# lmomgev: L-moments of the Generalized Extreme Value Distribution In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

This function estimates the L-moments of the Generalized Extreme Value distribution given the parameters (ξ, α, and κ) from pargev. The L-moments in terms of the parameters are

λ_1 = ξ + \frac{α}{κ}(1-Γ(1+κ)) \mbox{,}

λ_2 = \frac{α}{κ}(1-2^{-κ})Γ(1+κ) \mbox{,}

τ_3 = \frac{2(1-3^{-κ})}{1-2^{-κ}} - 3 \mbox{, and}

τ_4 = \frac{5(1-4^{-κ})-10(1-3^{-κ})+6(1-2^{-κ})}{1-2^{-κ}} \mbox{.}

## Usage

 1 lmomgev(para)

## Arguments

 para The parameters of the distribution.

## Value

An R list is returned.

 lambdas Vector of the L-moments. First element is λ_1, second element is λ_2, and so on. ratios Vector of the L-moment ratios. Second element is τ, third element is τ_3 and so on. trim Level of symmetrical trimming used in the computation, which is 0. leftrim Level of left-tail trimming used in the computation, which is NULL. rightrim Level of right-tail trimming used in the computation, which is NULL. source An attribute identifying the computational source of the L-moments: “lmomgev”.

W.H. Asquith

## References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105–124.

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 lmr <- lmoms(c(123,34,4,654,37,78)) lmomgev(pargev(lmr)) ## Not run: # The Gumbel is a limiting version of the maxima regardless of parent. The GLO, # PE3 (twice), and GPA are studied here. A giant number of events to simulate is made. # Then numbers of events per year before the annual maxima are computed are specified. # The Gumbel is a limiting version of the maxima regardless of parent. The GLO, # PE3 (twice), and GPA are studied here. A giant number of events to simulate is made. # Then numbers of events per year before the annual maxima are computed are specified. nevents <- 100000 nev_yr <- c(1,2,3,4,5,6,10,15,20,30,50,100,200,500); n <- length(nev_yr) pdf("Gumbel_in_the_limit.pdf", useDingbats=FALSE) # Draw the usually L-moment ratio diagram but only show a few of the # three parameter families. plotlmrdia(lmrdia(), xlim=c(-.5,0.5), ylim=c(0,0.3), nopoints=TRUE, autolegend=TRUE, noaep4=TRUE, nogov=TRUE, xleg=0.1, yleg=0.3) gum <- lmrdia()$gum # extract the L-skew and L-kurtosis of the Gumbel points(gum[1], gum[2], pch=10, cex=3, col=2) # draw the Gumbel para <- parglo(vec2lmom(c(1,.1,0))) # generalized logistic t3 <- t4 <- rep(NA, n) # define for(k in 1:n) { # generate GLO time series of annual maxima with k-events per year lmr <- lmoms(sapply(1:nevents/nev_yr[k], function(i) max(rlmomco(nev_yr[k], para)))) t3[k] <- lmr$ratios[3]; t4[k] <- lmr$ratios[4] } lines(t3, t4, lwd=0.8); points(t3, t4, lwd=0.8, pch=21, bg=3) para <- parglo(vec2lmom(c(1,.1,0.3))) # generalized logistic t3 <- t4 <- rep(NA, n) # define for(k in 1:n) { # generate GLO time series of annual maxima with k-events per year lmr <- lmoms(sapply(1:nevents/nev_yr[k], function(i) max(rlmomco(nev_yr[k], para)))) t3[k] <- lmr$ratios[3]; t4[k] <- lmr$ratios[4] } lines(t3, t4, lwd=0.8); points(t3, t4, lwd=0.8, pch=21, bg=3) para <- parglo(vec2lmom(c(1,.1,-0.3))) # generalized logistic t3 <- t4 <- rep(NA, n) # define for(k in 1:n) { # generate GLO time series of annual maxima with k-events per year lmr <- lmoms(sapply(1:nevents/nev_yr[k], function(i) max(rlmomco(nev_yr[k], para)))) t3[k] <- lmr$ratios[3]; t4[k] <- lmr$ratios[4] } lines(t3, t4, lwd=0.8); points(t3, t4, lwd=0.8, pch=21, bg=3) para <- parpe3(vec2lmom(c(1,.1,.4))) # Pearson type III t3 <- t4 <- rep(NA, n) # reset for(k in 1:n) { # generate PE3 time series of annual maxima with k-events per year lmr <- lmoms(sapply(1:nevents/k, function(i) max(rlmomco(nev_yr[k], para)))) t3[k] <- lmr$ratios[3]; t4[k] <- lmr$ratios[4] } lines(t3, t4, lwd=0.8); points(t3, t4, lwd=0.8, pch=21, bg=6) para <- parpe3(vec2lmom(c(1,.1,0))) # Pearson type III t3 <- t4 <- rep(NA, n) # reset for(k in 1:n) { # generate another PE3 time series of annual maxima with k-events per year lmr <- lmoms(sapply(1:nevents/k, function(i) max(rlmomco(nev_yr[k], para)))) t3[k] <- lmr$ratios[3]; t4[k] <- lmr$ratios[4] } lines(t3, t4, lwd=0.8); points(t3, t4, lwd=0.8, pch=21, bg=6) para <- parpe3(vec2lmom(c(1,.1,-.4))) # Pearson type III t3 <- t4 <- rep(NA, n) # reset for(k in 1:n) { # generate PE3 time series of annual maxima with k-events per year lmr <- lmoms(sapply(1:nevents/k, function(i) max(rlmomco(nev_yr[k], para)))) t3[k] <- lmr$ratios[3]; t4[k] <- lmr$ratios[4] } lines(t3, t4, lwd=0.8); points(t3, t4, lwd=0.8, pch=21, bg=6) para <- pargpa(vec2lmom(c(1,.1,0))) # generalized Pareto t3 <- t4 <- rep(NA, n) # reset for(k in 1:n) { # generate GPA time series of annual maxima with k-events per year lmr <- lmoms(sapply(1:nevents/k, function(i) max(rlmomco(nev_yr[k], para)))) t3[k] <- lmr$ratios[3]; t4[k] <- lmr$ratios[4] } lines(t3, t4, lwd=0.8); points(t3, t4, lwd=0.8, pch=21, bg=4) para <- pargpa(vec2lmom(c(1,.1,.4))) # generalized Pareto t3 <- t4 <- rep(NA, n) # reset for(k in 1:n) { # generate GPA time series of annual maxima with k-events per year lmr <- lmoms(sapply(1:nevents/k, function(i) max(rlmomco(nev_yr[k], para)))) t3[k] <- lmr$ratios[3]; t4[k] <- lmr$ratios[4] } lines(t3, t4, lwd=0.8); points(t3, t4, lwd=0.8, pch=21, bg=4) para <- pargpa(vec2lmom(c(1,.1,-.4))) # generalized Pareto t3 <- t4 <- rep(NA, n) # reset for(k in 1:n) { # generate GPA time series of annual maxima with k-events per year lmr <- lmoms(sapply(1:nevents/k, function(i) max(rlmomco(nev_yr[k], para)))) t3[k] <- lmr$ratios[3]; t4[k] <- lmr\$ratios[4] } lines(t3, t4, lwd=0.8); points(t3, t4, lwd=0.8, pch=21, bg=4) dev.off() # ## End(Not run)