# Gumbel (Type I) distribution

## Where do you meet this distribution?

- Extreme value theory (EVT)
- Risk management – Operational risk

## Shape of Distribution

### Basic Properties

- Two parameters are required (How can you get these).
- Continuous distribution defined on entire range
- This distribution is always asymmetric.

### Probability

- Cumulative distribution function
- Probability density function
- How to compute these on Excel.

1 2 3 4 5 6

7

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 - Function reference : NTGUMBELDIST

### Quantile

- Inverse function of cumulative distribution function
- How to compute this on Excel.

1 2 3 4 5 6

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
where is Euler’s constant.

- How to compute this on Excel

1 2 3 4 5 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
where is Riemann zeta function.

Standard Deviation is a positive square root of Variance.

- How to compute this on Excel

1 2 3 4 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)

- Skewness of the distribution is given as
where is Riemann zeta function.

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

- Kurtosis of the distribution is

## Random Numbers

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

- If you already have parameters of the distribution
- Generating random numbers based on Mersenne Twister algorithm: NTRANDGUMBEL
- Computing probability : NTGUMBELDIST
- Computing mean : NTGUMBELMEAN
- Computing standard deviation : NTGUMBELSTDEV
- Computing skewness : NTGUMBELSKEW
- Computing kurtosis : NTGUMBELKURT
- Computing moments above at once : NTGUMBELMOM

- If you know mean and standard deviation of the distribution
- Estimating parameters of the distribution:NTGUMBELRPARAM

## Reference

- Wolfram Mathworld – Gumbel Distribution
- Wikipedia – Gumbel distribution
- Statistics Online Computational Resource