1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.6. Probability Distributions 1.3.6.6. Gallery of Distributions
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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
where is the location parameter and is the scale parameter. The case where = 0 and = 1 is called the standard Gumbel distribution. The equation for the standard Gumbel distribution (minimum) reduces to
The following is the 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
where is the location parameter and is the scale parameter. The case where = 0 and = 1 is called the standard Gumbel distribution. The equation for the standard Gumbel distribution (maximum) reduces to
The following is the plot of the Gumbel probability density function for the maximum case.
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. |
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Cumulative Distribution Function |
The formula for the cumulative distribution
function of the Gumbel distribution (minimum) is
The following is the plot of the Gumbel cumulative distribution function for the minimum case.
The formula for the cumulative distribution function of the Gumbel distribution (maximum) is
The following is the plot of the Gumbel cumulative distribution function for the maximum case.
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Percent Point Function |
The formula for the percent point
function of the Gumbel distribution (minimum) is
The following is the plot of the Gumbel percent point function for the minimum case.
The formula for the percent point function of the Gumbel distribution (maximum) is
The following is the plot of the Gumbel percent point function for the maximum case.
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Hazard Function |
The formula for the hazard
function of the Gumbel distribution (minimum) is
The following is the plot of the Gumbel hazard function for the minimum case.
The formula for the hazard function of the Gumbel distribution (maximum) is
The following is the plot of the Gumbel hazard function for the maximum case.
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Cumulative Hazard Function |
The formula for the cumulative hazard
function of the Gumbel distribution (minimum) is
The following is the plot of the Gumbel cumulative hazard function for the minimum case.
The formula for the cumulative hazard function of the Gumbel distribution (maximum) is
The following is the plot of the Gumbel cumulative hazard function for the maximum case.
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Survival Function |
The formula for the survival
function of the Gumbel distribution (minimum) is
The following is the plot of the Gumbel survival function for the minimum case.
The formula for the survival function of the Gumbel distribution (maximum) is
The following is the plot of the Gumbel survival function for the maximum case.
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Inverse Survival Function |
The formula for the inverse
survival function of the Gumbel distribution (minimum) is
The following is the plot of the Gumbel inverse survival function for the minimum case.
The formula for the inverse survival function of the Gumbel distribution (maximum) is
The following is the plot of the Gumbel inverse survival function for the maximum case.
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Common Statistics |
The formulas below are for the maximum order statistic case.
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Parameter Estimation |
The method of moments estimators of the Gumbel (maximum) distribution
are
where and s are the sample mean and standard deviation, respectively. The equations for the maximum likelihood estimation of the shape and scale parameters are discussed in Chapter 15 of Evans, Hastings, and Peacock and Chapter 22 of Johnson, Kotz, and Balakrishnan. These equations need to be solved numerically and this is typically accomplished by using statistical software packages. |
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Software | Some general purpose statistical software programs, including Dataplot, support at least some of the probability functions for the extreme value type I distribution. |