Inverse Weibull Distribution

The inverse Weibull distribution, also known as the Fréchet distribution, is a special case of the generalized extreme value distribution. It is used in various fields such as reliability engineering and hydrology.

Definition

The probability density function (PDF) is given by:

f(x)=αs(x−ms)−α−1e−(x−ms)−α

The cumulative distribution function (CDF) is:

F(x)=e−(x−ms)−α

The distribution is defined for values where x is greater than or equal to the location parameter m. The distribution requires a positive shape parameter α and a positive scale parameter s.

Applications

The inverse Weibull distribution is commonly used in reliability engineering to model the life of products and materials. It is also used in hydrology to model extreme events such as floods.

In material science, the inverse Weibull distribution helps predict the failure times of composite materials under stress. Environmental scientists use it to analyze extreme wind speeds and maximum precipitation levels. The distribution is also valuable in risk assessment for insurance companies when modeling catastrophic events.

Properties

Statistical Properties
PropertyValue
Meanm+sΓ(1−1α),α>1
Medianm+sαln⁡2
Modem+s(α1+α)1/α
Variances2[Γ(1−2α)−Γ2(1−1α)],α>2
SkewnessΓ(1−3α)−3Γ(1−1α)Γ(1−2α)+2Γ(1−1α)3[Γ(1−2α)−Γ(1−1α)2]3/2,α>3
Excess KurtosisΓ(1−4α)−4Γ(1−1α)Γ(1−3α)+3Γ(1−2α)2[Γ(1−2α)−Γ2(1−1α)]2−6,α>4

Notable properties include:

  • The distribution has heavy right tails for all parameter values.

Relationships to Other Distributions

  • If X follows a Weibull distribution, then Y = 1/X follows an inverse Weibull distribution.

  • The inverse Weibull distribution is a special case of the generalized extreme value distribution.

  • It is also known as the Fréchet distribution or Type II extreme value distribution. The Gumbel distribution is Type I, and the Weibull distribution is Type III.

The InverseWeibullDistribution class

The inverse Weibull distribution is implemented by the InverseWeibullDistribution class. It has two constructors.

The first constructor takes from one to three parameters. The first parameter is the shape parameter of the distribution. It must be greater than zero. The second is the scale parameter, which is optional with a default value of 1. The third is the location parameter, which is also optional with a default value of 0.

C# 
var invWeibull1 = new InverseWeibullDistribution(1.0);
var invWeibull2 = new InverseWeibullDistribution(1.0, 2.0);
var invWeibull3 = new InverseWeibullDistribution(1.0, 2.0, 3.0);

The second constructor is used to estimate the distribution from a sample. The first parameter is a Vector<T> that specifies the sample. The second parameter, which is optional, specifies the estimation method. This can be matching moments (the default) or maximum likelihood. Note that when estimating the distribution, the location parameter is always assumed to be zero.

C# 
var invWeibull4 = new InverseWeibullDistribution(new double[] 
    { 1.0, 2.0, 2.5, 4.0 });

The InverseWeibullDistribution class has three properties that return the LocationParameter, ScaleParameter, and ShapeParameter, respectively.

References

For more information on the inverse Weibull distribution, refer to the following sources:

  • Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 19. Wiley Series in Probability and Statistics.

  • Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 17. Wiley Series in Probability and Statistics.

  • Rinne, H. (2009). The Weibull Distribution: A Handbook. Chapman and Hall/CRC.

  • Nadarajah, S., & Kotz, S. (2008). Inverse exponential and inverse gamma distributions. Bulletin of the Brazilian Mathematical Society, 39(3), 461-488.

See Also