# L-moments of the Kumaraswamy Distribution 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 [REMOVE_ME]$$ \lambda_1 = \beta B(\eta, \beta) \mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \lambda_2 = \beta [B(\eta, \beta) - 2B(\eta, 2\beta)] \mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \tau_3 = \frac{B(\eta,\beta) - 6B(\eta,2\beta) + 6B(\eta,3\beta)}{B(\eta,\beta) - 2B(\eta,2\beta)} \mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \tau_4 = \frac{B(\eta,\beta) - 12B(\eta,2\beta) + 30B(\eta,3\beta) - 40B(\eta,4\beta)}{B(\eta,\beta) - 2B(\eta,2\beta)} \mbox{, and} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \tau_5 = \frac{B(\eta,\beta) - 20B(\eta,2\beta) + 90B(\eta,3\beta) - 140B(\eta,4\beta) + 70B(\eta,5\beta)}{B(\eta,\beta) - 2B(\eta,2\beta)} \mbox{.} [REMOVE_ME_2]$$ where $B(a,b)$ is the complete beta function or `beta()`. ## 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()`. ```r lmomkur(para) ``` ## Arguments - `para`: The parameters of the distribution. ## Returns An `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 . ## References Jones, M.C., 2009, Kumaraswamy's distribution---A beta-type distribution with some tractability advantages: Statistical Methodology, v. 6, pp. 70--81. ## Author(s) W.H. Asquith ## See Also `parkur`, `cdfkur`, `pdfkur`, `quakur` ## Examples ```r 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) ```

lmomkur function