Mabsadd m .68634 .18679 -0.90428 1.24464 Mabsadd L .68634 .18679 -3.80423 1.23607 Mabsadd L .68634 .18679 -1.46316 -0.47541 Mabsadd L .68634 .18679 -2.35114 -3.23607 Mabsadd L .68634 .18679 0 -1.53846 Mabsadd L .68634 .18679 2.35114 -3.23607 Mabsadd L .68634 .18679 1.46316 -0.47541 Mabsadd L .68634 .18679 3.80423 1.23607 Mabsadd L .68634 .18679 .90428 1.24464 Mabsadd L .68634 .18679 0 4 Mabsadd L closepath F .72774 .17434 0 4 Mabsadd m .72774 .17434 -0.90428 1.24464 Mabsadd L .72774 .17434 -3.80423 1.23607 Mabsadd L .72774 .17434 -1.46316 -0.47541 Mabsadd L .72774 .17434 -2.35114 -3.23607 Mabsadd L .72774 .17434 0 -1.53846 Mabsadd L .72774 .17434 2.35114 -3.23607 Mabsadd L .72774 .17434 1.46316 -0.47541 Mabsadd L .72774 .17434 3.80423 1.23607 Mabsadd L .72774 .17434 .90428 1.24464 Mabsadd L .72774 .17434 0 4 Mabsadd L closepath F .76915 .17544 0 4 Mabsadd m .76915 .17544 -0.90428 1.24464 Mabsadd L .76915 .17544 -3.80423 1.23607 Mabsadd L .76915 .17544 -1.46316 -0.47541 Mabsadd L .76915 .17544 -2.35114 -3.23607 Mabsadd L .76915 .17544 0 -1.53846 Mabsadd L .76915 .17544 2.35114 -3.23607 Mabsadd L .76915 .17544 1.46316 -0.47541 Mabsadd L .76915 .17544 3.80423 1.23607 Mabsadd L .76915 .17544 .90428 1.24464 Mabsadd L .76915 .17544 0 4 Mabsadd L closepath F .81056 .17544 0 4 Mabsadd m .81056 .17544 -0.90428 1.24464 Mabsadd L .81056 .17544 -3.80423 1.23607 Mabsadd L .81056 .17544 -1.46316 -0.47541 Mabsadd L .81056 .17544 -2.35114 -3.23607 Mabsadd L .81056 .17544 0 -1.53846 Mabsadd L .81056 .17544 2.35114 -3.23607 Mabsadd L .81056 .17544 1.46316 -0.47541 Mabsadd L .81056 .17544 3.80423 1.23607 Mabsadd L .81056 .17544 .90428 1.24464 Mabsadd L .81056 .17544 0 4 Mabsadd L closepath F .85197 .23099 0 4 Mabsadd m .85197 .23099 -0.90428 1.24464 Mabsadd L .85197 .23099 -3.80423 1.23607 Mabsadd L .85197 .23099 -1.46316 -0.47541 Mabsadd L .85197 .23099 -2.35114 -3.23607 Mabsadd L .85197 .23099 0 -1.53846 Mabsadd L .85197 .23099 2.35114 -3.23607 Mabsadd L .85197 .23099 1.46316 -0.47541 Mabsadd L .85197 .23099 3.80423 1.23607 Mabsadd L .85197 .23099 .90428 1.24464 Mabsadd L .85197 .23099 0 4 Mabsadd L closepath F .89337 .25567 0 4 Mabsadd m .89337 .25567 -0.90428 1.24464 Mabsadd L .89337 .25567 -3.80423 1.23607 Mabsadd L .89337 .25567 -1.46316 -0.47541 Mabsadd L .89337 .25567 -2.35114 -3.23607 Mabsadd L .89337 .25567 0 -1.53846 Mabsadd L .89337 .25567 2.35114 -3.23607 Mabsadd L .89337 .25567 1.46316 -0.47541 Mabsadd L .89337 .25567 3.80423 1.23607 Mabsadd L .89337 .25567 .90428 1.24464 Mabsadd L .89337 .25567 0 4 Mabsadd L closepath F .93478 .22556 0 4 Mabsadd m .93478 .22556 -0.90428 1.24464 Mabsadd L .93478 .22556 -3.80423 1.23607 Mabsadd L .93478 .22556 -1.46316 -0.47541 Mabsadd L .93478 .22556 -2.35114 -3.23607 Mabsadd L .93478 .22556 0 -1.53846 Mabsadd L .93478 .22556 2.35114 -3.23607 Mabsadd L .93478 .22556 1.46316 -0.47541 Mabsadd L .93478 .22556 3.80423 1.23607 Mabsadd L .93478 .22556 .90428 1.24464 Mabsadd L .93478 .22556 0 4 Mabsadd L closepath F .97619 .26315 0 4 Mabsadd m .97619 .26315 -0.90428 1.24464 Mabsadd L .97619 .26315 -3.80423 1.23607 Mabsadd L .97619 .26315 -1.46316 -0.47541 Mabsadd L .97619 .26315 -2.35114 -3.23607 Mabsadd L .97619 .26315 0 -1.53846 Mabsadd L .97619 .26315 2.35114 -3.23607 Mabsadd L .97619 .26315 1.46316 -0.47541 Mabsadd L .97619 .26315 3.80423 1.23607 Mabsadd L .97619 .26315 .90428 1.24464 Mabsadd L .97619 .26315 0 4 Mabsadd L closepath F .001 w [ .5 3 ] 0 Mabsdash .06522 .58626 m .10663 .59447 L .14803 .59657 L .18944 .60332 L .23085 .58633 L .27226 .55315 L .31366 .52827 L .35507 .53433 L .39648 .53713 L .43789 .54653 L .4793 .548 L .5207 .5056 L .56211 .45952 L .60352 .47497 L .64493 .43801 L .68634 .28605 L .72774 .26151 L .76915 .25193 L .81056 .25193 L .85197 .29809 L .89337 .32034 L .93478 .27877 L .97619 .31732 L s .0001 w [ ] 0 setdash .06522 .58626 -2.25 0 Mabsadd m .06522 .58626 0 3 Mabsadd L .06522 .58626 2.25 0 Mabsadd L .06522 .58626 0 -3 Mabsadd L .06522 .58626 -2.25 0 Mabsadd L closepath F .10663 .59447 -2.25 0 Mabsadd m .10663 .59447 0 3 Mabsadd L .10663 .59447 2.25 0 Mabsadd L .10663 .59447 0 -3 Mabsadd L .10663 .59447 -2.25 0 Mabsadd L closepath F .14803 .59657 -2.25 0 Mabsadd m .14803 .59657 0 3 Mabsadd L .14803 .59657 2.25 0 Mabsadd L .14803 .59657 0 -3 Mabsadd L .14803 .59657 -2.25 0 Mabsadd L closepath F .18944 .60332 -2.25 0 Mabsadd m .18944 .60332 0 3 Mabsadd L .18944 .60332 2.25 0 Mabsadd L .18944 .60332 0 -3 Mabsadd L .18944 .60332 -2.25 0 Mabsadd L closepath F .23085 .58633 -2.25 0 Mabsadd m .23085 .58633 0 3 Mabsadd L .23085 .58633 2.25 0 Mabsadd L .23085 .58633 0 -3 Mabsadd L .23085 .58633 -2.25 0 Mabsadd L closepath F .27226 .55315 -2.25 0 Mabsadd m .27226 .55315 0 3 Mabsadd L .27226 .55315 2.25 0 Mabsadd L .27226 .55315 0 -3 Mabsadd L .27226 .55315 -2.25 0 Mabsadd L closepath F .31366 .52827 -2.25 0 Mabsadd m .31366 .52827 0 3 Mabsadd L .31366 .52827 2.25 0 Mabsadd L .31366 .52827 0 -3 Mabsadd L .31366 .52827 -2.25 0 Mabsadd L closepath F .35507 .53433 -2.25 0 Mabsadd m .35507 .53433 0 3 Mabsadd L .35507 .53433 2.25 0 Mabsadd L .35507 .53433 0 -3 Mabsadd L .35507 .53433 -2.25 0 Mabsadd L closepath F .39648 .53713 -2.25 0 Mabsadd m .39648 .53713 0 3 Mabsadd L .39648 .53713 2.25 0 Mabsadd L .39648 .53713 0 -3 Mabsadd L .39648 .53713 -2.25 0 Mabsadd L closepath F .43789 .54653 -2.25 0 Mabsadd m .43789 .54653 0 3 Mabsadd L .43789 .54653 2.25 0 Mabsadd L .43789 .54653 0 -3 Mabsadd L .43789 .54653 -2.25 0 Mabsadd L closepath F .4793 .548 -2.25 0 Mabsadd m .4793 .548 0 3 Mabsadd L .4793 .548 2.25 0 Mabsadd L .4793 .548 0 -3 Mabsadd L .4793 .548 -2.25 0 Mabsadd L closepath F .5207 .5056 -2.25 0 Mabsadd m .5207 .5056 0 3 Mabsadd L .5207 .5056 2.25 0 Mabsadd L .5207 .5056 0 -3 Mabsadd L .5207 .5056 -2.25 0 Mabsadd L closepath F .56211 .45952 -2.25 0 Mabsadd m .56211 .45952 0 3 Mabsadd L .56211 .45952 2.25 0 Mabsadd L .56211 .45952 0 -3 Mabsadd L .56211 .45952 -2.25 0 Mabsadd L closepath F .60352 .47497 -2.25 0 Mabsadd m .60352 .47497 0 3 Mabsadd L .60352 .47497 2.25 0 Mabsadd L .60352 .47497 0 -3 Mabsadd L .60352 .47497 -2.25 0 Mabsadd L closepath F .64493 .43801 -2.25 0 Mabsadd m .64493 .43801 0 3 Mabsadd L .64493 .43801 2.25 0 Mabsadd L .64493 .43801 0 -3 Mabsadd L .64493 .43801 -2.25 0 Mabsadd L closepath F .68634 .28605 -2.25 0 Mabsadd m .68634 .28605 0 3 Mabsadd L .68634 .28605 2.25 0 Mabsadd L .68634 .28605 0 -3 Mabsadd L .68634 .28605 -2.25 0 Mabsadd L closepath F .72774 .26151 -2.25 0 Mabsadd m .72774 .26151 0 3 Mabsadd L .72774 .26151 2.25 0 Mabsadd L .72774 .26151 0 -3 Mabsadd L .72774 .26151 -2.25 0 Mabsadd L closepath F .76915 .25193 -2.25 0 Mabsadd m .76915 .25193 0 3 Mabsadd L .76915 .25193 2.25 0 Mabsadd L .76915 .25193 0 -3 Mabsadd L .76915 .25193 -2.25 0 Mabsadd L closepath F .81056 .25193 -2.25 0 Mabsadd m .81056 .25193 0 3 Mabsadd L .81056 .25193 2.25 0 Mabsadd L .81056 .25193 0 -3 Mabsadd L .81056 .25193 -2.25 0 Mabsadd L closepath F .85197 .29809 -2.25 0 Mabsadd m .85197 .29809 0 3 Mabsadd L .85197 .29809 2.25 0 Mabsadd L .85197 .29809 0 -3 Mabsadd L .85197 .29809 -2.25 0 Mabsadd L closepath F .89337 .32034 -2.25 0 Mabsadd m .89337 .32034 0 3 Mabsadd L .89337 .32034 2.25 0 Mabsadd L .89337 .32034 0 -3 Mabsadd L .89337 .32034 -2.25 0 Mabsadd L closepath F .93478 .27877 -2.25 0 Mabsadd m .93478 .27877 0 3 Mabsadd L .93478 .27877 2.25 0 Mabsadd L .93478 .27877 0 -3 Mabsadd L .93478 .27877 -2.25 0 Mabsadd L closepath F .97619 .31732 -2.25 0 Mabsadd m .97619 .31732 0 3 Mabsadd L .97619 .31732 2.25 0 Mabsadd L .97619 .31732 0 -3 Mabsadd L .97619 .31732 -2.25 0 Mabsadd L closepath F .001 w [ .5 4 5 4 ] 0 Mabsdash .06522 .11291 m .10663 .05325 L .14803 .0606 L .18944 .06178 L .23085 .05018 L .27226 .06551 L .31366 .0705 L .35507 .07929 L .39648 .08013 L .43789 .08068 L .4793 .07986 L .5207 .07756 L .56211 .07389 L .60352 .08016 L .64493 .08302 L .68634 .06436 L .72774 .07213 L .76915 .07296 L .81056 .07296 L .85197 .068 L .89337 .0819 L .93478 .08001 L .97619 .07791 L s .0001 w [ ] 0 setdash .06522 .11291 -2.5 2.5 Mabsadd m .06522 .11291 2.5 2.5 Mabsadd L .06522 .11291 2.5 -2.5 Mabsadd L .06522 .11291 -2.5 -2.5 Mabsadd L .06522 .11291 -2.5 2.5 Mabsadd L s .10663 .05325 -2.5 2.5 Mabsadd m .10663 .05325 2.5 2.5 Mabsadd L .10663 .05325 2.5 -2.5 Mabsadd L .10663 .05325 -2.5 -2.5 Mabsadd L .10663 .05325 -2.5 2.5 Mabsadd L s .14803 .0606 -2.5 2.5 Mabsadd m .14803 .0606 2.5 2.5 Mabsadd L .14803 .0606 2.5 -2.5 Mabsadd L .14803 .0606 -2.5 -2.5 Mabsadd L .14803 .0606 -2.5 2.5 Mabsadd L s .18944 .06178 -2.5 2.5 Mabsadd m .18944 .06178 2.5 2.5 Mabsadd L .18944 .06178 2.5 -2.5 Mabsadd L .18944 .06178 -2.5 -2.5 Mabsadd L .18944 .06178 -2.5 2.5 Mabsadd L s .23085 .05018 -2.5 2.5 Mabsadd m .23085 .05018 2.5 2.5 Mabsadd L .23085 .05018 2.5 -2.5 Mabsadd L .23085 .05018 -2.5 -2.5 Mabsadd L .23085 .05018 -2.5 2.5 Mabsadd L s .27226 .06551 -2.5 2.5 Mabsadd m .27226 .06551 2.5 2.5 Mabsadd L .27226 .06551 2.5 -2.5 Mabsadd L .27226 .06551 -2.5 -2.5 Mabsadd L .27226 .06551 -2.5 2.5 Mabsadd L s .31366 .0705 -2.5 2.5 Mabsadd m .31366 .0705 2.5 2.5 Mabsadd L .31366 .0705 2.5 -2.5 Mabsadd L .31366 .0705 -2.5 -2.5 Mabsadd L .31366 .0705 -2.5 2.5 Mabsadd L s .35507 .07929 -2.5 2.5 Mabsadd m .35507 .07929 2.5 2.5 Mabsadd L .35507 .07929 2.5 -2.5 Mabsadd L .35507 .07929 -2.5 -2.5 Mabsadd L .35507 .07929 -2.5 2.5 Mabsadd L s .39648 .08013 -2.5 2.5 Mabsadd m .39648 .08013 2.5 2.5 Mabsadd L .39648 .08013 2.5 -2.5 Mabsadd L .39648 .08013 -2.5 -2.5 Mabsadd L .39648 .08013 -2.5 2.5 Mabsadd L s .43789 .08068 -2.5 2.5 Mabsadd m .43789 .08068 2.5 2.5 Mabsadd L .43789 .08068 2.5 -2.5 Mabsadd L .43789 .08068 -2.5 -2.5 Mabsadd L .43789 .08068 -2.5 2.5 Mabsadd L s .4793 .07986 -2.5 2.5 Mabsadd m .4793 .07986 2.5 2.5 Mabsadd L .4793 .07986 2.5 -2.5 Mabsadd L .4793 .07986 -2.5 -2.5 Mabsadd L .4793 .07986 -2.5 2.5 Mabsadd L s .5207 .07756 -2.5 2.5 Mabsadd m .5207 .07756 2.5 2.5 Mabsadd L .5207 .07756 2.5 -2.5 Mabsadd L .5207 .07756 -2.5 -2.5 Mabsadd L .5207 .07756 -2.5 2.5 Mabsadd L s .56211 .07389 -2.5 2.5 Mabsadd m .56211 .07389 2.5 2.5 Mabsadd L .56211 .07389 2.5 -2.5 Mabsadd L .56211 .07389 -2.5 -2.5 Mabsadd L .56211 .07389 -2.5 2.5 Mabsadd L s .60352 .08016 -2.5 2.5 Mabsadd m .60352 .08016 2.5 2.5 Mabsadd L .60352 .08016 2.5 -2.5 Mabsadd L .60352 .08016 -2.5 -2.5 Mabsadd L .60352 .08016 -2.5 2.5 Mabsadd L s .64493 .08302 -2.5 2.5 Mabsadd m .64493 .08302 2.5 2.5 Mabsadd L .64493 .08302 2.5 -2.5 Mabsadd L .64493 .08302 -2.5 -2.5 Mabsadd L .64493 .08302 -2.5 2.5 Mabsadd L s .68634 .06436 -2.5 2.5 Mabsadd m .68634 .06436 2.5 2.5 Mabsadd L .68634 .06436 2.5 -2.5 Mabsadd L .68634 .06436 -2.5 -2.5 Mabsadd L .68634 .06436 -2.5 2.5 Mabsadd L s .72774 .07213 -2.5 2.5 Mabsadd m .72774 .07213 2.5 2.5 Mabsadd L .72774 .07213 2.5 -2.5 Mabsadd L .72774 .07213 -2.5 -2.5 Mabsadd L .72774 .07213 -2.5 2.5 Mabsadd L s .76915 .07296 -2.5 2.5 Mabsadd m .76915 .07296 2.5 2.5 Mabsadd L .76915 .07296 2.5 -2.5 Mabsadd L .76915 .07296 -2.5 -2.5 Mabsadd L .76915 .07296 -2.5 2.5 Mabsadd L s .81056 .07296 -2.5 2.5 Mabsadd m .81056 .07296 2.5 2.5 Mabsadd L .81056 .07296 2.5 -2.5 Mabsadd L .81056 .07296 -2.5 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{{906.375, 961.625}, {424.25, \ 369.063}} -> {-33.8486, -5.09971, 0.0724809, 0.0724809}, {{906.375, 961.625}, \ {360.25, 305.063}} -> {-33.8486, -7.4191, 0.0724809, 0.0724809}, {{906.375, \ 961.625}, {296.25, 241.063}} -> {-33.8486, -9.73849, 0.0724809, 0.0724809}, \ {{906.375, 961.625}, {232.188, 177}} -> {-33.8486, -12.0601, 0.0724809, \ 0.0724809}, {{906.375, 961.625}, {168.188, 113}} -> {-33.8486, -14.3795, \ 0.0724809, 0.0724809}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["5. CONCLUSION", "Section"], Cell["\<\ It is often said that economists are the only scientists who do not collect \ their own data. The wealth of government provided statistics allows them to \ proceed directly to testing their models. The only drawback is that \ sometimes the data do not reflect precisely what the economist seeks to \ investigate. As we argue above, the picture created by data provided by the \ Census on individual incomes is not a pure reflection of market outcomes. \ The inclusion of transfer payments in income measures is helpful for \ researchers charting poverty and inequality but it adds confusion to the \ question of which statistical distribution reflects most accurately the \ results of economic forces. Similarly, the absence of capital gains from the \ measures creates an incomplete picture of an underlying economic reality in \ which a growing proportion of the population earns an income from invested \ savings. The Beaman distribution is a parsimonious descriptive model of the size \ distribution of income that stands up well relative to existing models and \ has the added advantage that it can account for negative incomes. \ Furthermore, since the official measures of income with which we are testing \ the models include various transfer payments, the fit of the Beaman \ distribution might be even better than it appears. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["REFERENCES", "Section"], Cell[TextData[{ "A", StyleBox["MOROSA, ", FontSize->10], "L.", StyleBox[" ", FontSize->10], "(1925): \[OpenCurlyDoubleQuote]Ricerche Intorno alla Curva dei Redditi,\ \[CloseCurlyDoubleQuote] ", StyleBox["Annali di Mathematica Pura ed Applicata, ", FontSlant->"Italic"], "Ser. 4-21, pp. 123-159.\n\n", "B", StyleBox["ANDOURIAN", FontSize->9], ", ", "R", "., ", "J.B", StyleBox[". ", FontSize->9], "M", StyleBox["C", FontSize->10], "D", StyleBox["ONALD", FontSize->10], " ", StyleBox["and", FontSize->10], " ", "R. S. T", StyleBox["URLEY", FontSize->9], " (2002): \[OpenCurlyDoubleQuote]A Comparison of Parametric Models of \ Income Distribution Across Countries and Over Time\[CloseCurlyDoubleQuote] \ (June 2002), Luxembourg Income Study Working Paper No. 305.\n\nF", StyleBox["ISK, ", FontSize->10], "P. R. (1961): \[OpenCurlyDoubleQuote]The Graduation of Income \ Distributions,\[CloseCurlyDoubleQuote] ", StyleBox["Econometrica", FontSlant->"Italic"], ", Vol. 29, No. 2 (Apr., 1961), pp. 171-185.\n\nF", StyleBox["RIEDMAN", FontSize->10], StyleBox[", ", FontSize->10], "M. (1953): \[OpenCurlyDoubleQuote]Choice, Chance and the Personal \ Distribution of Income,\[CloseCurlyDoubleQuote] ", StyleBox["The Journal of Political Economy", FontSlant->"Italic"], ", Vol. 61, No. 4 (Aug., 1953), pp. 277-290.\n\nG", StyleBox["IBRAT", FontSize->10], ", R. (1931): ", StyleBox["Les In\[EAcute]galit\[EAcute]s \[EAcute]conomiques,", FontSlant->"Italic"], " Paris: Recueil Sirey.\n\nM", StyleBox["C", FontSize->10], "D", StyleBox["ONALD", FontSize->10], ", J. B. (1984): \[OpenCurlyDoubleQuote]Some Generalized Functions for the \ Size Distribution of Income,\[CloseCurlyDoubleQuote] ", StyleBox["Econometrica", FontSlant->"Italic"], ", Vol. 52, No. 3 (May, 1984), pp. 647-664.\n\n_____________ ", StyleBox["AND ", FontSize->10], "Y. J. X", StyleBox["U ", FontSize->10], "(1995): \[OpenCurlyDoubleQuote]A Generalization of the Beta Distribution \ with Applications,\[CloseCurlyDoubleQuote] ", StyleBox["Journal of Econometrics", FontSlant->"Italic"], ", Vol. 66, No. , pp. 133-152.\n", StyleBox["\n", FontSlant->"Italic"], "R", StyleBox["EITH", FontSize->10], ", J. E. (1986): \"Some Rules of Market Behavior,\" unpublished manuscript.\ \n\n------------- (2003): \[OpenCurlyDoubleQuote]Adam Smith\ \[CloseCurlyQuote]s Visible Hands,\[CloseCurlyDoubleQuote] post to NKS Forum, \ ", StyleBox["forum.wolframscience.com", FontSlant->"Italic"], ", November 10, 2003.\n\nS", StyleBox["ALEM, ", FontSize->10], "A. B. Z. ", StyleBox["AND ", FontSize->10], "T. D. M", StyleBox["OUNT ", FontSize->10], "(1974): \[OpenCurlyDoubleQuote]A Convenient Descriptive Model of Income \ Distribution,\[CloseCurlyDoubleQuote] ", StyleBox["Econometrica", FontSlant->"Italic"], ", Vol. 42, No. 6 (Nov., 1974), pp. 1115-1127.\n\nUS", StyleBox[" ", FontSize->10], "C", StyleBox["ENSUS ", FontSize->10], "B", StyleBox["UREAU ", FontSize->10], "(2003): \[OpenCurlyDoubleQuote]Income in the United States: 2002,\ \[CloseCurlyDoubleQuote] ", StyleBox["Current Population Reports", FontSlant->"Italic"], ", P60-221.\n\n------------------------ (1995-2004): ", StyleBox["Statistical Abstracts of the United States", FontSlant->"Italic"], ",", StyleBox[" ", FontSlant->"Italic"], "Washington D.C.: Government Printing Office.\n\nW", StyleBox["EIBULL, ", FontSize->10], "W. (1951): \[OpenCurlyDoubleQuote]A Statistical Distribution Function of \ Wide Applicability,\[CloseCurlyDoubleQuote] ", StyleBox["Journal of Applied Mechanics", FontSlant->"Italic"], ", Sept. 1951, pp. 293-297.\n", StyleBox["\n", FontSize->10] }], "Text"] }, Open ]], Cell[TextData[StyleBox["March 5, 2005", FontSize->10]], "Section"] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 685}}, WindowSize->{1016, 651}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, Magnification->1.25 ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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