# Trimmed L-moments of the Slash Distribution This function estimates the trimmed L-moments of the Slash distribution given the parameters ($\xi$ and $\alpha$) from `parsla`. The relation between the TL-moments (`trim=1`) and the parameters have been numerically determined and are $\lambda^{(1)}_1 = \xi$, $\lambda^{(1)}_2 = 0.93686275\alpha$, $\tau^{(1)}_3 = 0$, $\tau^{(1)}_4 = 0.30420472$, $\tau^{(1)}_5 = 0$, and $\tau^{(1)}_6 = 0.18900723$. These TL-moments (trim=1) are symmetrical for the first L-moments defined because $\mathrm{E}[X_{1:n}]$ and $\mathrm{E}[X_{n:n}]$ are undefined expectations for the Slash. ```r lmomsla(para) ``` ## Arguments - `para`: The parameters of the distribution. ## Returns An `list` is returned. - **lambdas**: Vector of the trimmed L-moments. First element is $\lambda^{(1)}_1$, second element is $\lambda^{(1)}_2$, and so on. - **ratios**: Vector of the L-moment ratios. Second element is $\tau^{(1)}$, third element is $\tau^{(1)}_3$ and so on. - **trim**: Level of symmetrical trimming used in the computation, which is `1`. - **leftrim**: Level of left-tail trimming used in the computation, which is `1`. - **rightrim**: Level of right-tail trimming used in the computation, which is `1`. - **source**: An attribute identifying the computational source of the L-moments: lmomsla - **trim**: Level of symmetrical trimming used. ## References Rogers, W.H., and Tukey, J.W., 1972, Understanding some long-tailed symmetrical distributions: Statistica Neerlandica, v. 26, no. 3, pp. 211--226. ## Author(s) W.H. Asquith ## See Also `parsla`, `cdfsla`, `pdfsla`, `quasla` ## Examples ```r ## Not run: # This example was used to numerically back into the TL-moments and the # relation between \alpha and \lambda_2. "lmomtrim1" <- function(para) { bigF <- 0.9999 minX <- para$para[1] - para$para[2]*qnorm(1 - bigF) / qunif(1 - bigF) maxX <- para$para[1] + para$para[2]*qnorm( bigF) / qunif(1 - bigF) minF <- cdfsla(minX, para); maxF <- cdfsla(maxX, para) lmr <- theoTLmoms(para, nmom = 6, leftrim = 1, rightrim = 1) } U <- -10; i <- 0 As <- seq(.1,abs(10),by=.2) L1s <- L2s <- T3s <- T4s <- T5s <- T6s <- vector(mode="numeric", length=length(As)) for(A in As) { i <- i + 1 lmr <- lmomtrim1(vec2par(c(U, A), type="sla")) L1s[i] <- lmr$lambdas[1]; L2s[i] <- lmr$lambdas[2] T3s[i] <- lmr$ratios[3]; T4s[i] <- lmr$ratios[4] T5s[i] <- lmr$ratios[5]; T6s[i] <- lmr$ratios[6] } print(summary(lm(L2s~As-1))$coe) print(mean(T4s)) print(mean(T6s)) # ## End(Not run) ## Not run: alpha <- 30 tlmr <- theoTLmoms(vec2par(c(100, alpha), type="cau"), nmom=6, trim=1) print( c(tlmr$lambdas[2] / alpha, tlmr$ratios[c(4,6)]), 8 ) # ## End(Not run) ```

lmomsla function