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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,ba, b are required.

    a<ba<b

    These parameters are minimum and maximum value of variable respectively.

  • Continuous distribution defined on bounded range a≤x≤ba\leq x \leq b

  • This distribution is always symmetric.

Probability​

  • Cumulative distribution function

    F(x)=a3[(x−b)3+(b−a)3]F(x)=\frac{a}{3}\left[(x-b)^3+(b-a)^3\right]

    where

    α=12(b−a)3,β=a+b2\alpha=\frac{12}{(b-a)^3},\beta=\frac{a+b}{2}

  • Probability density function

    f(x)=a(x−b)2f(x)=a(x-b)^2

  • How to compute these on Excel.

AB
1DataDescription
22Value for which you want the distribution
31Value of parameter A
45Value of parameter B
5=12/((A4-A3)^3)Vertical scale
6(A3+A4)/2Mean of the distribution
7FormulaDescription (Result)
8=A5*((A2-A6)^3+(A6-A5)^3)/3Cumulative distribution function for the terms above
9=A5*(A2-A6)^2Probability density function for the terms above

Quantile​

  • Inverse of cumulative distribution function

    F−1(P)=[3Pα−(β−α)3]1/3+βF^{-1}(P)=\left[\frac{3P}{\alpha}-(\beta-\alpha)^3\right]^{1/3}+\beta

    where

    α=12(b−a)3,β=a+b2\alpha=\frac{12}{(b-a)^3},\beta=\frac{a+b}{2}

  • How to compute this on Excel.

AB
1DataDescription
20.5Probability associated with the distribution
31Value of parameter A
45Value of parameter B
5=12/((A4-A3)^3)Vertical scale
6=(A3+A4)/2Mean of the distribution
7FormulaDescription (Result)
8=(3*A2/A5-(A6-A5)^3)^(1/3)+A6Inverse 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

    a+b2\frac{a+b}{2}

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter A
32Value of parameter B
4FormulaDescription (Result)
5(A2+A3)/2Mean of the distribution for the terms above, (5)

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

  • Variance of the distribution is given as

    320(b−a)2\frac{3}{20}(b-a)^2

    Standard Deviation is a positive square root of Variance.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter A
32Value of parameter B
4FormulaDescription (Result)
5=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 00.

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

  • Kurtosis of the distribution is given as

    3112(a−b)4\frac{3}{112}(a-b)^4

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter A
32Value of parameter B
4FormulaDescription (Result)
5=3*(A3-A2)^4/112Kurtosis 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=[3Uα−(β−α)3]1/3+βx=\left[\frac{3U}{\alpha}-(\beta-\alpha)^3\right]^{1/3}+\beta

    where

    α=12(b−a)3,β=a+b2\alpha=\frac{12}{(b-a)^3},\beta=\frac{a+b}{2}

  • How to generate random numbers on Excel.

AB
1DataDescription
21Value of parameter A
35Value of parameter B
4=12/((A3-A2)^3)Vertical scale
5=(A2+A3)/2Mean of the distribution
6FormulaDescription (Result)
7=(3*NTRAND(100)/A2-(A3-A2)^3)^(1/3)+A3100 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​