lmomco (version 2.4.14)

lmomkur: L-moments of the Kumaraswamy Distribution

Description

This function estimates the L-moments of the Kumaraswamy distribution given the parameters ($\alpha$ and $\beta$) from parkur. The L-moments in terms of the parameters with $\eta =1+1/\alpha$ are $${\lambda }_1=\beta B(\eta ,\beta )\text{,}$$ $${\lambda }_2=\beta [B(\eta ,\beta )-2B(\eta ,2\beta )]\text{,}$$ $${\tau }_3=\frac{B(\eta ,\beta )-6B(\eta ,2\beta )+6B(\eta ,3\beta )}{B(\eta ,\beta )-2B(\eta ,2\beta )}\text{,}$$ $${\tau }_4=\frac{B(\eta ,\beta )-12B(\eta ,2\beta )+30B(\eta ,3\beta )-40B(\eta ,4\beta )}{B(\eta ,\beta )-2B(\eta ,2\beta )}\text{, and}$$ $${\tau }_5=\frac{B(\eta ,\beta )-20B(\eta ,2\beta )+90B(\eta ,3\beta )-140B(\eta ,4\beta )+70B(\eta ,5\beta )}{B(\eta ,\beta )-2B(\eta ,2\beta )}\text{.}$$ where $B(a,b)$ is the complete beta function or beta().

Usage

lmomkur(para)

Value

An R

list is returned.

lambdas

Vector of the L-moments. First element is ${\lambda }_1$, second element is ${\lambda }_2$, and so on.

ratios

Vector of the L-moment ratios. Second element is $\tau$, third element is ${\tau }_3$ 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”.

Arguments

para

The parameters of the distribution.

Author

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

lmr <- lmoms(c(0.25, 0.4, 0.6, 0.65, 0.67, 0.9))
lmomkur(parkur(lmr))
if (FALSE) {
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")
}