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”.

Author(s)

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.

See Also

parkur, cdfkur, pdfkur, quakur

Examples

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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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