21.4.7.3. Extreme Value Distribution

Suppose that \(\{X_{1}^{1},X_{2}^{1},X_{3}^{1},...,X_{n}^{1}\}\) are the measured set from the n trials of a material strength test. The test results are listed with ascending order. Then, assume that these measuring are repeated \(m\) times such as \(\{X_{1}^{1},X_{2}^{1},X_{3}^{1},...,X_{n}^{1}\}\), \(\{X_{1}^{2},X_{2}^{2},X_{3}^{2},...,X_{n}^{2}\}\),…, \(\{X_{1}^{m},X_{2}^{m},X_{3}^{m},...,X_{n}^{m}\}\). From the measured data set, if probable distribution functions \({{F}_{{{X}_{1}}}}(X)\) and \({{F}_{{{X}_{2}}}}(X)\) are made from the minimum value set \(\{X_{1}^{1},X_{1}^{2},X_{1}^{3},...,X_{1}^{m}\}\) and the maximum value set \(\{X_{n}^{1},X_{n}^{2},X_{n}^{3},...,X_{n}^{m}\}\), these PDFs are referred to as Extreme Value Distributions for the minimum value and the maximum value of \(X\), respectively.

If the random variables \({{X}_{i}},i=1,2,...,n\) are independent and follow a common distribution \({{F}_{X}}(x)\), the shape of the distribution function of the extreme value \({{X}_{1}}\) and \({{X}_{n}}\) becomes increasingly insensitive to the exact shape of the common distribution function \({{F}_{X}}(x)\), as \(n\to \propto\). These asymptotic distributions often describe the behavior of the random variable \({{X}_{1}}\) and \({{X}_{n}}\) reasonably well even when the exact shape of the parent distribution function \({{F}_{X}}(x)\) is not known precisely. These asymptotic distribution forms are classified into three types based on the general features of the tail part of the distributions.

21.4.7.3.1. Type-I Extreme Value Distribution

Maximum Value Distribution

This distribution is, referred to as the Gumbel distribution, useful when the right tail of the parent distribution \({{F}_{X}}(x)\) is not bounded, which is of an exponential type. In such a case, the distribution function can be expressed as

\({{F}_{X}}(x)=1-exp[-h(x)]\)

where \(h(x)\) increases with x monotonically. The distributions such as normal, log-normal and gamma distributions belong to this category. The extreme value distribution for the maximum value, \(Y\equiv {{X}_{n}}\), is given by

\({{F}_{Y}}(y)=exp[-exp(-{{a}_{n}}(y-{{w}_{n}}))],-\propto <y<\propto\)

where the parameters of distribution, \({{a}_{n}}\) and \({{w}_{n}}\), can be determined from the observation data. They are related to the mean and the standard deviation of the extreme value \(Y\) as

\({{a}_{n}}=\frac{1}{\sqrt{6}}\left( \frac{\pi }{{{\sigma }_{Y}}} \right)\) and \({{w}_{n}}={{\mu }_{Y}}-\frac{\gamma }{{{a}_{n}}}\)

Where \(\gamma =0.5772\) is the Euler’s constant.

Minimum Value Distribution

This distribution is useful whenever the left tail of the parent distribution is unbounded and decreases to zero towards the left in an exponential form. In this case, the distribution function for the minimum value, \(Z\equiv {{X}_{1}}\), is given by

\({{F}_{Z}}(z)=1-exp(-{{a}_{1}}(z-{{w}_{1}})),-\propto <z<\propto\)

where, \({{a}_{1}}\) and \({{w}_{1}}\) are the parameters of the distribution. They are related to the mean and the standard deviation of the extreme value \(Z\) as

\({{a}_{1}}=\frac{1}{\sqrt{6}}\left( \frac{\pi }{{{\sigma }_{z}}} \right)\) and \({{w}_{1}}={{\mu }_{z}}+\frac{\gamma }{{{a}_{1}}}\)

Where \(\gamma =0.5772\) is the Euler’s constant.

21.4.7.3.2. Type-II Extreme Value Distribution

Maximum Value Distribution

The type-II asymptotic distribution for the maximum values is useful whenever the parent distribution \({{F}_{X}}(x)\) is defined over the range \(0<x<\propto\) and approaches one as \(x\to \propto\) according to the relation.

\({{F}_{X}}(x)=1-\alpha {{\left( \frac{1}{x} \right)}^{m}};x>0\)

where, \(a>0\) and \(m>0\) are the parameters of the distribution. The extreme value distribution for the maximum value, \(Y\equiv {{X}_{n}}\), is given by

\({{F}_{Y}}(y)=\exp \left[ -{{\left( \frac{{{v}_{n}}}{y} \right)}^{k}} \right]\)

The corresponding PDF is

\({{F}_{Y}}(y)=\frac{k}{{{v}_{n}}}{{\left( \frac{{{v}_{n}}}{y} \right)}^{k+1}}\exp \left[ -{{\left( \frac{{{v}_{n}}}{y} \right)}^{k}} \right],y\ge 0,k>2\)

where \({{v}_{n}}\) and \(k\) are the parameters of the distribution; \({{v}_{n}}\) is the characteristic maximum value of the underlying variable \(X\) and \(k\), the shape parameter, is a measure of dispersion. The Type-II asymptotic form is obtained as \(n\) goes to infinity from an initial distribution that has a polynomial tail in the direction of the extreme value, which requires a polynomial tail. Therefore, a lognormal distribution converges to a Type-II asymptotic form for the maximum value. However, the Type-I converges from an exponential tail.

For the Type-II distribution of maxima, the mean, standard deviation, and COV of \(Y\) are related to the distribution parameter \({{v}_{n}}\) and \(k\) as follows:

\({{\mu }_{Y}}={{v}_{n}}\Gamma \left( 1-\frac{1}{k} \right),k>1\)

\(\sigma _{Y}^{2}=v_{n}^{2}\left[ \Gamma \left( 1-\frac{2}{k} \right)-{{\Gamma }^{2}}\left( 1-\frac{1}{k} \right) \right],k>2\)

and

\(1+\delta _{Y}^{2}=\frac{\Gamma \left( 1-\frac{2}{k} \right)}{{{\Gamma }^{2}}\left( 1-\frac{1}{k} \right)},k>2\)

In these equations, \(\Gamma\) is the gamma function. It is useful for representing annual maximum winds and other meteorological and hydrological phenomena.

Minimum Value Distribution

The Type-II asymptotic distribution for the minimum values is useful whenever the parent distribution \({{F}_{X}}(x)\) is defined over the range \(-\propto <x<0\). In this case, the distribution function for the minimum value, \(Z\equiv {{X}_{1}}\), is given by

\({{F}_{Z}}(z)=1-\exp \left[ -{{\left( \frac{v{}_{1}}{z} \right)}^{k}} \right];z\le 0\)

where the parameter \({{v}_{1}}\) is the characteristic minimum value of the initial variable \(X\) and \(k\) is the shape parameter, an inverse measure of dispersion.

For the Type-II distribution of minima, the mean, standard deviation, and COV of \(Z\) are related to the distribution parameter \({{v}_{1}}\) and \(k\) as follows:

\({{\mu }_{Z}}={{v}_{1}}\Gamma \left( 1-\frac{1}{k} \right),k>1\)

\(\sigma _{Z}^{2}=v_{1}^{2}\left[ \Gamma \left( 1-\frac{2}{k} \right)-{{\Gamma }^{2}}\left( 1-\frac{1}{k} \right) \right],k>2\)

and

\(1+\delta _{z}^{2}=\frac{\Gamma \left( 1-\frac{2}{k} \right)}{{{\Gamma }^{2}}\left( 1-\frac{1}{k} \right)},k>2\)

In these equations, \(\Gamma\) is the gamma function. It is not commonly used since the required parent distribution shape is not commonly observed in practical applications.

21.4.7.3.3. Type-III Extreme Value Distribution Value

Maximum Value Distribution

The Type-III asymptotic distribution for the maximum values is useful whenever the parent distribution \({{F}_{X}}(x)\) is defined over the range \(-\propto <x<\varpi\) and has the form

\({{F}_{X}}(x)=1-a{{(\varpi -x)}^{m}};x<\varpi\)

where \(a>0\) and \(m>0\) are the parameters of the distribution. The above distribution denotes a uniform distribution when \(m=1\) and a triangular distribution when \(m=2\). The Type-III extreme value distribution for the maximum value, \(Y\equiv {{X}_{n}}\), is given by

\({{F}_{Y}}(y)=\exp \left[ -{{\left( \frac{\varpi -y}{\varpi -{{w}_{n}}} \right)}^{m}} \right];y\le \varpi\)

The corresponding PDF is

\({{f}_{Y}}(y)=\frac{k}{\varpi -{{w}_{n}}}{{\left( \frac{\varpi -y}{\varpi -{{w}_{n}}} \right)}^{k-1}}\exp \left[ -{{\left( \frac{\varpi -y}{\varpi -{{w}_{n}}} \right)}^{k}} \right];y\le \varpi\)

where \(\varpi\) is the upper bound of the initial distribution, \({{w}_{n}}\) is the characteristic maximum value of \(X\) and \(k\) is a shape parameter.

The mean and variance of \({{Y}_{n}}\) are related to the parameters \({{w}_{n}}\) and \(k\) as follows:

\({{\mu }_{Y}}=\varpi (\varpi -{{w}_{n}})\Gamma \left( 1+\frac{1}{k} \right)\)

And

\(\sigma _{Y}^{2}=Var(\varpi -Y)={{(\varpi -{{w}_{n}})}^{2}}\left[ \Gamma \left( 1+\frac{2}{k} \right)-{{\Gamma }^{2}}\left( 1+\frac{1}{k} \right) \right]\)

Minimum Value Distribution

The Type-III minimum value distribution can be used whenever the parent distribution is defined over the range \(\varepsilon \le x<\propto\) and has the form

\({{F}_{X}}(x)=a{{(x-\varepsilon )}^{m}};x\ge \varepsilon\)

where \(a>0\) and \(m>0\) are the parameters of the distribution. In this case, the Type-III asymptotic distribution for the minimum value, \(Z\equiv {{X}_{1}}\), is given by

\({{F}_{Z}}(z)=1-\exp \left[ -{{\left( \frac{z-\varepsilon }{v-\varepsilon } \right)}^{m}} \right];z\ge \varepsilon ,m>0,v>\varepsilon\)

By differentiating the distribution function, the CDF can be obtained as

\({{f}_{z}}(z)=\frac{m}{v-\varepsilon }{{\left( \frac{z-\varepsilon }{v-\varepsilon } \right)}^{m-1}}\exp \left[ -{{\left( \frac{z-\varepsilon }{v-\varepsilon } \right)}^{m}} \right];z\ge \varepsilon\)

In this case mean and variance of \(Z\) are given by

\({{\mu }_{z}}=\varepsilon +(v-\varepsilon )\Gamma \left( 1+\frac{1}{m} \right)\)

And

\(\sigma _{z}^{2}={{(v-\varepsilon )}^{2}}\left\{ \Gamma \left( 1+\frac{2}{m} \right)-{{\Gamma }^{2}}\left( 1+\frac{1}{m} \right) \right\}\)