Gumbel (Type II) distribution

Where do you meet this distribution?

• Meteorology, Seismology : Model of extreme event (Extreme value theory)
• Risk management -- Operational risk

Shape of Distribution

Basic Properties

• Two parameters $a, b$ are required.

  $$a>0, b>0$$

• Continuous distribution defined on semi-infinite range $x>0$

• This distribution is always asymmetric.

Probability

• Cumulative distribution function

  $$F(x)=\exp\left(-bx^{-a}\right)$$

• Probability density function

  $$f(x)=abx^{-a-1}\exp\left(-bx^{-a}\right)$$

• How to compute these on Excel.

	A	B
1	Data	Description
2	0.5	Value for which you want the distribution
3	8	Value of parameter A
4	2	Value of parameter B
5	Formula	Description (Result)
6	=EXP(-A4*A2^(-A3))	Cumulative distribution function for the terms above
7	=A3*A4*A2^(-A3-1)*EXP(-A4*A2^(-A3))	Probability density function for the terms above

Quantile

• Inverse function of cumulative distribution function

  $$F^{-1}(P)=\left(-\frac{\ln P}{b}\right)^{-1/a}$$

• How to compute this on Excel.

	A	B
1	Data	Description
2	0.7	Probability associated with the distribution
3	1.7	Value of parameter A
4	0.9	Value of parameter B
5	Formula	Description (Result)
6	=POWER(-LN(A2)/A4,-1/A3)	Inverse of the cumulative distribution function for the terms above

Random Numbers

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

  $$x=\left(-\frac{\ln U}{b}\right)^{-1/a}$$

• How to generate random numbers on Excel.

	A	B
1	Data	Description
2	8	Value of parameter A
3	2	Value of parameter B
4	Formula	Description (Result)
5	=POWER(-A2*LN(-NTRAND(100)/A3),1/A2)	100 Gumbel (Type II) 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

Not supported yet

Reference

• Wikipedia -- Gumbel (Type II) distribution
• Statistics Online Computational Resource