# lmomkur: L-moments of the Kumaraswamy Distribution

### Description

This function estimates the L-moments of the Kumaraswamy distribution given the parameters (α and β) from parkur. The L-moments in terms of the parameters with η = 1 + 1/α are

λ_1 = β B(η, β) \mbox{,}

λ_2 = β [B(η, β) - 2B(η, 2β)] \mbox{,}

τ_3 = \frac{B(η,β) - 6B(η,2β) + 6B(η,3β)}{B(η,β) - 2B(η,2β)} \mbox{,}

τ_4 = \frac{B(η,β) - 12B(η,2β) + 30B(η,3β) - 40B(η,4β)}{B(η,β) - 2B(η,2β)} \mbox{, and}

τ_5 = \frac{B(η,β) - 20B(η,2β) + 90B(η,3β) - 140B(η,4β) + 70B(η,5β)}{B(η,β) - 2B(η,2β)} \mbox{.}

where B(a,b) is the complete beta function or beta().

### Usage

 1 lmomkur(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: “lmomkur”.

W.H. Asquith

### References

Jones, M.C., 2009, Kumaraswamy's distribution—A beta-type distribution with some tractability advantages: Statistical Methodology, v. 6, pp. 70–81.

parkur, cdfkur, pdfkur, quakur

### Examples

  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 lmr <- lmoms(c(0.25, 0.4, 0.6, 0.65, 0.67, 0.9)) lmomkur(parkur(lmr)) ## Not run: A <- B <- exp(seq(-3,5, by=.05)) logA <- logB <- T3 <- T4 <- c(); i <- 0 for(a in A) { for(b in B) { i <- i + 1 parkur <- list(para=c(a,b), type="kur"); lmr <- lmomkur(parkur) logA[i] <- log(a); logB[i] <- log(b) T3[i] <- lmr$ratios[3]; T4[i] <- lmr$ratios[4] } } library(lattice) contourplot(T3~logA+logB, cuts=20, lwd=0.5, label.style="align", xlab="LOG OF ALPHA", ylab="LOG OF BETA", xlim=c(-3,5), ylim=c(-3,5), main="L-SKEW FOR KUMARASWAMY DISTRIBUTION") contourplot(T4~logA+logB, cuts=10, lwd=0.5, label.style="align", xlab="LOG OF ALPHA", ylab="LOG OF BETA", xlim=c(-3,5), ylim=c(-3,5), main="L-KURTOSIS FOR KUMARASWAMY DISTRIBUTION") ## End(Not run) 

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