ShortCourseMalta

1 0 gumbel type limit as 0 2 0 frchet type 3

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: gs to one of the following three types. Background on Extreme Value Analysis (EVA) Extremal Types Theorem I. Gumbel G(z ) = exp − exp − z−b a II. Fréchet z ≤ b, 0, G (z ) = exp − , −∞<z <∞ z −b −α a , z > b; III. Weibull G (z ) = exp − − 1, with parameters a, b and α > 0. z −b α a , z < b, z≥b Background on Extreme Value Analysis (EVA) Extremal Types Theorem The three types can be written as a single family of distributions, known as the generalized extreme value (GEV) distribution. G(z ) = exp − 1 + ξ z−µ σ −1/ξ , + where y+ = max{y, 0}, −∞ < µ, ξ < ∞ and σ > 0. Background on Extreme Value Analysis (EVA) GEV distribution Three parameters: location (µ), scale (σ ) and shape (ξ ). 1. ξ = 0 (Gumbel type, limit as ξ −→ 0) 2. ξ > 0 (Fréchet type) 3. ξ < 0 (Weibull type) Background on Extreme Value Analysis (EVA) Gumbel type • Light tail • Domain of attraction for many common distributions (e.g., normal, lognormal, exponential, gamma) 0.0 0.1 pdf 0.2 0.3 0.4 0.5 Gumbel 0 1 2 3 4 5 Background on Extreme Value Analysis (EVA) Fréchet type • Heavy tail • E [X r ] = ∞ for r ≥ 1/ξ (i.e., infinite variance if ξ ≥ 1/2) • Of interest for precipitation, streamflow, economic impacts pdf 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Frechet 0 1 2 3 4 5 Background on Extreme Value Analysis (EVA) Weibull type • Bounded upper tail at µ − σ ξ • Of interest for temperature, wind speed, sea level 0.0 0.1 0.2 pdf 0.3 0.4 0.5 Weibull 0 1 2 3 4 5 6 Background on Extreme Value Analysis (EVA) Normal vs. GEV ## The probability of exceeding increasingly high values ## (as they double). # Normal pnorm( c(1,2,4,8,16,32), lower.tail=FALSE) # Gumbel pgev( c(1,2,4,8,16,32), lower.tail=FALSE) # Fréchet pgev( c(1,2,4,8,16,32), shape=0.5, lower.tail=FALSE) # Weibull (note bounded upper tail!) pgev( c(1,2,4,8,16,32), shape=-0.5, lower.tail=FALSE) Background on Extreme Value Analysis (EVA) Normal vs. GEV # Find Pr{X < x} for X = 0, . . . , 20. cdfNorm < − pnorm( 0:20) cdfGum < − pgev( 0:20) cdfFrech < − pgev( 0:20, shape=0.5) cdfWeib < − pgev( 0:20, shape=-0.5) # Now find Pr{X = x} for X = 0, . . . , 20. pdfNorm < − dnorm( 0:20) pdfGum < − dgev( 0:20) pdfFrech < − dgev( 0:20, shape=0.5) pdfWeib < − dgev( 0:20, shape=-0.5) Background on Extreme Value Analysis (EVA) Normal vs. GEV par( mfrow=c(2,1), mar=c(5,4,0.5,0.5)) plot( 0:20, cdfNorm, ylim=c(0,1), type="l", xaxt="n", col="blue", lwd=2, xlab="", ylab="F(x)") lines( 0:20, cdfGum, col="green", lty=2, lwd=2) lines( 0:20, cdfFrech, col="red", lwd=2) lines( 0:20, cdfWeib, col="orange", lwd=2) legend( 10, 0.05, legend=c("Normal", "Gumbel", "Frechet", "Weibull"), col=c("blue", "green", "red", "orange"), lty=c(1,2,1,1), bty="n", lwd=2) Background on Extreme Value Analysis (EVA) Normal vs. GEV plot( 0:20, pdfNorm, ylim=c(0,1), typ="l"...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern