The problem of modeling extreme or rare events arises in many areas where such events can have very negative consequences. Some examples of rare events include extreme floods and snowfalls, high wind speeds, extreme temperatures, large fluctuations in exchange rates, and market crashes. To develop appropriate probabilistic models and assess the risks caused by these events, business analysts and engineers frequently use the extreme value distributions (EVD).
Extreme Value Theory
Gumbel Distribution (EVD Type I)
Fréchet Distribution (EVD Type II)
Weibull Distribution (EVD Type III)
Generalized Extreme Value Distribution
Using Extreme Value Distributions in EasyFit
Extreme value theory is a separate branch of statistics that deals with extreme events. This theory is based on the extremal types theorem, also called the three types theorem, stating that there are only three types of distributions that are needed to model the maximum or minimum of the collection of random observations from the same distribution.
In other words, if you generate N data sets from the same distribution, and create a new data set that includes the maximum values from these N data sets, the resulting data set can only be described by one of the three models - specifically, the Gumbel, Fréchet, and Weibull distributions.
These models, along with the Generalized Extreme Value distribution, are widely used in risk management, finance, insurance, economics, hydrology, material sciences, telecommunications, and many other industries dealing with extreme events.
One of the first scientists to apply the theory was a German mathematician Emil Gumbel (1891-1966). Gumbel's focus was primarily on applications of extreme value theory to engineering problems, in particular modeling of meteorological phenomena such as annual flood flows:
"It seems that the rivers know the theory. It only remains to convince the engineers of the validity of this analysis."
The Gumbel distribution, also known as the Extreme Value Type I distribution, is unbounded (defined on the entire real axis), and has the following probability density function:
where z=(x-μ)/σ, μ is the location parameter, and σ is the distribution scale (σ>0). The shape of the Gumbel model does not depend on the distribution parameters:
The graph above shows the Gumbel PDF for σ=1 and μ=0.
Maurice Fréchet (1878-1973) was a French mathematician who had identified one possible limit distribution for the largest order statistic in 1927. The Fréchet distribution, also known as the Extreme Value Type II distribution, is defined as
where α is the shape parameter (α>0), and β is the scale parameter (β>0). This distribution is bounded on the lower side (x>0) and has a heavy upper tail. The animation below shows the Fréchet PDF graph for β=1 and various values of α:
Waloddi Weibull (1887-1979) was a Swedish engineer and scientist well-known for his work on strength of materials and fatigue analysis. The Weibull distribution, also known as the Extreme Value Type III distribution, first appeared in his papers in 1939. The two-parameter version of this distribution has the density function
The Weibull distribution is defined for x>0, and both distribution parameters (α - shape, β - scale) are positive. The two-parameter Weibull distribution can be generalized by adding the location (shift) parameter γ:
In this model, the location parameter γ can take on any real value, and the distribution is defined for x>γ.
Even though the Weibull distribution was originally developed to address the problems arising in material sciences, it is widely used in many other areas thanks to its flexibility. When α=1, this distribution reduces to the Exponential model, and when α=2, it mimics the Rayleigh distribution which is mainly used in telecommunications. In addition, it resembles the Normal distribution when α=3.5:
It's worth noting that the Gumbel and Fréchet models described above relate to maxima (largest extreme value), while the Weibull model relates to minima (smallest extreme value). This form of the Weibull distribution is commonly used in practice.
The Generalized Extreme Value (GEV) distribution is a flexible three-parameter model that combines the Gumbel, Fréchet, and Weibull maximum extreme value distributions. It has the following PDF:
where z=(x-μ)/σ, and k, σ, μ are the shape, scale, and location parameters respectively. The scale must be positive (sigma>0), the shape and location can take on any real value. The range of definition of the GEV distribution depends on k:
Various values of the shape parameter yield the extreme value type I, II, and III distributions. Specifically, the three cases k=0, k>0, and k<0 correspond to the Gumbel, Fréchet, and "reversed" Weibull distributions. The reversed Weibull distribution is a quite rarely used model bounded on the upper side. For example, for k=−0.5, the GEV PDF graph has the form:
When fitting the GEV distribution to sample data, the sign of the shape parameter k will usually indicate which one of the three models best describes the random process you are dealing with.
EasyFit supports the entire family of extreme value distributions, including the Gumbel, Fréchet, Weibull, and GEV models. Like most distributions in EasyFit, you can fit these models to your data or use them in Excel-based Monte Carlo simulations.
The Gumbel distribution is available in two forms: Gumbel Max (maximum extreme value) and Gumbel Min (minimum extreme value), enabling you to model left-skewed and right-skewed data:
In addition to the "classical" two-parameter Fréchet distribution, EasyFit supports the three-parameter model which has the location parameter γ:
In this model, α and β have the same meaning as in the two-parameter model, but the distribution is defined for x>γ (γ can take on any real value). Similarly, EasyFit supports both two-parameter and three-parameter Weibull distributions.
Since the range of definition of the Generalized Extreme Value distribution depends on the shape parameter k, this model falls into the Advanced distributions class, according to the classification used in EasyFit.
The extreme value distributions can be easily fitted to your data using either automated or manual fitting capabilities of EasyFit.
In automated fitting mode, EasyFit will fit both forms of the Weibull and Fréchet distributions unless you specify otherwise in the Distribution Fitting Options dialog. A similar capability is available in manual fitting mode.
When fitting the three-parameter Fréchet or Weibull models, you can either have EasyFit estimate the three parameters from data, or manually specify the location parameter and estimate α and β only. This feature can be useful if γ is known and doesn't need to be estimated.
To compare the fit of the extreme value distributions and select the best fitting model, you can use the goodness of fit tests and distribution graphs displayed by EasyFit. In general, the GEV distribution frequently provides better fit than the Gumbel, Fréchet, and Weibull models.
Once you select the best fitting model, you can use it to perform specific calculations and make appropriate decisions based on the analysis results. Some typical applications include calculating probabilities and making estimates and projections.