lmomco (version 1.4.3)

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

$$\lambda_1 = \beta B(\eta, \beta) \mbox{,}$$ $$\lambda_2 = \beta [B(\eta, \beta) - 2B(\eta, 2\beta)] \mbox{,}$$ $$\tau_3 = \frac{B(\eta,\beta) - 6B(\eta,2\beta) + 6B(\eta,3\beta)}{B(\eta,\beta) - 2B(\eta,2\beta)} \mbox{,}$$ $$\tau_4 = \frac{B(\eta,\beta) - 12B(\eta,2\beta) + 30B(\eta,3\beta) - 40B(\eta,4\beta)}{B(\eta,\beta) - 2B(\eta,2\beta)} \mbox{, and}$$ $$\tau_5 = \frac{B(\eta,\beta) - 20B(\eta,2\beta) + 90B(\eta,3\beta) - 140B(\eta,4\beta) + 70B(\eta,5\beta)}{B(\eta,\beta) - 2B(\eta,2\beta)} \mbox{.}$$

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

Usage

lmomkur(para)

Arguments

para

The parameters of the distribution.

Value

• An R list is returned.
• L1
  Arithmetic mean.
• L2
  L-scale---analogous to standard deviation.
• LCV
  coefficient of L-variation---analogous to coe. of variation.
• TAU3
  The third L-moment ratio or L-skew--analogous to skew.
• TAU4
  The fourth L-moment ratio or L-kurtosis---analogous to kurtosis.
• TAU5
  The fifth L-moment ratio.
• L3
  The third L-moment.
• L4
  The fourth L-moment.
• L5
  The fifth L-moment.
• source
  An attribute identifying the computational source of the L-moments: lmomkur.

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, quakur, cdfkur

Examples

lmr <- lmom.ub(c(0.25, 0.4, 0.6, 0.65, 0.67, 0.9))
lmomkur(parkur(lmr))

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$TAU3; T4[i] <- lmr$TAU4
  }
}
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")