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Gumbel (Type I) distribution

Where do you meet this distribution?

Shape of Distribution

Basic Properties

  • Two parameters \alpha, \beta are required (How can you get these).
    \beta>0
  • Continuous distribution defined on entire range
  • This distribution is always asymmetric.

Probability

  • Cumulative distribution function
    F(x)=\exp\left[-\exp\left(-\frac{x-\alpha}{\beta}\right)\right]
  • Probability density function
    f(x)=\frac{1}{\beta}\exp\left(-\frac{x-\alpha}{\beta}\right)\exp\left[-\exp\left(-\frac{x-\alpha}{\beta}\right)\right]
  • How to compute these on Excel.
     
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    A B
    Data Description
    0.5 Value for which you want the distribution
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTGUMBELDIST(A2,A3,A4,TRUE) Cumulative distribution function for the terms above
    =NTGUMBELDIST(A2,A3,A4,FALSE) Probability density function for the terms above

    Sample distribution

  • Function reference : NTGUMBELDIST

Quantile

  • Inverse function of cumulative distribution function
    F^{-1}(P)=\alpha-\beta\ln\ln\frac{1}{P}
  • How to compute this on Excel.
     
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    A B
    Data Description
    0.7 Probability associated with the distribution
    1.7 Value of parameter Alpha
    0.9 Value of parameter Beta
    Formula Description (Result)
    =GUMBELINV(A2,A3,A4) Inverse of the cumulative distribution function for the terms above
  • Function reference : NTGUMBELINV

Characteristics

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

  • Mean of the distribution is given as
    \alpha+\gamma\beta

    where \gamma is Euler’s constant.

  • How to compute this on Excel
     
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    A B
    Data Description
    8 Value of parameter Alpha
    2 Value of parameter Beta
    Formula Description (Result)
    =NTGUMBELMEAN(A2,A3) Mean of the distribution for the terms above
  • Function reference : NTGUMBELMEAN

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

  • Variance of the distribution is given as
    \beta^2\zeta(2)

    where \zeta(\cdot) is Riemann zeta function.

    Standard Deviation is a positive square root of Variance.

  • How to compute this on Excel
     
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    A B
    Data Description
    2 Value of parameter B
    Formula Description (Result)
    =NTGUMBELSTDEV(A2) Standard deviation of the distribution for the terms above
  • Function reference : NTGUMBELSTDEV

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

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

  • Kurtosis of the distribution is 2.4

Random Numbers

  • Random number x is generated by inverse function method, which is for uniform random U,
    x=\alpha-\beta\ln\ln\frac{1}{U}
  • How to generate random numbers on Excel.
     
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    A B
    Data Description
    0.5 Value of parameter Alpha
    0.5 Value of parameter Beta
    Formula Description (Result)
    =NTRANDGUMBEL(100,A2,A3,0) 100 Gumbel Type I 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 A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

Reference

 


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