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Gumbel (Type I) Distribution

Where do you meet this distribution?​

Shape of Distribution​

Basic Properties​

  • Two parameters α,β\alpha, \beta are required (How can you get these).

    β>0\beta>0

  • Continuous distribution defined on entire range

  • This distribution is always asymmetric.

Probability​

  • Cumulative distribution function

    F(x)=exp⁡[−exp⁡(−x−αβ)]F(x)=\exp\left[-\exp\left(-\frac{x-\alpha}{\beta}\right)\right]

  • Probability density function

    f(x)=1βexp⁡(−x−αβ)exp⁡[−exp⁡(−x−αβ)]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.

AB
1DataDescription
20.5Value for which you want the distribution
38Value of parameter Alpha
42Value of parameter Beta
5FormulaDescription (Result)
6=NTGUMBELDIST(A2,A3,A4,TRUE)Cumulative distribution function for the terms above
7=NTGUMBELDIST(A2,A3,A4,FALSE)Probability density function for the terms above

Sample distribution

Quantile​

  • Inverse function of cumulative distribution function

    F−1(P)=α−βln⁡ln⁡1PF^{-1}(P)=\alpha-\beta\ln\ln\frac{1}{P}

  • How to compute this on Excel.

AB
1DataDescription
20.7Probability associated with the distribution
31.7Value of parameter Alpha
40.9Value of parameter Beta
5FormulaDescription (Result)
6=GUMBELINV(A2,A3,A4)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

    α+γβ\alpha+\gamma\beta

    where γ\gamma is Euler's constant.

  • How to compute this on Excel

AB
1DataDescription
28Value of parameter Alpha
32Value of parameter Beta
4FormulaDescription (Result)
5=NTGUMBELMEAN(A2,A3)Mean of the distribution for the terms above

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

AB
1DataDescription
22Value of parameter Beta
3FormulaDescription (Result)
4=NTGUMBELSTDEV(A2)Standard deviation of the distribution for the terms above

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

  • Skewness of the distribution is given as

    −126ζ(3)π3=−1.139547099⋯-\frac{12\sqrt{6}\zeta(3)}{\pi^3}=-1.139547099\cdots

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

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

  • Kurtosis of the distribution is 2.42.4

Random Numbers​

  • Random number x is generated by inverse function method, which is for uniform random U,

    x=α−βln⁡ln⁡1Ux=\alpha-\beta\ln\ln\frac{1}{U}

  • How to generate random numbers on Excel.

AB
1DataDescription
20.5Value of parameter Alpha
30.5Value of parameter Beta
4FormulaDescription (Result)
5=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​