Contents
- Index
Smallest Extreme Value Family of Distributions
Shape:
The smallest extreme value family of distributions is made up of three distributions: Weibull, negative Fréchet and smallest extreme value. It covers any specified average, standard deviation and any skewness above -5.6051382. Together they form a 3-parameter family of distributions that is represented by a curve on a skewness-kurtosis plot as shown below. The Weibull distribution covers the portion of the curve with skewness above -1.139547. The negative Fréchet distribution covers the portion of the curve with skewness below -1.139547. The smallest extreme value distribution handles the remaining case of skewness equal to -1.139547.
Density Function - Smallest Extreme Value Distribution:
The density function of the smallest extreme value distribution is shown below:
The equation, parameters and bounds of the density function are:
Moments - Smallest Extreme Value Distribution:
The moments of the smallest extreme value distribution can be calculated from the parameters as shown below:
Density Function - Weibull:
The density function of the Weibull distribution is shown below:
The equation, parameters and bounds of the density function are:
Moments - Weibull:
The moments of the Weibull distribution can be calculated from the parameters as shown below:
Properties:
The exponential distribution is a special case of the Weibull distribution and thus falls on the Weibull curve in the skewness-kurtosis plot.
The Weibull and smallest extreme value distributions are the distributions of minimums. Under certain restrictions, the minimum of distributions without lower bounds tends to the smallest extreme value distribution and the minimum of distributions bounded below tends to the Weibull.