Constructing Lorenz-like curves

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In economics, the Lorenz curve measures economic inequality in countries. The probability density function given by wikipedia is:



enter image description here



The further the Lorenz curve is from the line of equality the more inequality there exists. For a project I'm looking to construct a function family or parametric function family



enter image description here



that resembles the Lorenz distribution, in the unit square, $ Bbb R^2 (0,1) times(0,1), $ with a variable and a smoothly varying parameter that is symmetric about the line $y=x$ and $ y=1-x $, such that the function smoothly transitions from itself to its inverse as the parameter is varied and as it crosses the line $y=x$. The family $ f_n(x)= x^n $ almost works but it is not symmetric about the line $y=1-x$. Also I'm not sure if the Lorenz curve admits an inverse but if it did that would help me a lot and would probably go a long way in answering my question.



Thanks.




calculus




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    up vote
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    down vote

    favorite












    In economics, the Lorenz curve measures economic inequality in countries. The probability density function given by wikipedia is:



    enter image description here



    The further the Lorenz curve is from the line of equality the more inequality there exists. For a project I'm looking to construct a function family or parametric function family



    enter image description here



    that resembles the Lorenz distribution, in the unit square, $ Bbb R^2 (0,1) times(0,1), $ with a variable and a smoothly varying parameter that is symmetric about the line $y=x$ and $ y=1-x $, such that the function smoothly transitions from itself to its inverse as the parameter is varied and as it crosses the line $y=x$. The family $ f_n(x)= x^n $ almost works but it is not symmetric about the line $y=1-x$. Also I'm not sure if the Lorenz curve admits an inverse but if it did that would help me a lot and would probably go a long way in answering my question.



    Thanks.




    calculus




    share|cite|improve this question






















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      In economics, the Lorenz curve measures economic inequality in countries. The probability density function given by wikipedia is:



      enter image description here



      The further the Lorenz curve is from the line of equality the more inequality there exists. For a project I'm looking to construct a function family or parametric function family



      enter image description here



      that resembles the Lorenz distribution, in the unit square, $ Bbb R^2 (0,1) times(0,1), $ with a variable and a smoothly varying parameter that is symmetric about the line $y=x$ and $ y=1-x $, such that the function smoothly transitions from itself to its inverse as the parameter is varied and as it crosses the line $y=x$. The family $ f_n(x)= x^n $ almost works but it is not symmetric about the line $y=1-x$. Also I'm not sure if the Lorenz curve admits an inverse but if it did that would help me a lot and would probably go a long way in answering my question.



      Thanks.




      calculus




      share|cite|improve this question












      In economics, the Lorenz curve measures economic inequality in countries. The probability density function given by wikipedia is:



      enter image description here



      The further the Lorenz curve is from the line of equality the more inequality there exists. For a project I'm looking to construct a function family or parametric function family



      enter image description here



      that resembles the Lorenz distribution, in the unit square, $ Bbb R^2 (0,1) times(0,1), $ with a variable and a smoothly varying parameter that is symmetric about the line $y=x$ and $ y=1-x $, such that the function smoothly transitions from itself to its inverse as the parameter is varied and as it crosses the line $y=x$. The family $ f_n(x)= x^n $ almost works but it is not symmetric about the line $y=1-x$. Also I'm not sure if the Lorenz curve admits an inverse but if it did that would help me a lot and would probably go a long way in answering my question.



      Thanks.





      calculus





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      asked Aug 9 at 18:46









      George Thomas

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          This system of symmetric superellipse equations works well:



          $ x^s+y^s=1 $



          $ (1-x)^s+(1-y)^s=1 $



          $s in Bbb R (1,infty). $






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            1 Answer
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            active

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            up vote
            0
            down vote



            accepted










            This system of symmetric superellipse equations works well:



            $ x^s+y^s=1 $



            $ (1-x)^s+(1-y)^s=1 $



            $s in Bbb R (1,infty). $






            share|cite|improve this answer


























              up vote
              0
              down vote



              accepted










              This system of symmetric superellipse equations works well:



              $ x^s+y^s=1 $



              $ (1-x)^s+(1-y)^s=1 $



              $s in Bbb R (1,infty). $






              share|cite|improve this answer
























                up vote
                0
                down vote



                accepted







                up vote
                0
                down vote



                accepted






                This system of symmetric superellipse equations works well:



                $ x^s+y^s=1 $



                $ (1-x)^s+(1-y)^s=1 $



                $s in Bbb R (1,infty). $






                share|cite|improve this answer














                This system of symmetric superellipse equations works well:



                $ x^s+y^s=1 $



                $ (1-x)^s+(1-y)^s=1 $



                $s in Bbb R (1,infty). $







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Aug 16 at 12:03

























                answered Aug 9 at 19:37









                George Thomas

                76417




                76417






















                     

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