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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions

1.3.6.6.16.

Extreme Value Type I Distribution

Probability Density Function The extreme value type I distribution has two forms. One is based on the smallest extreme and the other is based on the largest extreme. We call these the minimum and maximum cases, respectively. Formulas and plots for both cases are given. The extreme value type I distribution is also referred to as the Gumbel distribution.

The general formula for the probability density function of the Gumbel (minimum) distribution is

\( f(x) = \frac{1} {\beta} e^{\frac{x-\mu}{\beta}}e^{-e^{\frac{x-\mu} {\beta}}} \)

where μ is the location parameter and β is the scale parameter. The case where μ = 0 and β = 1 is called the standard Gumbel distribution. The equation for the standard Gumbel distribution (minimum) reduces to

\( f(x) = e^{x}e^{-e^{x}} \)

The following is the plot of the Gumbel probability density function for the minimum case.

plot of the Gumbel probability density function for the minimum
 case

The general formula for the probability density function of the Gumbel (maximum) distribution is

\( f(x) = \frac{1}{\beta} e^{-\frac{x-\mu}{\beta}}e^{-e^{-\frac{x-\mu} {\beta}}} \)

where μ is the location parameter and β is the scale parameter. The case where μ = 0 and β = 1 is called the standard Gumbel distribution. The equation for the standard Gumbel distribution (maximum) reduces to

\( f(x) = e^{-x}e^{-e^{-x}} \)

The following is the plot of the Gumbel probability density function for the maximum case.

\( f(x) = e^{-x}e^{-e^{-x}} \)

Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.

Cumulative Distribution Function The formula for the cumulative distribution function of the Gumbel distribution (minimum) is

\( F(x) = 1 - e^{-e^{x}} \)

The following is the plot of the Gumbel cumulative distribution function for the minimum case.

plot of the Gumbel cumulative distribution function for
 the minimum case

The formula for the cumulative distribution function of the Gumbel distribution (maximum) is

\( F(x) = e^{-e^{-x}} \)

The following is the plot of the Gumbel cumulative distribution function for the maximum case.

plot of the Gumbel cumulative distribution function for the
 maximum case

Percent Point Function The formula for the percent point function of the Gumbel distribution (minimum) is

\( G(p) = \ln(\ln(\frac{1} {1 - p})) \)

The following is the plot of the Gumbel percent point function for the minimum case.

plot of the Gumbel percent point function for the minimum
 case

The formula for the percent point function of the Gumbel distribution (maximum) is

\( G(p) = -\ln(\ln(\frac{1} {p})) \)

The following is the plot of the Gumbel percent point function for the maximum case.

plot of the Gumbel percent point function for the
 maximum case

Hazard Function The formula for the hazard function of the Gumbel distribution (minimum) is

\( h(x) = e^{x} \)

The following is the plot of the Gumbel hazard function for the minimum case.

plot of the Gumbel hazard function for the minimum case

The formula for the hazard function of the Gumbel distribution (maximum) is

\( h(x) = \frac{e^{-x}} {e^{e^{-x}} - 1} \)

The following is the plot of the Gumbel hazard function for the maximum case.

plot of the Gumbel hazard function for the maximum case

Cumulative Hazard Function The formula for the cumulative hazard function of the Gumbel distribution (minimum) is

\( H(x) = e^{x} \)

The following is the plot of the Gumbel cumulative hazard function for the minimum case.

plot of the Gumbel cumulative hazard function for the minimum case

The formula for the cumulative hazard function of the Gumbel distribution (maximum) is

\( H(x) = -\ln(1 - e^{-e^{-x}}) \)

The following is the plot of the Gumbel cumulative hazard function for the maximum case.

plot of the Gumbel cumulative hazard function for the maximum case

Survival Function The formula for the survival function of the Gumbel distribution (minimum) is

\( S(x) = e^{-e^{x}} \)

The following is the plot of the Gumbel survival function for the minimum case.

plot of the Gumbel survival function for the minimum case

The formula for the survival function of the Gumbel distribution (maximum) is

\( S(x) = 1 - e^{-e^{-x}} \)

The following is the plot of the Gumbel survival function for the maximum case.

plot of the Gumbel survival function for the maximum case

Inverse Survival Function The formula for the inverse survival function of the Gumbel distribution (minimum) is

\( Z(p) = \ln(\ln(\frac{1} {p})) \)

The following is the plot of the Gumbel inverse survival function for the minimum case.

plot of the Gumbel inverse survival function for the minimum case

The formula for the inverse survival function of the Gumbel distribution (maximum) is

\( Z(p) = -\ln(\ln(\frac{1} {1-p})) \)

The following is the plot of the Gumbel inverse survival function for the maximum case.

plot of the Gumbel inverse survival function for the maximum case

Common Statistics The formulas below are for the maximum order statistic case.

Mean \( \mu + 0.5772\beta \)

The constant 0.5772 is Euler's number.

Median \( \mu - \beta\ln(\ln(2)) \)
Mode μ
Range \(-\infty \mbox{ to } \infty\)
Standard Deviation \( \frac{\beta\pi} {\sqrt{6}} \)
Skewness 1.13955
Kurtosis 5.4
Coefficient of Variation \( \frac {\beta\pi} {\sqrt{6}(\mu + 0.5772\beta)} \)

Parameter Estimation The method of moments estimators of the Gumbel (maximum) distribution are

\( \tilde{\beta} = \frac{s\sqrt{6}} {\pi} \)

\( \tilde{\mu} = \bar{X} - 0.5772 \tilde{\beta} = \bar{X} - 0.45006 s \)

where \( \bar{X} \) and s are the sample mean and standard deviation, respectively.

The method of moments estimators of the Gumbel (minimum) distribution are

\( \tilde{\beta} = \frac{s\sqrt{6}} {\pi} \)

\( \tilde{\mu} = \bar{X} + 0.5772 \tilde{\beta} = \bar{X} + 0.45006 s \)

where \( \bar{X} \) and s are the sample mean and standard deviation, respectively.

The maximum likelihood estimates for the maximum case are the solution to the following simultaneous equations

\( \bar{x} - \frac{\sum_{i=1}^{n}{x_i \exp(-x_i/\hat{\beta})}} {\sum_{i=1}^{n}{\exp(-x_i/\hat{\beta})}} - \hat{\beta} = 0 \)

\( -\hat{\beta} \log \left( \frac{1}{n} \sum_{i=1}^{n}{\exp(-x_i/\hat{\beta})} \right) - \hat{\mu} = 0 \)

For the minimum case, replace \(-x_i\) with \(x_i\) in the above equations.

These equations need to be solved numerically and this is typically accomplished by using statistical software packages.

Software Some general purpose statistical software programs support at least some of the probability functions for the extreme value type I distribution.
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