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U-quadratic distribution

Where do you meet this distribution?

  • Economic

    “Job Insecurity isn’t Always Efficient” by David J. Balan and Dan Hanner

Shape of Distribution

Basic Properties

  • Two parameters a, b are required.
    a<b

    These parameters are minimum and maximum value of variable respectively.

  • Continuous distribution defined on bounded range a\leq x \leq b
  • This distribution is always symmetric.

Probability

  • Cumulative distribution function
    F(x)=\frac{a}{3}\left[(x-b)^3+(b-a)^3\right]

    where

    \alpha=\frac{12}{(b-a)^3},\;\beta=\frac{a+b}{2}
  • Probability density function
    f(x)=a(x-b)^2
  • How to compute these on Excel.
     
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    A B
    Data Description
    2 Value for which you want the distribution
    1 Value of parameter A
    5 Value of parameter B
    =12/((A4-A3)^3) Vertical scale
    =(A3+A4)/2 Mean of the distribution
    Formula Description (Result)
    =A5*((A2-A6)^3+(A6-A5)^3)/3 Cumulative distribution function for the terms above
    =A5*(A2-A6)^2 Probability density function for the terms above

Quantile

  • Inverse of cumulative distribution function
    F^{-1}(P)=\left[\frac{3P}{\alpha}-(\beta-\alpha)^3\right]^{1/3}+\beta

    where

    \alpha=\frac{12}{(b-a)^3},\;\beta=\frac{a+b}{2}
  • How to compute this on Excel.
     
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    A B
    Data Description
    0.5 Probability associated with the distribution
    1 Value of parameter A
    5 Value of parameter B
    =12/((A4-A3)^3) Vertical scale
    =(A3+A4)/2 Mean of the distribution
    Formula Description (Result)
    =(3*A2/A5-(A6-A5)^3)^(1/3)+A6 Inverse of the cumulative distribution function for the terms above

Characteristics

Mean – Where is the “center” of the distribution? (Definition)

  • Mean of the distribution is given as
    \frac{a+b}{2}
  • How to compute this on Excel
     
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    A B
    Data Description
    8 Value of parameter A
    2 Value of parameter B
    Formula Description (Result)
    =(A2+A3)/2 Mean of the distribution for the terms above

Standard Deviation – How wide does the distribution spread? (Definition)

  • Variance of the distribution is given as
    \frac{3}{20}(b-a)^2

    Standard Deviation is a positive square root of Variance.

  • How to compute this on Excel
     
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    A B
    Data Description
    8 Value of parameter A
    2 Value of parameter B
    Formula Description (Result)
    =SQRT(3)*(A3-A2)/(2*SQRT(5)) Standard deviation of the distribution for the terms above

Skewness – Which side is the distribution distorted into? (Definition)

  • Skewness of the distribution is 0.

Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

  • Kurtosis of the distribution is given as
    \frac{3}{112}(a-b)^4
  • How to compute this on Excel
     
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    A B
    Data Description
    8 Value of parameter A
    2 Value of parameter B
    Formula Description (Result)
    =3*(A3-A2)^4/112 Kurtosis of the distribution for the terms above

Random Numbers

  • Random number x is generated by inverse function method, which is for uniform random U,
    x=\left[\frac{3U}{\alpha}-(\beta-\alpha)^3\right]^{1/3}+\beta

    where

    \alpha=\frac{12}{(b-a)^3},\;\beta=\frac{a+b}{2}
  • How to generate random numbers on Excel.
     
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    A B
    Data Description
    1 Value of parameter A
    5 Value of parameter B
    =12/((A3-A2)^3) Vertical scale
    =(A2+A3)/2 Mean of the distribution
    Formula Description (Result)
    =(3*NTRAND(100)/A2-(A3-A2)^3)^(1/3)+A3 100 U-quadratic deviates based on Mersenne-Twister algorithm for which the parameters above

    Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A7:A106 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

Not supported yet

Reference

 

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